id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
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proofwiki-21100 | Symmetry of Invariant Metric on Vector Space | <onlyinclude>
Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $d$ be an invariant metric on $X$.
Then we have:
:$\map d {x, y} = \map d {-x, -y}$
for each $x, y \in X$. | Let $x, y \in X$.
We have:
{{begin-eqn}}
{{eqn | l = \map d {x, y}
| r = \map d {x + \paren {-y - x}, y + \paren {-y - x} }
| c = {{Defof|Invariant Metric on Vector Space}}
}}
{{eqn | r = \map d {-y, -x}
}}
{{eqn | r = \map d {-x, -y}
| c = {{Metric-space-axiom|3}}
}}
{{end-eqn}}
{{qed}}
Category:Invariant Metric... | <onlyinclude>
Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $d$ be an [[Definition:Invariant Metric on Vector Space|invariant metric]] on $X$.
Then we have:
:$\map d {x, y} = \map d {-x, -y}$
for each $x, y \in X$. | Let $x, y \in X$.
We have:
{{begin-eqn}}
{{eqn | l = \map d {x, y}
| r = \map d {x + \paren {-y - x}, y + \paren {-y - x} }
| c = {{Defof|Invariant Metric on Vector Space}}
}}
{{eqn | r = \map d {-y, -x}
}}
{{eqn | r = \map d {-x, -y}
| c = {{Metric-space-axiom|3}}
}}
{{end-eqn}}
{{qed}}
[[Category:Invariant Me... | Symmetry of Invariant Metric on Vector Space | https://proofwiki.org/wiki/Symmetry_of_Invariant_Metric_on_Vector_Space | https://proofwiki.org/wiki/Symmetry_of_Invariant_Metric_on_Vector_Space | [
"Invariant Metrics on Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Invariant Metric on Vector Space"
] | [
"Category:Invariant Metrics on Vector Spaces"
] |
proofwiki-21101 | Subadditivity of Invariant Metric on Vector Space | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $d$ be an invariant metric on $X$.
Then:
:$\map d {n x, {\mathbf 0}_X} \le n \map d {x, {\mathbf 0}_X}$
for each $n \in \N$ and $x \in X$. | We have:
{{begin-eqn}}
{{eqn | l = \map d {n x, {\mathbf 0}_X}
| r = \sum_{k \mathop = 1}^n \map d {k x, \paren {k - 1} x}
| c = {{Metric-space-axiom|2}}
}}
{{eqn | o = \le
| r = \sum_{k \mathop = 1}^n \map d {k x - \paren {k - 1} x, \paren {k - 1} x - \paren {k - 1} x}
| c = {{Defof|Invariant Metric on Vector... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $d$ be an [[Definition:Invariant Metric on Vector Space|invariant metric]] on $X$.
Then:
:$\map d {n x, {\mathbf 0}_X} \le n \map d {x, {\mathbf 0}_X}$
for each $n \in \N$ and $x \in X$. | We have:
{{begin-eqn}}
{{eqn | l = \map d {n x, {\mathbf 0}_X}
| r = \sum_{k \mathop = 1}^n \map d {k x, \paren {k - 1} x}
| c = {{Metric-space-axiom|2}}
}}
{{eqn | o = \le
| r = \sum_{k \mathop = 1}^n \map d {k x - \paren {k - 1} x, \paren {k - 1} x - \paren {k - 1} x}
| c = {{Defof|Invariant Metric on Vector... | Subadditivity of Invariant Metric on Vector Space | https://proofwiki.org/wiki/Subadditivity_of_Invariant_Metric_on_Vector_Space | https://proofwiki.org/wiki/Subadditivity_of_Invariant_Metric_on_Vector_Space | [
"Invariant Metrics on Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Invariant Metric on Vector Space"
] | [] |
proofwiki-21102 | Quotient Metric on Vector Space is Well-Defined | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $d$ be an invariant metric on $X$.
Let $N$ be a vector subspace of $X$.
Let $X/N$ be the quotient vector space of $X$ modulo $N$.
Let $\pi : X \to X/N$ be the quotient mapping.
Then the mapping $d_N : X/N \times X/N \to \hointr 0 \infty$ defined by:
:$\ds \m... | Let $x, y \in X$.
Then $\map d {x - y, z} \ge 0$ for all $z \in N$, and so:
:$\ds \inf_{z \mathop \in N} \map d {x - y, z}$ exists as a real number.
Let $x', y' \in X$ be such that $\map \pi x = \map \pi {x'}$ and $\map \pi y = \map \pi {y'}$.
We now need to show that if $x', y' \in X$ are such that:
:$\map \pi x = \ma... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $d$ be an [[Definition:Invariant Metric on Vector Space|invariant metric]] on $X$.
Let $N$ be a [[Definition:Vector Subspace|vector subspace]] of $X$.
Let $X/N$ be the [[Definition:Quotie... | Let $x, y \in X$.
Then $\map d {x - y, z} \ge 0$ for all $z \in N$, and so:
:$\ds \inf_{z \mathop \in N} \map d {x - y, z}$ exists as a [[Definition:Real Number|real number]].
Let $x', y' \in X$ be such that $\map \pi x = \map \pi {x'}$ and $\map \pi y = \map \pi {y'}$.
We now need to show that if $x', y' \in X$ are... | Quotient Metric on Vector Space is Well-Defined | https://proofwiki.org/wiki/Quotient_Metric_on_Vector_Space_is_Well-Defined | https://proofwiki.org/wiki/Quotient_Metric_on_Vector_Space_is_Well-Defined | [
"Quotient Metrics on Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Invariant Metric on Vector Space",
"Definition:Vector Subspace",
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Definition:Mapping"
] | [
"Definition:Real Number",
"Quotient Mapping is Linear Transformation",
"Kernel of Quotient Mapping",
"Category:Quotient Metrics on Vector Spaces"
] |
proofwiki-21103 | Quotient Metric on Vector Space is Invariant Pseudometric | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $d$ be an invariant metric on $X$.
Let $N$ be a vector subspace of $X$.
Let $X/N$ be the quotient vector space of $X$ modulo $N$.
Let $\pi : X \to X/N$ be the quotient mapping.
Let $d_N$ be the quotient metric on $X/N$ induced by $d$.
Then $d_N$ is an invaria... | === Proof of {{Metric-space-axiom|1}} ===
Let $x, y \in X$.
Then, we have:
:$\ds \map {d_N} {\map \pi x, \map \pi x} = \inf_{z \mathop \in N} \map d {x - x, z} = \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z}$
Since $N$ is a vector subspace, we have ${\mathbf 0}_X \in N$.
From {{Metric-space-axiom|1}}, we have $\map... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $d$ be an [[Definition:Invariant Metric on Vector Space|invariant metric]] on $X$.
Let $N$ be a [[Definition:Vector Subspace|vector subspace]] of $X$.
Let $X/N$ be the [[Definition:Quotie... | === Proof of {{Metric-space-axiom|1}} ===
Let $x, y \in X$.
Then, we have:
:$\ds \map {d_N} {\map \pi x, \map \pi x} = \inf_{z \mathop \in N} \map d {x - x, z} = \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z}$
Since $N$ is a [[Definition:Vector Subspace|vector subspace]], we have ${\mathbf 0}_X \in N$.
From {{Me... | Quotient Metric on Vector Space is Invariant Pseudometric | https://proofwiki.org/wiki/Quotient_Metric_on_Vector_Space_is_Invariant_Pseudometric | https://proofwiki.org/wiki/Quotient_Metric_on_Vector_Space_is_Invariant_Pseudometric | [
"Invariant Pseudometrics on Vector Spaces",
"Quotient Metrics on Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Invariant Metric on Vector Space",
"Definition:Vector Subspace",
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Definition:Quotient Metric on Vector Space",
"Definition:Invariant Pseudometric on Vector S... | [
"Definition:Vector Subspace"
] |
proofwiki-21104 | Quotient Metric on Vector Space is Invariant Metric iff Vector Subspace is Closed | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $d$ be an invariant metric on $X$.
Let $N$ be a vector subspace of $X$.
Let $X/N$ be the quotient vector space of $X$ modulo $N$.
Let $\pi : X \to X/N$ be the quotient mapping.
Let $d_N$ be the quotient metric on $X/N$ induced by $d$.
Then $d_N$ is an invaria... | From Quotient Metric on Vector Space is Invariant Pseudometric, $d_N$ is an invariant pseudometric.
It remains to show that {{Metric-space-axiom|4}} holds {{iff}} $N$ is closed.
Note that {{Metric-space-axiom|4}} holds {{iff}} for $x, y \in X$:
:$\map {d_N} {\map \pi x, \map \pi y} = 0$
implies that $\map \pi x = \map ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $d$ be an [[Definition:Invariant Metric on Vector Space|invariant metric]] on $X$.
Let $N$ be a [[Definition:Vector Subspace|vector subspace]] of $X$.
Let $X/N$ be the [[Definition:Quotie... | From [[Quotient Metric on Vector Space is Invariant Pseudometric]], $d_N$ is an [[Definition:Invariant Pseudometric on Vector Space|invariant pseudometric]].
It remains to show that {{Metric-space-axiom|4}} holds {{iff}} $N$ is [[Definition:Closed Set|closed]].
Note that {{Metric-space-axiom|4}} holds {{iff}} for $x,... | Quotient Metric on Vector Space is Invariant Metric iff Vector Subspace is Closed | https://proofwiki.org/wiki/Quotient_Metric_on_Vector_Space_is_Invariant_Metric_iff_Vector_Subspace_is_Closed | https://proofwiki.org/wiki/Quotient_Metric_on_Vector_Space_is_Invariant_Metric_iff_Vector_Subspace_is_Closed | [
"Quotient Metrics on Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Invariant Metric on Vector Space",
"Definition:Vector Subspace",
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Definition:Quotient Metric on Vector Space",
"Definition:Invariant Metric on Vector Space",... | [
"Quotient Metric on Vector Space is Invariant Pseudometric",
"Definition:Invariant Pseudometric on Vector Space",
"Definition:Closed Set",
"Quotient Mapping is Linear Transformation",
"Kernel of Quotient Mapping",
"Subset of Metric Space is Closed iff contains all Zero Distance Points",
"Definition:Clos... |
proofwiki-21105 | Generalized Sum Restricted to Non-zero Summands | Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\family{g_i}_{i \in I}$ be an indexed family of elements of $G$.
Let $J = \set{i \in I : g_i \ne 0_G}$
Let $h \in G$.
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$
{{iff}}:
:the generalized sum $\ds \sum_{j \mathop \... | === Necessary Condition ===
Let the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converge to $h$.
It will be shown that $\ds \sum_{j \mathop \in J} g_j$ converges to $h$.
{{:Generalized Sum Restricted to Non-zero Summands/Necessary Condition}}{{qed|lemma}} | Let $G$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Topological Semigroup|topological semigroup]] with [[Definition:Identity Element|identity]] $0_G$.
Let $\family{g_i}_{i \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] of $G$.
Let $J = \set{i \in... | === [[Generalized Sum Restricted to Non-zero Summands/Necessary Condition|Necessary Condition]] ===
Let the [[Definition:Generalized Sum|generalized sum]] $\ds \sum_{i \mathop \in I} g_i$ [[Definition:Convergent Net|converge]] to $h$.
It will be shown that $\ds \sum_{j \mathop \in J} g_j$ [[Definition:Convergent Net|... | Generalized Sum Restricted to Non-zero Summands | https://proofwiki.org/wiki/Generalized_Sum_Restricted_to_Non-zero_Summands | https://proofwiki.org/wiki/Generalized_Sum_Restricted_to_Non-zero_Summands | [
"Generalized Sums",
"Generalized Sum Restricted to Non-zero Summands"
] | [
"Definition:Commutative Semigroup",
"Definition:Topological Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Generalized Sum",
"Definition:Convergent... | [
"Generalized Sum Restricted to Non-zero Summands/Necessary Condition",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Convergent Net",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Convergent Net"
] |
proofwiki-21106 | Functional Completeness over Finite Number of Arguments | Every truth function is definable from the set of Binary Truth Functions. | Let $f : \Bbb B^k \to \Bbb B$ be an arbitrary truth function.
Suppose that $k = 0$.
Then, let:
:$b = \map f {}$
If $b = T$, then let:
:$\map g {p, q} = \map {f_\T} {p, q} = T$
otherwise, if $b = F$:
:$\map g {p, q} = \map {f_\F} {p, q} = F$
Finally, let $\map i {} = \tuple {\F, \F}$.
$i$ is clearly an injection, and:
:... | Every [[Definition:Truth Function|truth function]] is [[Definition:Definable Truth Function|definable]] from the [[Definition:Set|set]] of [[Binary Truth Functions]]. | Let $f : \Bbb B^k \to \Bbb B$ be an arbitrary [[Definition:Truth Function|truth function]].
Suppose that $k = 0$.
Then, let:
:$b = \map f {}$
If $b = T$, then let:
:$\map g {p, q} = \map {f_\T} {p, q} = T$
otherwise, if $b = F$:
:$\map g {p, q} = \map {f_\F} {p, q} = F$
Finally, let $\map i {} = \tuple {\F, \F}$.
... | Functional Completeness over Finite Number of Arguments | https://proofwiki.org/wiki/Functional_Completeness_over_Finite_Number_of_Arguments | https://proofwiki.org/wiki/Functional_Completeness_over_Finite_Number_of_Arguments | [] | [
"Definition:Truth Function",
"Definition:Definable Truth Function",
"Definition:Set",
"Binary Truth Functions"
] | [
"Definition:Truth Function",
"Definition:Definable Truth Function",
"Unary Truth Functions",
"Definition:Constant Mapping",
"Definition:Identity Mapping",
"Definition:Logical Not",
"Definition:Definable Truth Function",
"Definition:Mathematical Induction",
"Definition:Identity Mapping",
"Definitio... |
proofwiki-21107 | Quotient Metric on Vector Space induces Quotient Topology | Let $K$ be a topological field.
Let $X$ be a vector space over $K$.
Let $d$ be an invariant metric such that the induced topology $\tau$ makes $\struct {X, \tau}$ a topological vector space.
Let $N$ be a closed linear subspace of $X$.
Let $X/N$ be the quotient vector space of $X$ modulo $N$.
Let $\struct {X/N, \tau_N}$... | From Quotient Metric on Vector Space is Invariant Metric iff Vector Subspace is Closed, $d_N$ is a metric and hence:
:$\struct {X/N, d_N}$ is a metric space.
Let $\pi : X \to X/N$ be the quotient mapping.
We first show that:
:$\pi \sqbrk {\set {x \in X : \map d {x, {\mathbf 0}_X} < r} } = \set {\map \pi x \in X/N : \ma... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $d$ be an [[Definition:Invariant Metric on Vector Space|invariant metric]] such that the [[Definition:Topology Induced by Metric|induced topology]] $\tau$ makes $\struct {X, \tau}$ a [[... | From [[Quotient Metric on Vector Space is Invariant Metric iff Vector Subspace is Closed]], $d_N$ is a [[Definition:Metric|metric]] and hence:
:$\struct {X/N, d_N}$ is a [[Definition:Metric Space|metric space]].
Let $\pi : X \to X/N$ be the [[Definition:Quotient Mapping|quotient mapping]].
We first show that:
:$\pi \... | Quotient Metric on Vector Space induces Quotient Topology | https://proofwiki.org/wiki/Quotient_Metric_on_Vector_Space_induces_Quotient_Topology | https://proofwiki.org/wiki/Quotient_Metric_on_Vector_Space_induces_Quotient_Topology | [
"Quotient Topological Vector Spaces",
"Quotient Metrics on Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Vector Space",
"Definition:Invariant Metric on Vector Space",
"Definition:Topology Induced by Metric",
"Definition:Topological Vector Space",
"Definition:Closed Linear Subspace",
"Definition:Quotient Vector Space",
"Definition:Quotient Topological Vector Spa... | [
"Quotient Metric on Vector Space is Invariant Metric iff Vector Subspace is Closed",
"Definition:Metric Space/Metric",
"Definition:Metric Space",
"Definition:Quotient Mapping",
"Definition:Linear Subspace",
"Definition:Invariant Metric on Vector Space",
"Definition:Infimum of Set/Real Numbers",
"Kerne... |
proofwiki-21108 | Translation of Open Ball in Invariant Pseudometric on Vector Space | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $d$ be an invariant pseudometric on $X$.
For $x \in X$ and $\epsilon > 0$, let $\map {B_\epsilon} x$ be the open ball centered at $x$ with radius $\epsilon$.
Let $y \in X$.
Then:
:$\map {B_\epsilon} x + y = \map {B_\epsilon} {x + y}$ | Let $z \in X$.
Then we have $z \in \map {B_\epsilon} x + y$ {{iff}} $z = u + y$ for $u \in \map {B_\epsilon} x$.
That is, {{iff}} $z - y \in \map {B_\epsilon} x$.
This is equivalent to:
:$\map d {z - y, x} < \epsilon$
Since $d$ is invariant, this is equivalent to:
:$\map d {z, x + y} < \epsilon$
So, we have $z \in \... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $d$ be an [[Definition:Invariant Pseudometric on Vector Space|invariant pseudometric]] on $X$.
For $x \in X$ and $\epsilon > 0$, let $\map {B_\epsilon} x$ be the [[Definition:Open Ball|open... | Let $z \in X$.
Then we have $z \in \map {B_\epsilon} x + y$ {{iff}} $z = u + y$ for $u \in \map {B_\epsilon} x$.
That is, {{iff}} $z - y \in \map {B_\epsilon} x$.
This is equivalent to:
:$\map d {z - y, x} < \epsilon$
Since $d$ is [[Definition:Invariant Pseudometric on Vector Space|invariant]], this is equivalent... | Translation of Open Ball in Invariant Pseudometric on Vector Space | https://proofwiki.org/wiki/Translation_of_Open_Ball_in_Invariant_Pseudometric_on_Vector_Space | https://proofwiki.org/wiki/Translation_of_Open_Ball_in_Invariant_Pseudometric_on_Vector_Space | [
"Invariant Pseudometrics on Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Invariant Pseudometric on Vector Space",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius"
] | [
"Definition:Invariant Pseudometric on Vector Space",
"Category:Invariant Pseudometrics on Vector Spaces"
] |
proofwiki-21109 | Topological Vector Space over Connected Topological Field is Connected | Let $K$ be a connected topological field.
Let $X$ be a topological vector space over $K$.
Then $X$ is connected. | From the definition of a topological vector space, the mapping $\circ_X : K \times X \to X$ defined by:
:$\map {\circ_X} {\lambda, x} = \lambda x$
for $\tuple {\lambda, x} \in K \times X$ is continuous.
Let $x \in X$.
From Horizontal Section of Continuous Function is Continuous, the mapping $c_x : K \to X$ defined by:... | Let $K$ be a [[Definition:Connected Topological Space|connected]] [[Definition:Topological Field|topological field]].
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $K$.
Then $X$ is [[Definition:Connected Topological Space|connected]]. | From the definition of a [[Definition:Topological Vector Space|topological vector space]], the [[Definition:Mapping|mapping]] $\circ_X : K \times X \to X$ defined by:
:$\map {\circ_X} {\lambda, x} = \lambda x$
for $\tuple {\lambda, x} \in K \times X$ is [[Definition:Continuous Mapping|continuous]].
Let $x \in X$.
Fr... | Topological Vector Space over Connected Topological Field is Connected | https://proofwiki.org/wiki/Topological_Vector_Space_over_Connected_Topological_Field_is_Connected | https://proofwiki.org/wiki/Topological_Vector_Space_over_Connected_Topological_Field_is_Connected | [
"Topological Vector Spaces",
"Connected Topological Spaces"
] | [
"Definition:Connected Topological Space",
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Connected Topological Space"
] | [
"Definition:Topological Vector Space",
"Definition:Mapping",
"Definition:Continuous Mapping",
"Horizontal Section of Continuous Function is Continuous",
"Definition:Mapping",
"Definition:Continuous Mapping",
"Definition:Connected Topological Space",
"Definition:Connected Topological Space",
"Continu... |
proofwiki-21110 | Quantum-Charge Ratio | The ratio of Planck's constant to the elementary charge is given by:
{{begin-eqn}}
{{eqn | l = \dfrac h \E
| o = \approx
| r = 4 \cdotp 13566 \, 7697 \times 10^{-15}
| c = joule seconds per coulombs
| cc= {{OEIS|A343571}}
}}
{{eqn | o = \approx
| r = 4 \cdotp 13566 \, 7697 \times 10^{-7}
... | We have:
{{begin-eqn}}
{{eqn | l = h
| r = 6 \cdotp 62607 \, 015 \times 10^{-34} \, \mathrm {J \, s}
| c = that is: joule seconds
| cc= {{Defof|Planck's Constant|subdef = Value}}
}}
{{eqn | l = \E
| r = 1 \cdotp 60217 \, 6634 \times 10^{−19} \, \mathrm C
| c = that is: coulombs
| cc... | The [[Definition:Ratio|ratio]] of [[Definition:Planck's Constant|Planck's constant]] to the [[Definition:Elementary Charge|elementary charge]] is given by:
{{begin-eqn}}
{{eqn | l = \dfrac h \E
| o = \approx
| r = 4 \cdotp 13566 \, 7697 \times 10^{-15}
| c = [[Definition:Joule|joule]] [[Definition:Se... | We have:
{{begin-eqn}}
{{eqn | l = h
| r = 6 \cdotp 62607 \, 015 \times 10^{-34} \, \mathrm {J \, s}
| c = that is: [[Definition:Joule|joule]] [[Definition:Second of Time|seconds]]
| cc= {{Defof|Planck's Constant|subdef = Value}}
}}
{{eqn | l = \E
| r = 1 \cdotp 60217 \, 6634 \times 10^{−19} \... | Quantum-Charge Ratio | https://proofwiki.org/wiki/Quantum-Charge_Ratio | https://proofwiki.org/wiki/Quantum-Charge_Ratio | [
"Planck's Constant",
"Elementary Charge"
] | [
"Definition:Ratio",
"Definition:Planck's Constant",
"Definition:Electric Charge/Quantum",
"Definition:SI/Energy/Joule",
"Definition:Time/Unit/Second",
"Definition:Coulomb",
"Definition:CGS/Energy/Erg",
"Definition:Time/Unit/Second",
"Definition:Abcoulomb",
"Definition:CGS/Energy/Erg",
"Definitio... | [
"Definition:SI/Energy/Joule",
"Definition:Time/Unit/Second",
"Definition:Coulomb",
"Definition:CGS/Energy/Erg",
"Definition:Time/Unit/Second",
"Definition:Abcoulomb",
"Definition:CGS/Energy/Erg",
"Definition:Time/Unit/Second",
"Definition:Statcoulomb"
] |
proofwiki-21111 | Birkhoff-Kakutani Theorem/Topological Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Then $\struct {X, \tau}$ is pseudometrizable {{iff}} $\struct {X, \tau}$ is first-countable.
Further, if $\struct {X, \tau}$ is pseudometrizable then there exists an invariant pseudometric $d$ on $X$ such that:
:$(1): \quad$ ... | === Sufficient Condition ===
Suppose that $\struct {X, \tau}$ is first-countable and Hausdorff.
Let $\sequence {U_n}_{n \mathop \in \N}$ be a local basis for ${\mathbf 0}_X$ in $\struct {X, \tau}$.
Let $V_1 = U_1$.
From Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods: Corollar... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Then $\struct {X, \tau}$ is [[Definition:Pseudometrizable Topology|pseudometrizable]] {{iff}} $\struct {X, \tau}$ is [[Definition:First-Countable Space|first-countable]].
Further, i... | === Sufficient Condition ===
Suppose that $\struct {X, \tau}$ is [[Definition:First-Countable Space|first-countable]] and [[Definition:Hausdorff Space|Hausdorff]].
Let $\sequence {U_n}_{n \mathop \in \N}$ be a [[Definition:Local Basis|local basis]] for ${\mathbf 0}_X$ in $\struct {X, \tau}$.
Let $V_1 = U_1$.
From ... | Birkhoff-Kakutani Theorem/Topological Vector Space | https://proofwiki.org/wiki/Birkhoff-Kakutani_Theorem/Topological_Vector_Space | https://proofwiki.org/wiki/Birkhoff-Kakutani_Theorem/Topological_Vector_Space | [
"Topological Vector Spaces",
"Birkhoff-Kakutani Theorem"
] | [
"Definition:Topological Vector Space",
"Definition:Pseudometrizable Topology",
"Definition:First-Countable Space",
"Definition:Pseudometrizable Topology",
"Definition:Invariant Pseudometric on Vector Space",
"Definition:Topology Induced by Pseudometric",
"Definition:Open Ball",
"Definition:Balanced Se... | [
"Definition:First-Countable Space",
"Definition:T2 Space",
"Definition:Local Basis",
"Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods/Corollary 2",
"Definition:Open Neighborhood",
"Definition:Local Basis",
"Definition:Set",
"Definition:Real Number",
"Definit... |
proofwiki-21112 | Compton Wavelength of Electron | The Compton wavelength of the electron is:
{{begin-eqn}}
{{eqn | l = \lambda_\E
| o = \approx
| r = 2 \cdotp 42631 \, 02386 \, 7(73) \times 10^{-12} \, \mathrm m
}}
{{eqn | o = \approx
| r = 2 \cdotp 42631 \, 02386 \, 7(73) \times 10^{-10} \, \mathrm {cm}
}}
{{end-eqn}} | By definition, the Compton wavelength $\lambda_\E$ of an electron is given as:
:$\lambda_\E = \dfrac h {m_\E c}$
where:
:$m_\E$ denotes the mass of the electron
:$h$ denotes Planck's constant
:$c$ denotes the speed of light.
Then we have:
{{begin-eqn}}
{{eqn | l = h
| r = 6 \cdotp 62607 \, 015 \times 10^{-34} \, ... | The [[Definition:Compton Wavelength|Compton wavelength]] of the [[Definition:Electron|electron]] is:
{{begin-eqn}}
{{eqn | l = \lambda_\E
| o = \approx
| r = 2 \cdotp 42631 \, 02386 \, 7(73) \times 10^{-12} \, \mathrm m
}}
{{eqn | o = \approx
| r = 2 \cdotp 42631 \, 02386 \, 7(73) \times 10^{-10} \, ... | By definition, the [[Definition:Compton Wavelength|Compton wavelength]] $\lambda_\E$ of an [[Definition:Electron|electron]] is given as:
:$\lambda_\E = \dfrac h {m_\E c}$
where:
:$m_\E$ denotes the [[Definition:Mass|mass]] of the [[Definition:Electron|electron]]
:$h$ denotes [[Definition:Planck's Constant|Planck's cons... | Compton Wavelength of Electron | https://proofwiki.org/wiki/Compton_Wavelength_of_Electron | https://proofwiki.org/wiki/Compton_Wavelength_of_Electron | [
"Compton Wavelength",
"Electrons"
] | [
"Definition:Compton Wavelength",
"Definition:Electron"
] | [
"Definition:Compton Wavelength",
"Definition:Electron",
"Definition:Mass",
"Definition:Electron",
"Definition:Planck's Constant",
"Definition:Speed of Light",
"Definition:SI/Energy/Joule",
"Definition:Time/Unit/Second",
"Definition:Metric System/Mass/Kilogram",
"Definition:Electron/Mass",
"Defin... |
proofwiki-21113 | Reduced Compton Wavelength of Electron | The Compton wavelength of the electron is:
{{begin-eqn}}
{{eqn | l = \lambdabar_\E
| o = \approx
| r = 3 \cdotp 86159 \, 267 \times 10^{-13} \, \mathrm m
}}
{{eqn | o = \approx
| r = 3 \cdotp 86159 \, 267 \times 10^{-11} \, \mathrm {cm}
}}
{{end-eqn}} | By definition, the reduced Compton wavelength $\lambdabar_\E$ of an electron is given as:
:$\lambdabar_\E = \dfrac {\lambda_\E} {2 \pi}$
where $\lambda_\E$ denotes the Compton wavelength of the electron.
Then we have:
{{begin-eqn}}
{{eqn | l = \lambda_\E
| o = \approx
| r = 2 \cdotp 42631 \, 02386 \, 7(73) ... | The [[Definition:Compton Wavelength|Compton wavelength]] of the [[Definition:Electron|electron]] is:
{{begin-eqn}}
{{eqn | l = \lambdabar_\E
| o = \approx
| r = 3 \cdotp 86159 \, 267 \times 10^{-13} \, \mathrm m
}}
{{eqn | o = \approx
| r = 3 \cdotp 86159 \, 267 \times 10^{-11} \, \mathrm {cm}
}}
{{e... | By definition, the [[Definition:Reduced Compton Wavelength|reduced Compton wavelength]] $\lambdabar_\E$ of an [[Definition:Electron|electron]] is given as:
:$\lambdabar_\E = \dfrac {\lambda_\E} {2 \pi}$
where $\lambda_\E$ denotes the [[Definition:Compton Wavelength|Compton wavelength]] of the [[Definition:Electron|elec... | Reduced Compton Wavelength of Electron | https://proofwiki.org/wiki/Reduced_Compton_Wavelength_of_Electron | https://proofwiki.org/wiki/Reduced_Compton_Wavelength_of_Electron | [
"Reduced Compton Wavelength",
"Electrons"
] | [
"Definition:Compton Wavelength",
"Definition:Electron"
] | [
"Definition:Reduced Compton Wavelength",
"Definition:Electron",
"Definition:Compton Wavelength",
"Definition:Electron",
"Compton Wavelength of Electron"
] |
proofwiki-21114 | Translation of Local Basis in Topological Vector Space | Let $K$ be a topological field.
Let $\struct {X, \tau}$ be a topological vector space over $K$.
Let $\sequence {U_\alpha}_{\alpha \mathop \in A}$ be a local basis at ${\mathbf 0}_X$.
Let $x \in X$.
Then $\sequence {U_\alpha + x}_{\alpha \in A}$ is an local basis at $x$. | From Translation of Open Set in Topological Vector Space is Open, $U_\alpha + x$ is an open neighborhood of $x$ in $\struct {X, \tau}$ for each $\alpha \in A$.
Let $U$ be an open neighborhood of $x$ in $\struct {X, \tau}$.
From Translation of Open Set in Topological Vector Space is Open, $U - x$ is an open neighborhoo... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $K$.
Let $\sequence {U_\alpha}_{\alpha \mathop \in A}$ be a [[Definition:Local Basis|local basis]] at ${\mathbf 0}_X$.
Let $x \in X$.
Then $\sequence {... | From [[Translation of Open Set in Topological Vector Space is Open]], $U_\alpha + x$ is an [[Definition:Open Neighborhood|open neighborhood]] of $x$ in $\struct {X, \tau}$ for each $\alpha \in A$.
Let $U$ be an [[Definition:Open Neighborhood|open neighborhood]] of $x$ in $\struct {X, \tau}$.
From [[Translation of Op... | Translation of Local Basis in Topological Vector Space | https://proofwiki.org/wiki/Translation_of_Local_Basis_in_Topological_Vector_Space | https://proofwiki.org/wiki/Translation_of_Local_Basis_in_Topological_Vector_Space | [
"Translation of Subsets of Vector Spaces",
"Local Bases",
"Topological Vector Spaces",
"Translation of Subsets of Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Local Basis",
"Definition:Local Basis"
] | [
"Translation of Open Set in Topological Vector Space is Open",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Translation of Open Set in Topological Vector Space is Open",
"Definition:Open Neighborhood",
"Definition:Local Basis",
"Definition:Local Basis",
"Category:Local Bases",
"C... |
proofwiki-21115 | Dilation of Subset of Vector Space Distributes over Sum/General Case | Let $\family {E_\alpha}_{\alpha \mathop \in A}$ be an $A$-indexed family of sets.
Let $\lambda \in K$.
Then:
:$\ds \lambda \sum_{\alpha \mathop \in A} E_\alpha = \sum_{\alpha \mathop \in A} \paren {\lambda E_\alpha}$ | Let $x \in X$.
We have:
:$\ds x \in \lambda \sum_{\alpha \mathop \in A} E_\alpha$
{{iff}} there exists:
:a finite subset $F \subseteq E_\alpha$
:$x_\alpha \in F$ for each $\alpha \in F$
such that:
:$\ds x = \lambda \sum_{\alpha \in F} x_\alpha$
This is equivalent to:
:$\ds x = \sum_{\alpha \in F} \lambda x_\alpha$
for ... | Let $\family {E_\alpha}_{\alpha \mathop \in A}$ be an [[Definition:Indexed Family of Sets|$A$-indexed family of sets]].
Let $\lambda \in K$.
Then:
:$\ds \lambda \sum_{\alpha \mathop \in A} E_\alpha = \sum_{\alpha \mathop \in A} \paren {\lambda E_\alpha}$ | Let $x \in X$.
We have:
:$\ds x \in \lambda \sum_{\alpha \mathop \in A} E_\alpha$
{{iff}} there exists:
:a [[Definition:Finite Subset|finite subset]] $F \subseteq E_\alpha$
:$x_\alpha \in F$ for each $\alpha \in F$
such that:
:$\ds x = \lambda \sum_{\alpha \in F} x_\alpha$
This is equivalent to:
:$\ds x = \sum_{\alph... | Dilation of Subset of Vector Space Distributes over Sum/General Case | https://proofwiki.org/wiki/Dilation_of_Subset_of_Vector_Space_Distributes_over_Sum/General_Case | https://proofwiki.org/wiki/Dilation_of_Subset_of_Vector_Space_Distributes_over_Sum/General_Case | [
"Dilation of Subset of Vector Space Distributes over Sum"
] | [
"Definition:Indexing Set/Family of Sets"
] | [
"Definition:Finite Subset",
"Definition:Finite Subset",
"Category:Dilation of Subset of Vector Space Distributes over Sum"
] |
proofwiki-21116 | Dilation of Subset of Vector Space Distributes over Sum/Finite Case | Let $A_1, \ldots, A_n \subseteq X$ and $\lambda \in \GF$.
Then:
:$\ds \lambda \sum_{j \mathop = 1}^n A_j = \sum_{j = 1}^n \paren {\lambda A_j}$
where:
:$\ds \lambda \paren \ldots$ denotes dilation by $\lambda$
:$\ds \sum_{j \mathop = 1}^n A_j$ denotes the linear combination of subsets of a vector space. | Let $x \in X$.
We have:
:$\ds x \in \lambda \sum_{j \mathop = 1}^n A_j$
{{iff}} there exists $x_j \in A_j$ for each $j \in \set {1, 2, \ldots, n}$ such that:
:$\ds x = \lambda \sum_{j \mathop = 1}^n x_j$
This is equivalent to:
:$\ds x = \sum_{j \mathop = 1}^n \lambda x_j \in \sum_{j = 1}^n \paren {\lambda A_j}$ for som... | Let $A_1, \ldots, A_n \subseteq X$ and $\lambda \in \GF$.
Then:
:$\ds \lambda \sum_{j \mathop = 1}^n A_j = \sum_{j = 1}^n \paren {\lambda A_j}$
where:
:$\ds \lambda \paren \ldots$ denotes [[Definition:Dilation of Subset of Vector Space|dilation by $\lambda$]]
:$\ds \sum_{j \mathop = 1}^n A_j$ denotes [[Definition:Li... | Let $x \in X$.
We have:
:$\ds x \in \lambda \sum_{j \mathop = 1}^n A_j$
{{iff}} there exists $x_j \in A_j$ for each $j \in \set {1, 2, \ldots, n}$ such that:
:$\ds x = \lambda \sum_{j \mathop = 1}^n x_j$
This is equivalent to:
:$\ds x = \sum_{j \mathop = 1}^n \lambda x_j \in \sum_{j = 1}^n \paren {\lambda A_j}$ for s... | Dilation of Subset of Vector Space Distributes over Sum/Finite Case | https://proofwiki.org/wiki/Dilation_of_Subset_of_Vector_Space_Distributes_over_Sum/Finite_Case | https://proofwiki.org/wiki/Dilation_of_Subset_of_Vector_Space_Distributes_over_Sum/Finite_Case | [
"Dilation of Subset of Vector Space Distributes over Sum"
] | [
"Definition:Linear Combination of Subsets of Vector Space/Dilation",
"Definition:Linear Combination of Subsets of Vector Space"
] | [
"Category:Dilation of Subset of Vector Space Distributes over Sum"
] |
proofwiki-21117 | Linear Combination of Balanced Sets is Balanced | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $\family {E_\alpha}_{\alpha \mathop \in A}$ be an $A$-indexed family of balanced sets.
Let $\lambda_\alpha \in \GF$ for each $\alpha \mathop \in A$.
Then:
:$\ds \sum_{\alpha \mathop \in A} \lambda_\alpha E_\alpha$ is balanced. | Let $s \in \C$ have $\cmod s \le 1$.
Then, we have:
{{begin-eqn}}
{{eqn | l = s \sum_{\alpha \mathop \in A} \lambda_\alpha E_\alpha
| r = \sum_{\alpha \mathop \in A} \lambda_\alpha \paren {s E_\alpha}
| c = Dilation of Subset of Vector Space Distributes over Sum: General Case
}}
{{eqn | o = \subseteq
| r = \sum_{... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $\family {E_\alpha}_{\alpha \mathop \in A}$ be an [[Definition:Indexed Family of Sets|$A$-indexed family]] of [[Definition:Balanced Set|balanced sets]].
Let $\lambda_\alpha \in \GF$ for each $\alpha \mathop \in A$.
... | Let $s \in \C$ have $\cmod s \le 1$.
Then, we have:
{{begin-eqn}}
{{eqn | l = s \sum_{\alpha \mathop \in A} \lambda_\alpha E_\alpha
| r = \sum_{\alpha \mathop \in A} \lambda_\alpha \paren {s E_\alpha}
| c = [[Dilation of Subset of Vector Space Distributes over Sum/General Case|Dilation of Subset of Vector Space Di... | Linear Combination of Balanced Sets is Balanced | https://proofwiki.org/wiki/Linear_Combination_of_Balanced_Sets_is_Balanced | https://proofwiki.org/wiki/Linear_Combination_of_Balanced_Sets_is_Balanced | [
"Balanced Sets"
] | [
"Definition:Vector Space",
"Definition:Indexing Set/Family of Sets",
"Definition:Balanced Set",
"Definition:Balanced Set"
] | [
"Dilation of Subset of Vector Space Distributes over Sum/General Case",
"Definition:Balanced Set",
"Category:Balanced Sets"
] |
proofwiki-21118 | Compton Wavelength of Proton | The Compton wavelength of the proton is:
{{begin-eqn}}
{{eqn | l = \lambda_{\mathrm p}
| o = \approx
| r = 1 \cdotp 32140 \, 98553 \, 9(40) \times 10^{-15} \, \mathrm m
}}
{{eqn | o = \approx
| r = 1 \cdotp 32140 \, 98553 \, 9(40) \times 10^{-13} \, \mathrm {cm}
}}
{{end-eqn}} | By definition, the Compton wavelength $\lambda_{\mathrm p}$ of a proton is given as:
:$\lambda_{\mathrm p} = \dfrac h {m_{\mathrm p} c}$
where:
:$m_{\mathrm p}$ denotes the mass of the proton
:$h$ denotes Planck's constant
:$c$ denotes the speed of light.
Then we have:
{{begin-eqn}}
{{eqn | l = h
| r = 6 \cdotp 6... | The [[Definition:Compton Wavelength|Compton wavelength]] of the [[Definition:Proton|proton]] is:
{{begin-eqn}}
{{eqn | l = \lambda_{\mathrm p}
| o = \approx
| r = 1 \cdotp 32140 \, 98553 \, 9(40) \times 10^{-15} \, \mathrm m
}}
{{eqn | o = \approx
| r = 1 \cdotp 32140 \, 98553 \, 9(40) \times 10^{-13... | By definition, the [[Definition:Compton Wavelength|Compton wavelength]] $\lambda_{\mathrm p}$ of a [[Definition:Proton|proton]] is given as:
:$\lambda_{\mathrm p} = \dfrac h {m_{\mathrm p} c}$
where:
:$m_{\mathrm p}$ denotes the [[Definition:Mass|mass]] of the [[Definition:Proton|proton]]
:$h$ denotes [[Definition:Plan... | Compton Wavelength of Proton | https://proofwiki.org/wiki/Compton_Wavelength_of_Proton | https://proofwiki.org/wiki/Compton_Wavelength_of_Proton | [
"Compton Wavelength",
"Protons"
] | [
"Definition:Compton Wavelength",
"Definition:Proton"
] | [
"Definition:Compton Wavelength",
"Definition:Proton",
"Definition:Mass",
"Definition:Proton",
"Definition:Planck's Constant",
"Definition:Speed of Light",
"Definition:SI/Energy/Joule",
"Definition:Time/Unit/Second",
"Definition:Metric System/Mass/Kilogram",
"Definition:Proton/Mass",
"Definition:... |
proofwiki-21119 | Reduced Compton Wavelength of Proton | The Compton wavelength of the proton is:
{{begin-eqn}}
{{eqn | l = \lambdabar_{\mathrm p}
| o = \approx
| r = 2 \cdotp 10308 \, 9103 \times 10^{-16} \, \mathrm m
}}
{{eqn | o = \approx
| r = 2 \cdotp 10308 \, 9103 \times 10^{-14} \, \mathrm {cm}
}}
{{end-eqn}} | By definition, the reduced Compton wavelength $\lambdabar_{\mathrm p}$ of a proton is given as:
:$\lambdabar_{\mathrm p} = \dfrac {\lambda_{\mathrm p} } {2 \pi}$
where $\lambda_{\mathrm p}$ denotes the Compton wavelength of the proton.
Then we have:
{{begin-eqn}}
{{eqn | l = \lambda_{\mathrm p}
| o = \approx
... | The [[Definition:Compton Wavelength|Compton wavelength]] of the [[Definition:Proton|proton]] is:
{{begin-eqn}}
{{eqn | l = \lambdabar_{\mathrm p}
| o = \approx
| r = 2 \cdotp 10308 \, 9103 \times 10^{-16} \, \mathrm m
}}
{{eqn | o = \approx
| r = 2 \cdotp 10308 \, 9103 \times 10^{-14} \, \mathrm {cm}... | By definition, the [[Definition:Reduced Compton Wavelength|reduced Compton wavelength]] $\lambdabar_{\mathrm p}$ of a [[Definition:Proton|proton]] is given as:
:$\lambdabar_{\mathrm p} = \dfrac {\lambda_{\mathrm p} } {2 \pi}$
where $\lambda_{\mathrm p}$ denotes the [[Definition:Compton Wavelength|Compton wavelength]] o... | Reduced Compton Wavelength of Proton | https://proofwiki.org/wiki/Reduced_Compton_Wavelength_of_Proton | https://proofwiki.org/wiki/Reduced_Compton_Wavelength_of_Proton | [
"Reduced Compton Wavelength",
"Protons"
] | [
"Definition:Compton Wavelength",
"Definition:Proton"
] | [
"Definition:Reduced Compton Wavelength",
"Definition:Proton",
"Definition:Compton Wavelength",
"Definition:Proton",
"Compton Wavelength of Proton"
] |
proofwiki-21120 | Pseudometric Space is First-Countable | Let $M = \struct {A, d}$ be a pseudometric space.
Then $M$ is first-countable. | Let $x \in A$.
Let:
:$\BB = \set {\map {B_{1/n} } x: n \in \N_{>0} }$
where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$ in $M$.
By Surjection from Natural Numbers iff Countable, we have that $\BB$ is countable.
By the definition of a first-countable space, it suffices to show that $\BB$ is a local bas... | Let $M = \struct {A, d}$ be a [[Definition:Pseudometric Space|pseudometric space]].
Then $M$ is [[Definition:First-Countable Space|first-countable]]. | Let $x \in A$.
Let:
:$\BB = \set {\map {B_{1/n} } x: n \in \N_{>0} }$
where $\map {B_\epsilon} x$ denotes the [[Definition:Open Ball|open $\epsilon$-ball of $x$ in $M$]].
By [[Surjection from Natural Numbers iff Countable]], we have that $\BB$ is [[Definition:Countable Set|countable]].
By the definition of a [[De... | Pseudometric Space is First-Countable | https://proofwiki.org/wiki/Pseudometric_Space_is_First-Countable | https://proofwiki.org/wiki/Pseudometric_Space_is_First-Countable | [
"Pseudometric Spaces",
"First-Countable Spaces"
] | [
"Definition:Pseudometric/Pseudometric Space",
"Definition:First-Countable Space"
] | [
"Definition:Open Ball",
"Surjection from Natural Numbers iff Countable",
"Definition:Countable Set",
"Definition:First-Countable Space",
"Definition:Local Basis",
"Open Ball is Open Set/Pseudometric Space",
"Definition:Element",
"Definition:Open Neighborhood/Point",
"Definition:Open Neighborhood/Poi... |
proofwiki-21121 | Open Ball is Open Set/Pseudometric Space | Let $M = \struct {A, d}$ be a pseudometric space.
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball of $x$ in $M$.
Then $\map {B_\epsilon} x$ is an open set of $M$. | Let $y \in \map {B_\epsilon} x$.
From Open Ball of Point Inside Open Ball, there exists $\delta \in \R_{>0}$ such that:
:$\map {B_\delta} y \subseteq \map {B_\epsilon} x$
The result follows from the definition of open set.
{{qed}}
Category:Open Ball is Open Set
Category:Open Sets (Pseudometric Spaces)
cfo383emog4gc3sb9... | Let $M = \struct {A, d}$ be a [[Definition:Pseudometric Space|pseudometric space]].
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} x$ be an [[Definition:Open Ball|open $\epsilon$-ball]] of $x$ in $M$.
Then $\map {B_\epsilon} x$ is an [[Definition:Open Set (Pseudometric Space)|open set]] of $M$. | Let $y \in \map {B_\epsilon} x$.
From [[Open Ball of Point Inside Open Ball]], there exists $\delta \in \R_{>0}$ such that:
:$\map {B_\delta} y \subseteq \map {B_\epsilon} x$
The result follows from the definition of [[Definition:Open Set (Pseudometric Space)|open set]].
{{qed}}
[[Category:Open Ball is Open Set]]
[[... | Open Ball is Open Set/Pseudometric Space | https://proofwiki.org/wiki/Open_Ball_is_Open_Set/Pseudometric_Space | https://proofwiki.org/wiki/Open_Ball_is_Open_Set/Pseudometric_Space | [
"Open Ball is Open Set",
"Open Sets (Pseudometric Spaces)"
] | [
"Definition:Pseudometric/Pseudometric Space",
"Definition:Open Ball",
"Definition:Open Set/Pseudometric Space"
] | [
"Open Ball of Point Inside Open Ball",
"Definition:Open Set/Pseudometric Space",
"Category:Open Ball is Open Set",
"Category:Open Sets (Metric Spaces)"
] |
proofwiki-21122 | Dimension of Rydberg Constant | The '''Rydberg constant''' has the dimension $\mathsf {L^{-1} }$. | By definition, the Rydberg constant is:
:$R_\infty = \dfrac {m_\E \E^4} {8 \varepsilon_0^2 h^3 c}$
where:
:$m_\E$ denotes the electron rest mass
:$\E$ denotes the elementary charge
:$\varepsilon_0$ denotes the vacuum permittivity
:$h$ denotes Planck's constant
:$c$ denotes the speed of light.
We have:
{{begin-eqn}}
{{e... | The '''[[Definition:Rydberg Constant|Rydberg constant]]''' has the [[Definition:Dimension of Measurement|dimension]] $\mathsf {L^{-1} }$. | By definition, the [[Definition:Rydberg Constant|Rydberg constant]] is:
:$R_\infty = \dfrac {m_\E \E^4} {8 \varepsilon_0^2 h^3 c}$
where:
:$m_\E$ denotes the [[Definition:Mass of Electron|electron rest mass]]
:$\E$ denotes the [[Definition:Elementary Charge|elementary charge]]
:$\varepsilon_0$ denotes the [[Definition... | Dimension of Rydberg Constant | https://proofwiki.org/wiki/Dimension_of_Rydberg_Constant | https://proofwiki.org/wiki/Dimension_of_Rydberg_Constant | [
"Rydberg Constant",
"Dimensions of Measurement"
] | [
"Definition:Rydberg Constant",
"Definition:Dimension (Measurement)"
] | [
"Definition:Rydberg Constant",
"Definition:Electron/Mass",
"Definition:Electric Charge/Quantum",
"Definition:Vacuum Permittivity",
"Definition:Planck's Constant",
"Definition:Speed of Light",
"Category:Rydberg Constant",
"Category:Dimensions of Measurement"
] |
proofwiki-21123 | Null Sequence in Metrizable Topological Vector Space Dominates some Sequence of Scalars Tending to Infinity | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a metrizable topological vector space over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence with $x_n \to {\mathbf 0}_X$.
Then there exists a sequence of positive real numbers $\sequence {\gamma_n}_{n \mathop \in \N}$ such that:
:$\gamma_n \to \infty... | From Birkhoff-Kakutani Theorem: Topological Vector Space, there exists an invariant metric $d$ on $X$ that induces $\tau$.
Then:
:$\map d {x_n, { {\mathbf 0}_X} } \to 0$
Pick $n_1 \in \N$ such that:
:$\map d {x_n, { {\mathbf 0}_X} } < 1$ for $n > n_1$.
Inductively, for $k \ge 2$, pick $n_k > n_{k - 1}$ such that:
:$\ma... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Metrizable Topology|metrizable]] [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] with $x_n \to {\mathbf 0}_X$.
Then there exists a [[Def... | From [[Birkhoff-Kakutani Theorem/Topological Vector Space|Birkhoff-Kakutani Theorem: Topological Vector Space]], there exists an [[Definition:Invariant Metric on Vector Space|invariant metric]] $d$ on $X$ that [[Definition:Topology Induced by Metric|induces]] $\tau$.
Then:
:$\map d {x_n, { {\mathbf 0}_X} } \to 0$
Pic... | Null Sequence in Metrizable Topological Vector Space Dominates some Sequence of Scalars Tending to Infinity | https://proofwiki.org/wiki/Null_Sequence_in_Metrizable_Topological_Vector_Space_Dominates_some_Sequence_of_Scalars_Tending_to_Infinity | https://proofwiki.org/wiki/Null_Sequence_in_Metrizable_Topological_Vector_Space_Dominates_some_Sequence_of_Scalars_Tending_to_Infinity | [
"Topological Vector Spaces"
] | [
"Definition:Metrizable Space",
"Definition:Topological Vector Space",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Positive/Real Number"
] | [
"Birkhoff-Kakutani Theorem/Topological Vector Space",
"Definition:Invariant Metric on Vector Space",
"Definition:Topology Induced by Metric",
"Subadditivity of Invariant Metric on Vector Space"
] |
proofwiki-21124 | Characterization of Continuous Linear Transformation from Metrizable Topological Vector Space to Topological Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau_X}$ be a metrizable topological vector space over $\GF$.
Let $\struct {Y, \tau_Y}$ be a topological vector space over $\GF$.
Let $T : X \to Y$ be a linear transformation.
{{TFAE}}
:$(1): \quad$ $T$ is continuous
:$(2): \quad$ $T$ is bounded
:$(3): \quad$ for every sequ... | === $(1)$ implies $(2)$ ===
This is precisely the result Continuous Linear Transformation between Topological Vector Spaces is Bounded.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau_X}$ be a [[Definition:Metrizable Topology|metrizable]] [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $\struct {Y, \tau_Y}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $T : X \to Y$ be a [[D... | === $(1)$ implies $(2)$ ===
This is precisely the result [[Continuous Linear Transformation between Topological Vector Spaces is Bounded]].
{{qed|lemma}} | Characterization of Continuous Linear Transformation from Metrizable Topological Vector Space to Topological Vector Space | https://proofwiki.org/wiki/Characterization_of_Continuous_Linear_Transformation_from_Metrizable_Topological_Vector_Space_to_Topological_Vector_Space | https://proofwiki.org/wiki/Characterization_of_Continuous_Linear_Transformation_from_Metrizable_Topological_Vector_Space_to_Topological_Vector_Space | [
"Topological Vector Spaces"
] | [
"Definition:Metrizable Space",
"Definition:Topological Vector Space",
"Definition:Topological Vector Space",
"Definition:Linear Transformation",
"Definition:Continuous Mapping",
"Definition:Bounded Linear Transformation/Topological Vector Space",
"Definition:Sequence",
"Definition:Sequence",
"Defini... | [
"Continuous Linear Transformation between Topological Vector Spaces is Bounded"
] |
proofwiki-21125 | Composition of Open Mappings is Open Mapping | Let $\struct {X, \tau_X}$, $\struct {Y, \tau_Y}$ and $\struct {Z, \tau_Z}$ be topological spaces.
Let $f : X \to Y$ and $g : Y \to Z$ be open mappings.
Then $g \circ f : X \to Z$ is an open mapping. | Let $U$ be an open set in $\struct {X, \tau_X}$.
Since $f : X \to Y$ is open, we have $f \sqbrk U \in \tau_Y$.
Since $g : Y \to Z$ is open, we have $g \sqbrk {f \sqbrk U} \in \tau_Z$.
That is, whenever $U \in \tau_X$, we have $\paren {g \circ f} \sqbrk U \in \tau_Z$.
So $g \circ f : X \to Z$ is an open mapping.
{{qed}}... | Let $\struct {X, \tau_X}$, $\struct {Y, \tau_Y}$ and $\struct {Z, \tau_Z}$ be [[Definition:Topological Space|topological spaces]].
Let $f : X \to Y$ and $g : Y \to Z$ be [[Definition:Open Mapping|open mappings]].
Then $g \circ f : X \to Z$ is an [[Definition:Open Mapping|open mapping]]. | Let $U$ be an [[Definition:Open Set|open set]] in $\struct {X, \tau_X}$.
Since $f : X \to Y$ is [[Definition:Open Mapping|open]], we have $f \sqbrk U \in \tau_Y$.
Since $g : Y \to Z$ is [[Definition:Open Mapping|open]], we have $g \sqbrk {f \sqbrk U} \in \tau_Z$.
That is, whenever $U \in \tau_X$, we have $\paren {g ... | Composition of Open Mappings is Open Mapping | https://proofwiki.org/wiki/Composition_of_Open_Mappings_is_Open_Mapping | https://proofwiki.org/wiki/Composition_of_Open_Mappings_is_Open_Mapping | [
"Open Mappings"
] | [
"Definition:Topological Space",
"Definition:Open Mapping",
"Definition:Open Mapping"
] | [
"Definition:Open Set",
"Definition:Open Mapping",
"Definition:Open Mapping",
"Definition:Open Mapping",
"Category:Open Mappings"
] |
proofwiki-21126 | Factorization of Open Linear Transformation between Topological Vector Spaces | Let $K$ be a topological field.
Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_X}$ be topological vector spaces over $K$.
Let $T : X \to Y$ be a linear transformation.
Let $N$ be a vector subspace of $X$ with $N \subseteq \ker T$.
Let $\struct {X/N, \tau_N}$ be the quotient topological vector space of $X$ modulo $N$.... | === Necessary Condition ===
Suppose that $T$ is open.
From Condition for Mapping from Quotient Vector Space to be Well-Defined, there exists a linear transformation $\Lambda : X/N \to Y$ such that $T x = \map \Lambda {\map \pi x}$ for each $x \in X$.
It remains to show that $\Lambda$ is open.
Let $E \subseteq X/N$.
We... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_X}$ be [[Definition:Topological Vector Space|topological vector spaces]] over $K$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Let $N$ be a [[Definition:Vector Subsp... | === Necessary Condition ===
Suppose that $T$ is [[Definition:Open Mapping|open]].
From [[Condition for Mapping from Quotient Vector Space to be Well-Defined]], there exists a [[Definition:Linear Transformation|linear transformation]] $\Lambda : X/N \to Y$ such that $T x = \map \Lambda {\map \pi x}$ for each $x \in X$... | Factorization of Open Linear Transformation between Topological Vector Spaces | https://proofwiki.org/wiki/Factorization_of_Open_Linear_Transformation_between_Topological_Vector_Spaces | https://proofwiki.org/wiki/Factorization_of_Open_Linear_Transformation_between_Topological_Vector_Spaces | [
"Open Mappings",
"Topological Vector Spaces",
"Quotient Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Linear Transformation",
"Definition:Vector Subspace",
"Definition:Quotient Topological Vector Space",
"Definition:Quotient Mapping",
"Definition:Open Mapping",
"Definition:Open Mapping",
"Definition:Linear Transformat... | [
"Definition:Open Mapping",
"Condition for Mapping from Quotient Vector Space to be Well-Defined",
"Definition:Linear Transformation",
"Definition:Open Mapping",
"Definition:Open Set",
"Definition:Quotient Topology",
"Definition:Continuous Mapping",
"Definition:Open Set",
"Definition:Open Mapping",
... |
proofwiki-21127 | First Isomorphism Theorem/Vector Spaces | Let $K$ be a field.
Let $X$ and $Y$ be vector spaces over $K$.
Let $T : X \to Y$ be a linear transformation.
Let $\ker T$ be the kernel of $T$.
Let $X/\ker T$ be the quotient vector space of $X$ modulo $\ker T$.
Then $X/\ker T$ is isomorphic to $\Img T$ as a vector space. | From Image of Linear Transformation is Submodule, we assure ourselves that $\Img T$ is indeed a vector space over $K$.
Let $\pi : X \to X/\ker T$ be the quotient mapping.
From Condition for Mapping from Quotient Vector Space to be Well-Defined, there exists a linear transformation $\Lambda : X/\ker T \to \Img T$ such ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ and $Y$ be [[Definition:Vector Space|vector spaces]] over $K$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Let $\ker T$ be the [[Definition:Kernel of Linear Transformation|kernel]] of $T$.
Let $X/\ker T$ be th... | From [[Image of Linear Transformation is Submodule]], we assure ourselves that $\Img T$ is indeed a [[Definition:Vector Space|vector space]] over $K$.
Let $\pi : X \to X/\ker T$ be the [[Definition:Quotient Mapping|quotient mapping]].
From [[Condition for Mapping from Quotient Vector Space to be Well-Defined]], ther... | First Isomorphism Theorem/Vector Spaces | https://proofwiki.org/wiki/First_Isomorphism_Theorem/Vector_Spaces | https://proofwiki.org/wiki/First_Isomorphism_Theorem/Vector_Spaces | [
"First Isomorphism Theorem",
"Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Linear Transformation",
"Definition:Kernel of Linear Transformation",
"Definition:Quotient Vector Space",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism"
] | [
"Image of Linear Transformation is Submodule",
"Definition:Vector Space",
"Definition:Quotient Mapping",
"Condition for Mapping from Quotient Vector Space to be Well-Defined",
"Definition:Linear Transformation",
"Definition:Injection",
"Definition:Surjection",
"Linear Transformation is Injective iff K... |
proofwiki-21128 | Open Ball of Point Inside Open Ball/Pseudometric Space | Let $M = \struct {A, d}$ be a pseudometric space.
Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M = \struct {A, d}$.
Let $y \in \map {B_\epsilon} x$.
Then:
:$\exists \delta \in \R: \map {B_\delta} y \subseteq \map {B_\epsilon} x$
That is, for every point in an open $\epsilon$-ball in a pseudometric space, th... | Let $\delta = \epsilon - \map d {x, y}$.
From the definition of open ball, this is strictly positive, since $y \in \map {B_\epsilon} x$.
If $z \in \map {B_\delta} y$, then $\map d {x, y} < \delta$.
So:
:$\map d {x, z} \le \map d {x, y} + \map d {y, z} < \map d {x, y} + \delta = \epsilon$
Thus $z \in \map {B_\epsilon} x... | Let $M = \struct {A, d}$ be a [[Definition:Pseudometric Space|pseudometric space]].
Let $\map {B_\epsilon} x$ be an [[Definition:Open Ball|open $\epsilon$-ball]] in $M = \struct {A, d}$.
Let $y \in \map {B_\epsilon} x$.
Then:
:$\exists \delta \in \R: \map {B_\delta} y \subseteq \map {B_\epsilon} x$
That is, for e... | Let $\delta = \epsilon - \map d {x, y}$.
From the definition of [[Definition:Open Ball|open ball]], this is [[Definition:Strictly Positive|strictly positive]], since $y \in \map {B_\epsilon} x$.
If $z \in \map {B_\delta} y$, then $\map d {x, y} < \delta$.
So:
:$\map d {x, z} \le \map d {x, y} + \map d {y, z} < \map ... | Open Ball of Point Inside Open Ball/Pseudometric Space | https://proofwiki.org/wiki/Open_Ball_of_Point_Inside_Open_Ball/Pseudometric_Space | https://proofwiki.org/wiki/Open_Ball_of_Point_Inside_Open_Ball/Pseudometric_Space | [
"Open Balls",
"Open Ball of Point Inside Open Ball"
] | [
"Definition:Pseudometric/Pseudometric Space",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Pseudometric/Pseudometric Space",
"Definition:Open Ball",
"Definition:Open Ball"
] | [
"Definition:Open Ball",
"Definition:Strictly Positive",
"Category:Open Balls",
"Category:Open Ball of Point Inside Open Ball"
] |
proofwiki-21129 | Vector Subspace of Hausdorff Topological Vector Space is Hausdorff Topological Vector Space | Let $\struct {K, +_K, \circ_K, \tau_K}$ be a topological field.
Let $\struct {\struct {X, +_X, \circ_X}_K, \tau_X}$ be a Hausdorff topological vector space over $K$.
Let $\struct {Y, +_Y, \circ_Y}_K$ be a vector subspace of $X$.
where:
:$+_Y : Y \times Y \to Y$ is the restriction of $+_X$ to $Y \times Y$
:$\circ_Y : K... | From Vector Subspace of Topological Vector Space is Topological Vector Space, $\struct {\struct {Y, +_Y, \circ_Y}_K, \tau_Y}$ is a topological vector space over $K$.
Since $\struct {\struct {X, +_X, \circ_X}_K, \tau_X}$ is a Hausdorff topological vector space over $K$, it is {{apriori}} Hausdorff.
We are given that $\s... | Let $\struct {K, +_K, \circ_K, \tau_K}$ be a [[Definition:Topological Field|topological field]].
Let $\struct {\struct {X, +_X, \circ_X}_K, \tau_X}$ be a [[Definition:Hausdorff Topological Vector Space|Hausdorff topological vector space]] over $K$.
Let $\struct {Y, +_Y, \circ_Y}_K$ be a [[Definition:Vector Subspace|v... | From [[Vector Subspace of Topological Vector Space is Topological Vector Space]], $\struct {\struct {Y, +_Y, \circ_Y}_K, \tau_Y}$ is a [[Definition:Topological Vector Space|topological vector space]] over $K$.
Since $\struct {\struct {X, +_X, \circ_X}_K, \tau_X}$ is a [[Definition:Hausdorff Topological Vector Space|Ha... | Vector Subspace of Hausdorff Topological Vector Space is Hausdorff Topological Vector Space | https://proofwiki.org/wiki/Vector_Subspace_of_Hausdorff_Topological_Vector_Space_is_Hausdorff_Topological_Vector_Space | https://proofwiki.org/wiki/Vector_Subspace_of_Hausdorff_Topological_Vector_Space_is_Hausdorff_Topological_Vector_Space | [
"Hausdorff Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Hausdorff Topological Vector Space",
"Definition:Vector Subspace",
"Definition:Restriction/Mapping",
"Definition:Restriction/Mapping",
"Definition:Topological Subspace",
"Definition:Hausdorff Topological Vector Space"
] | [
"Vector Subspace of Topological Vector Space is Topological Vector Space",
"Definition:Topological Vector Space",
"Definition:Hausdorff Topological Vector Space",
"Definition:T2 Space",
"Definition:Given",
"Definition:Vector Subspace",
"Definition:Given",
"Definition:Topological Subspace",
"Definiti... |
proofwiki-21130 | Dimension of Bohr Radius | The '''Bohr radius''' has the dimension $\mathsf L$. | By definition, the Bohr radius is:
:$a_0 = \dfrac {\varepsilon_0 h^2} {\pi \E^2 m_\E}$
where:
:$\varepsilon_0$ denotes the vacuum permittivity
:$h$ denotes Planck's constant
:$\E$ denotes the elementary charge
:$m_\E$ denotes the electron rest mass
We have:
{{begin-eqn}}
{{eqn | l = \varepsilon_0
| o = \text {has... | The '''[[Definition:Bohr Radius|Bohr radius]]''' has the [[Definition:Dimension of Measurement|dimension]] $\mathsf L$. | By definition, the [[Definition:Bohr Radius|Bohr radius]] is:
:$a_0 = \dfrac {\varepsilon_0 h^2} {\pi \E^2 m_\E}$
where:
:$\varepsilon_0$ denotes the [[Definition:Vacuum Permittivity|vacuum permittivity]]
:$h$ denotes [[Definition:Planck's Constant|Planck's constant]]
:$\E$ denotes the [[Definition:Elementary Charge|e... | Dimension of Bohr Radius | https://proofwiki.org/wiki/Dimension_of_Bohr_Radius | https://proofwiki.org/wiki/Dimension_of_Bohr_Radius | [
"Bohr Radius",
"Dimensions of Measurement"
] | [
"Definition:Bohr Radius",
"Definition:Dimension (Measurement)"
] | [
"Definition:Bohr Radius",
"Definition:Vacuum Permittivity",
"Definition:Planck's Constant",
"Definition:Electric Charge/Quantum",
"Definition:Electron/Mass",
"Category:Bohr Radius",
"Category:Dimensions of Measurement"
] |
proofwiki-21131 | Interior of Translation of Set in Topological Vector Space is Translation of Interior | Let $K$ be a topological field.
Let $\struct {X, \tau}$ be a topological vector space over $K$.
Let $E \subseteq X$.
Let $x \in X$.
Then:
:$\paren {E + x}^\circ = E^\circ + x$ | Let $y \in \paren {E + x}^\circ$.
Then there exists an open neighborhood $U$ of $y$ such that $U \subseteq E + x$.
Then from Translation of Open Set in Topological Vector Space is Open, $U - x$ is an open neighborhood of $y - x$ such that $U - x \subseteq E$.
Hence $y - x \in E^\circ$.
So $y \in E^\circ + x$.
We theref... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $K$.
Let $E \subseteq X$.
Let $x \in X$.
Then:
:$\paren {E + x}^\circ = E^\circ + x$ | Let $y \in \paren {E + x}^\circ$.
Then there exists an [[Definition:Open Neighborhood|open neighborhood]] $U$ of $y$ such that $U \subseteq E + x$.
Then from [[Translation of Open Set in Topological Vector Space is Open]], $U - x$ is an [[Definition:Open Neighborhood|open neighborhood]] of $y - x$ such that $U - x \s... | Interior of Translation of Set in Topological Vector Space is Translation of Interior | https://proofwiki.org/wiki/Interior_of_Translation_of_Set_in_Topological_Vector_Space_is_Translation_of_Interior | https://proofwiki.org/wiki/Interior_of_Translation_of_Set_in_Topological_Vector_Space_is_Translation_of_Interior | [
"Set Interior",
"Set Interiors",
"Translation of Subsets of Vector Spaces",
"Set Interiors",
"Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space"
] | [
"Definition:Open Neighborhood",
"Translation of Open Set in Topological Vector Space is Open",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Translation of Open Set in Topological Vector Space is Open",
"Definition:Open Neighborhood",
"Category:Translation of Subsets of Vector Spaces"... |
proofwiki-21132 | Translation of Intersection of Subsets of Vector Space | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $\family {E_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of $X$.
Let $x \in X$.
Then:
:$\ds \paren {\bigcap_{\alpha \mathop \in I} E_\alpha} + x = \bigcap_{\alpha \mathop \in I} \paren {E_\alpha + x}$
where $E_\alpha + x$ denotes the transl... | Let $v \in X$.
We have:
:$\ds v \in \paren {\bigcap_{\alpha \mathop \in I} E_\alpha} + x$
{{iff}}:
:$\ds v - x \in \bigcap_{\alpha \mathop \in I} E_\alpha$
{{iff}}:
:$v - x \in E_\alpha$ for each $\alpha \in I$
{{iff}}:
:$v \in E_\alpha + x$ for each $\alpha \in I$
{{iff}}:
:$\ds v \in \bigcap_{\alpha \mathop \in I} \p... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $\family {E_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family of Subsets|indexed family]] of [[Definition:Subset|subsets]] of $X$.
Let $x \in X$.
Then:
:$\ds \paren {\big... | Let $v \in X$.
We have:
:$\ds v \in \paren {\bigcap_{\alpha \mathop \in I} E_\alpha} + x$
{{iff}}:
:$\ds v - x \in \bigcap_{\alpha \mathop \in I} E_\alpha$
{{iff}}:
:$v - x \in E_\alpha$ for each $\alpha \in I$
{{iff}}:
:$v \in E_\alpha + x$ for each $\alpha \in I$
{{iff}}:
:$\ds v \in \bigcap_{\alpha \mathop \in I} \... | Translation of Intersection of Subsets of Vector Space | https://proofwiki.org/wiki/Translation_of_Intersection_of_Subsets_of_Vector_Space | https://proofwiki.org/wiki/Translation_of_Intersection_of_Subsets_of_Vector_Space | [
"Translation of Subsets of Vector Spaces",
"Set Intersection"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Indexing Set/Family of Subsets",
"Definition:Subset",
"Definition:Translation of Subset of Vector Space"
] | [
"Category:Translation of Subsets of Vector Spaces",
"Category:Set Intersection"
] |
proofwiki-21133 | Reduced Gyromagnetic Ratio of Proton | The reduced gyromagnetic ratio of the proton is given by:
{{begin-eqn}}
{{eqn | l = \gamma_{\mathrm p} / 2 \pi
| o = \approx
| r = 4 \cdotp 25774 \, 785 \times 10^7 \, \mathrm {Hz \, T^{-1} }
| c =
}}
{{eqn | o = \approx
| r = 4 \cdotp 25774 \, 785 \times 10^3 \, \mathrm {Hz \, Gs^{-1} }
... | By definition, the reduced gyromagnetic ratio of a proton is given as:
:$\dfrac {\gamma_{\mathrm p} } {2 \pi}$
where $\gamma_{\mathrm p}$ denotes the gyromagnetic ratio of the proton.
Then we have:
{{begin-eqn}}
{{eqn | l = \gamma_{\mathrm p}
| o = \approx
| r = 2 \cdotp 67522 \, 18744 \, (11) \times 10^8 \... | The [[Definition:Reduced Gyromagnetic Ratio|reduced gyromagnetic ratio]] of the [[Definition:Proton|proton]] is given by:
{{begin-eqn}}
{{eqn | l = \gamma_{\mathrm p} / 2 \pi
| o = \approx
| r = 4 \cdotp 25774 \, 785 \times 10^7 \, \mathrm {Hz \, T^{-1} }
| c =
}}
{{eqn | o = \approx
| r = 4 \... | By definition, the [[Definition:Reduced Gyromagnetic Ratio|reduced gyromagnetic ratio]] of a [[Definition:Proton|proton]] is given as:
:$\dfrac {\gamma_{\mathrm p} } {2 \pi}$
where $\gamma_{\mathrm p}$ denotes the [[Definition:Gyromagnetic Ratio|gyromagnetic ratio]] of the [[Definition:Proton|proton]].
Then we have:
... | Reduced Gyromagnetic Ratio of Proton | https://proofwiki.org/wiki/Reduced_Gyromagnetic_Ratio_of_Proton | https://proofwiki.org/wiki/Reduced_Gyromagnetic_Ratio_of_Proton | [
"Reduced Gyromagnetic Ratio",
"Protons"
] | [
"Definition:Reduced Gyromagnetic Ratio",
"Definition:Proton"
] | [
"Definition:Reduced Gyromagnetic Ratio",
"Definition:Proton",
"Definition:Gyromagnetic Ratio",
"Definition:Proton",
"Gyromagnetic Ratio of Proton"
] |
proofwiki-21134 | Ratio of Proton Moment to Nuclear Magneton | The ratio of the proton moment to the nuclear magneton is given by:
{{begin-eqn}}
{{eqn | l = \dfrac {\mu_{\mathrm p} } {\mu_{\mathrm N} }
| r = 2 \cdotp 79284 \, 73446 \, 3(82)
| c =
}}
{{end-eqn}} | We have:
{{begin-eqn}}
{{eqn | l = \mu_{\mathrm N}
| r = 5 \cdotp 05078 \, 3699 \, (31) \times 10^{-27} \, \mathrm {J \, T^{-1} }
| c = {{Defof|Nuclear Magneton|subdef = Value}}
}}
{{eqn | l = \mu_{\mathrm p}
| r = 1 \cdotp 41060 \, 67973 \, 6 \, (60) \times 10^{-26} \, \mathrm {J \, T^{-1} }
| ... | The [[Definition:Ratio|ratio]] of the [[Definition:Proton Moment|proton moment]] to the [[Definition:Nuclear Magneton|nuclear magneton]] is given by:
{{begin-eqn}}
{{eqn | l = \dfrac {\mu_{\mathrm p} } {\mu_{\mathrm N} }
| r = 2 \cdotp 79284 \, 73446 \, 3(82)
| c =
}}
{{end-eqn}} | We have:
{{begin-eqn}}
{{eqn | l = \mu_{\mathrm N}
| r = 5 \cdotp 05078 \, 3699 \, (31) \times 10^{-27} \, \mathrm {J \, T^{-1} }
| c = {{Defof|Nuclear Magneton|subdef = Value}}
}}
{{eqn | l = \mu_{\mathrm p}
| r = 1 \cdotp 41060 \, 67973 \, 6 \, (60) \times 10^{-26} \, \mathrm {J \, T^{-1} }
|... | Ratio of Proton Moment to Nuclear Magneton | https://proofwiki.org/wiki/Ratio_of_Proton_Moment_to_Nuclear_Magneton | https://proofwiki.org/wiki/Ratio_of_Proton_Moment_to_Nuclear_Magneton | [
"Proton Moment",
"Nuclear Magneton"
] | [
"Definition:Ratio",
"Definition:Proton Moment",
"Definition:Nuclear Magneton"
] | [] |
proofwiki-21135 | Finite Generalized Sum Converges to Summation | Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\set{i_0, i_1, \ldots, i_n}$ be a finite enumeration of a finite set $I$.
Let $\family{g_i}_{i \in I}$ be an indexed family of elements of $G$.
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$
converges to:
:the summation over finite ... | Let $\FF$ be the set of finite subsets of $I$.
Let $h = \ds \sum_{i \mathop \in I} g_i$ be the summation over finite index $I$.
Let $U$ be an open subset of $G$ such that $h \in U$.
From Set is Subset of Itself:
:$I \in \FF$
Let:
:$J \in \FF : I \subseteq J$
Let $h'= \ds \sum_{j \mathop \in J} g_j$ be the summation ov... | Let $G$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Topological Semigroup|topological semigroup]] with [[Definition:Identity Element|identity]] $0_G$.
Let $\set{i_0, i_1, \ldots, i_n}$ be a [[Definition:Finite Enumeration|finite enumeration]] of a [[Definition:Finite Set|finite set]] $I$.
Le... | Let $\FF$ be the [[Definition:Set|set]] of [[Definition:Finite Set|finite]] [[Definition:Subset|subsets]] of $I$.
Let $h = \ds \sum_{i \mathop \in I} g_i$ be the [[Definition:Summation over Finite Index|summation over finite index]] $I$.
Let $U$ be an [[Definition:Open Set|open subset]] of $G$ such that $h \in U$.
... | Finite Generalized Sum Converges to Summation | https://proofwiki.org/wiki/Finite_Generalized_Sum_Converges_to_Summation | https://proofwiki.org/wiki/Finite_Generalized_Sum_Converges_to_Summation | [
"Generalized Sums"
] | [
"Definition:Commutative Semigroup",
"Definition:Topological Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Enumeration/Finite",
"Definition:Finite Set",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Convergent ... | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Summation over Finite Index",
"Definition:Open Set",
"Set is Subset of Itself",
"Definition:Summation over Finite Index",
"Definition:Set Equality",
"Definition:Convergent Net",
"Definition:Generalized Sum",
"Definition:... |
proofwiki-21136 | Inequality of Height of Proper Ideal | Let $A$ be a commutative ring with unity.
Let $I$ be a proper ideal in $A$.
Then:
:$\map {\operatorname {dim_{Krull} } } {A / I} + \map {\operatorname {ht} } I \le \map {\operatorname {dim_{Krull} } } A$
where:
:$A/I$ is the quotient ring of $A$ by $I$
:$\operatorname {dim_{Krull} }$ denotes the Krull dimension
:$\map ... | Let:
:$n := \map {\operatorname {dim_{Krull} } } {A / I}$
Then there are $\mathfrak q_0, \ldots, \mathfrak q_n \in \Spec {A / I}$ such that:
:$\mathfrak q_0 \subsetneqq \cdots \subsetneqq \mathfrak q_n$
Let $\pi : A \to A / I$ be the quotient epimorphism.
Let:
:$\tilde {\mathfrak q_i} := \pi^{-1} \sqbrk {\mathfrak q_i}... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $I$ be a [[Definition:Proper Ideal of Ring|proper ideal]] in $A$.
Then:
:$\map {\operatorname {dim_{Krull} } } {A / I} + \map {\operatorname {ht} } I \le \map {\operatorname {dim_{Krull} } } A$
where:
:$A/I$ is the [[Definition... | Let:
:$n := \map {\operatorname {dim_{Krull} } } {A / I}$
Then there are $\mathfrak q_0, \ldots, \mathfrak q_n \in \Spec {A / I}$ such that:
:$\mathfrak q_0 \subsetneqq \cdots \subsetneqq \mathfrak q_n$
Let $\pi : A \to A / I$ be the [[Definition:Quotient Epimorphism/Ring|quotient epimorphism]].
Let:
:$\tilde {\math... | Inequality of Height of Proper Ideal | https://proofwiki.org/wiki/Inequality_of_Height_of_Proper_Ideal | https://proofwiki.org/wiki/Inequality_of_Height_of_Proper_Ideal | [
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring/Proper Ideal",
"Definition:Quotient Ring",
"Definition:Krull Dimension of Ring",
"Definition:Height of Proper Ideal"
] | [
"Definition:Quotient Epimorphism/Ring"
] |
proofwiki-21137 | Generalized Hilbert Sequence Space is Metric Space | Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $H^\alpha$ be the generalized Hilbert sequence space of weight $\alpha$.
Then:
:$H^\alpha$ is a metric space. | Recall $H^\alpha$ is the structure $\struct{A, d_2}$ where:
:$A$ is the set of all real-valued functions $x : I \to \R$ such that:
::$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
::$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
:$d_2: A \times A \to \R$ is the real-valued func... | Let $\alpha$ be an [[Definition:Infinite Cardinal|infinite cardinal number]].
Let $I$ be an [[Definition:Indexed Set|indexed set]] of [[Definition:Cardinality|cardinality]] $\alpha$.
Let $H^\alpha$ be the [[Definition:Generalized Hilbert Sequence Space|generalized Hilbert sequence space of weight $\alpha$]].
Then:... | Recall $H^\alpha$ is the [[Definition:Structure|structure]] $\struct{A, d_2}$ where:
:$A$ is the [[Definition:Set|set]] of all [[Definition:Real-Valued Function|real-valued functions]] $x : I \to \R$ such that:
::$(1)\quad \set{i \in I: x_i \ne 0}$ is [[Definition:Countable|countable]]
::$(2)\quad$ the [[Definition:Gen... | Generalized Hilbert Sequence Space is Metric Space | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space | [
"Generalized Hilbert Sequence Spaces",
"Generalized Hilbert Sequence Space is Metric Space"
] | [
"Definition:Infinite Cardinal",
"Definition:Indexing Set/Indexed Set",
"Definition:Cardinality",
"Definition:Generalized Hilbert Sequence Space",
"Definition:Metric Space"
] | [
"Definition:Structure",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Countable Set",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Real-Valued Function",
"Definition:Set",
"Definition:Real-Valued Function"
] |
proofwiki-21138 | Equivalence of Definitions of Associated Prime of Module | Let $A$ be a commutative ring with unity.
Let $M$ be a module over $A$.
Let $\mathfrak p$ be a prime ideal in $A$.
{{TFAE|def = Associated Prime of Module}} | === Definition 1 implies Definition 2 ===
Suppose that $x \in M$ satisfies:
:$\map {\operatorname {Ann}_A} x = \mathfrak p$
Define a submodule of $M$ by:
:$N := \set { a x : a \in A }$
Define a module homomorphism $\phi : A \to N$ by:
:$a \mapsto a x$
Then the kernel of $\phi$ is:
:$\map \ker \phi = \map {\operatorname... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $M$ be a [[Definition:Module over Ring|module over $A$]].
Let $\mathfrak p$ be a [[Definition:Prime Ideal of Ring|prime ideal]] in $A$.
{{TFAE|def = Associated Prime of Module}} | === Definition 1 implies Definition 2 ===
Suppose that $x \in M$ satisfies:
:$\map {\operatorname {Ann}_A} x = \mathfrak p$
Define a [[Definition:Submodule|submodule]] of $M$ by:
:$N := \set { a x : a \in A }$
Define a [[Definition:Module Homomorphism|module homomorphism]] $\phi : A \to N$ by:
:$a \mapsto a x$
Then... | Equivalence of Definitions of Associated Prime of Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Associated_Prime_of_Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Associated_Prime_of_Module | [
"Module Theory",
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Module over Ring",
"Definition:Prime Ideal of Ring"
] | [
"Definition:Submodule",
"Definition:Linear Transformation",
"Definition:Kernel of Linear Transformation",
"First Isomorphism Theorem/Modules",
"Definition:Submodule"
] |
proofwiki-21139 | Summation over Finite Subset is Well-Defined | Let $\struct{G, +}$ be a commutative monoid.
Let $F \subseteq G$ be a finite subset of $G$.
Then the summation $\ds \sum_{g \mathop \in F} g$ is well-defined. | To show that summation over $F$ is well-defined it needs to be shown:
:$(1) \quad \exists$ a finite enumeration of $F$
:$(2) \quad \forall$ finite enumerations $e$ and $d$ of $F : \ds \sum_{i \mathop = 1}^n e_i = \sum_{i \mathop = 1}^n d_i$ | Let $\struct{G, +}$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Monoid|monoid]].
Let $F \subseteq G$ be a [[Definition:Finite Set|finite subset]] of $G$.
Then the [[Definition:Summation over Finite Subset|summation]] $\ds \sum_{g \mathop \in F} g$ is [[Definition:Well-Defined|well-defined]]. | To show that [[Definition:Summation over Finite Subset|summation]] over $F$ is [[Definition:Well-Defined|well-defined]] it needs to be shown:
:$(1) \quad \exists$ a [[Definition:Finite Enumeration|finite enumeration]] of $F$
:$(2) \quad \forall$ [[Definition:Finite Enumeration|finite enumerations]] $e$ and $d$ of $F : ... | Summation over Finite Subset is Well-Defined | https://proofwiki.org/wiki/Summation_over_Finite_Subset_is_Well-Defined | https://proofwiki.org/wiki/Summation_over_Finite_Subset_is_Well-Defined | [
"Summations"
] | [
"Definition:Commutative Semigroup",
"Definition:Monoid",
"Definition:Finite Set",
"Definition:Summation over Finite Subset",
"Definition:Well-Defined"
] | [
"Definition:Summation over Finite Subset",
"Definition:Well-Defined",
"Definition:Enumeration/Finite",
"Definition:Enumeration/Finite",
"Definition:Enumeration/Finite",
"Definition:Summation",
"Definition:Enumeration/Finite"
] |
proofwiki-21140 | Summation over Finite Index is Well-Defined | Let $\struct{G, +}$ be a commutative monoid.
Let $\family{g }_{i \mathop \in I}$ be an indexed subset of $G$ where the indexing set $I$ is finite.
Then the summation $\ds \sum_{i \mathop \in I} g_i$ is well-defined. | To show that summation over $I$ is well-defined it needs to be shown:
:$(1) \quad \exists$ a finite enumeration of $I$
:$(2) \quad \forall$ finite enumerations $e$ and $d$ of $I : \ds \sum_{k \mathop = 1}^n g_{e_k} = \sum_{k \mathop = 1}^n g_{d_k}$ | Let $\struct{G, +}$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Monoid|monoid]].
Let $\family{g }_{i \mathop \in I}$ be an [[Definition:Indexed Set|indexed]] [[Definition:Subset|subset]] of $G$ where the [[Definition:Indexing Set|indexing set]] $I$ is [[Definition:Finite Set|finite]].
Then th... | To show that [[Definition:Summation over Finite Index|summation]] over $I$ is [[Definition:Well-Defined|well-defined]] it needs to be shown:
:$(1) \quad \exists$ a [[Definition:Finite Enumeration|finite enumeration]] of $I$
:$(2) \quad \forall$ [[Definition:Finite Enumeration|finite enumerations]] $e$ and $d$ of $I : \... | Summation over Finite Index is Well-Defined | https://proofwiki.org/wiki/Summation_over_Finite_Index_is_Well-Defined | https://proofwiki.org/wiki/Summation_over_Finite_Index_is_Well-Defined | [
"Summations"
] | [
"Definition:Commutative Semigroup",
"Definition:Monoid",
"Definition:Indexing Set/Indexed Set",
"Definition:Subset",
"Definition:Indexing Set",
"Definition:Finite Set",
"Definition:Summation over Finite Index",
"Definition:Well-Defined"
] | [
"Definition:Summation over Finite Index",
"Definition:Well-Defined",
"Definition:Enumeration/Finite",
"Definition:Enumeration/Finite",
"Definition:Enumeration/Finite",
"Definition:Summation",
"Definition:Enumeration/Finite"
] |
proofwiki-21141 | Maximal Annihilator of Module is Associated Prime | Let $A$ be a commutative ring with unity.
Let $M$ be a module over $A$.
Let $\mathbf p$ be a maximal element of the set:
:$\set { \map {\operatorname {Ann}_A} x : x \in M , x \ne 0 }$
with respect to the subset relation.
Then $\mathfrak p$ is an associated prime of $M$. | {{ProofWanted}}
Category:Commutative Algebra
kgdimxn2fu957dbfshwgkioxttr07qw | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $M$ be a [[Definition:Module over Ring|module over $A$]].
Let $\mathbf p$ be a [[Definition:Maximal Element|maximal element]] of the [[Definition:Set|set]]:
:$\set { \map {\operatorname {Ann}_A} x : x \in M , x \ne 0 }$
with res... | {{ProofWanted}}
[[Category:Commutative Algebra]]
kgdimxn2fu957dbfshwgkioxttr07qw | Maximal Annihilator of Module is Associated Prime | https://proofwiki.org/wiki/Maximal_Annihilator_of_Module_is_Associated_Prime | https://proofwiki.org/wiki/Maximal_Annihilator_of_Module_is_Associated_Prime | [
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Module over Ring",
"Definition:Maximal/Element",
"Definition:Set",
"Definition:Subset Relation",
"Definition:Associated Prime of Module"
] | [
"Category:Commutative Algebra"
] |
proofwiki-21142 | Commutators are Identity iff Group is Abelian | Let $\struct {G, \circ}$ be a group whose identity is $e$.
For $g, h \in G$, let $\sqbrk {g, h}$ denote the commutator of $g$ and $h$.
Then $\struct {G, \circ}$ is abelian {{iff}}:
:$\forall g, h \in G: \sqbrk {g, h} = e$ | === Necessary Condition ===
Let $\struct {G, \circ}$ be such that:
:$\forall g, h \in G: \sqbrk {g, h} = e$
From Commutator is Identity iff Elements Commute:
:$\forall g, h \in G: g \circ h = h \circ g$
Hence $\struct {G, \circ}$ is abelian by definition.
{{qed|lemma}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
For $g, h \in G$, let $\sqbrk {g, h}$ denote the [[Definition:Commutator of Group Elements|commutator]] of $g$ and $h$.
Then $\struct {G, \circ}$ is [[Definition:Abelian Group|abelian]] {{iff}}:
:$\forall ... | === Necessary Condition ===
Let $\struct {G, \circ}$ be such that:
:$\forall g, h \in G: \sqbrk {g, h} = e$
From [[Commutator is Identity iff Elements Commute]]:
:$\forall g, h \in G: g \circ h = h \circ g$
Hence $\struct {G, \circ}$ is [[Definition:Abelian Group|abelian]] by definition.
{{qed|lemma}} | Commutators are Identity iff Group is Abelian | https://proofwiki.org/wiki/Commutators_are_Identity_iff_Group_is_Abelian | https://proofwiki.org/wiki/Commutators_are_Identity_iff_Group_is_Abelian | [
"Group Commutators",
"Abelian Groups"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutator/Group",
"Definition:Abelian Group"
] | [
"Commutator is Identity iff Elements Commute",
"Definition:Abelian Group",
"Definition:Abelian Group",
"Commutator is Identity iff Elements Commute"
] |
proofwiki-21143 | Derived Subgroup is Normal | Let $G$ be a group whose identity is $e$.
Let $\sqbrk {G, G}$ denote the derived subgroup of $G$.
Then $\sqbrk {G, G}$ is a normal subgroup of $G$. | Recall the definition of $\sqbrk {G, G}$:
:$\sqbrk {G, G}$ is the subgroup of $G$ generated by all its commutators.
Recall also the definition of the commutator of $g, h \in G$:
:$\sqbrk {g, h} = g^{-1} h^{-1} g h$
From Derived Subgroup is Subgroup we note that $\sqbrk {G, G}$ is indeed a subgroup of $G$.
Let $g, h \in... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\sqbrk {G, G}$ denote the [[Definition:Derived Subgroup|derived subgroup]] of $G$.
Then $\sqbrk {G, G}$ is a [[Definition:Normal Subgroup|normal subgroup]] of $G$. | Recall the definition of $\sqbrk {G, G}$:
:$\sqbrk {G, G}$ is the [[Definition:Generated Subgroup|subgroup of $G$ generated]] by all its [[Definition:Commutator of Group Elements|commutators]].
Recall also the definition of the [[Definition:Commutator of Group Elements|commutator]] of $g, h \in G$:
:$\sqbrk {g, h} = g... | Derived Subgroup is Normal | https://proofwiki.org/wiki/Derived_Subgroup_is_Normal | https://proofwiki.org/wiki/Derived_Subgroup_is_Normal | [
"Derived Subgroups",
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Derived Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Generated Subgroup",
"Definition:Commutator/Group",
"Definition:Commutator/Group",
"Derived Subgroup is Subgroup",
"Definition:Subgroup",
"Definition:Generated Subgroup",
"Definition:Generated Subgroup",
"Inverse of Group Commutator",
"Definition:Coset",
"Definition:Normal Subgroup"
] |
proofwiki-21144 | Abelianization of Group is Abelian | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $G^{\mathrm {ab} }$ denote the abelianization of $G$.
Then $G^{\mathrm {ab} }$ is an abelian group. | {{Recall|Abelianization of Group}}
{{:Definition:Abelianization of Group}}
From Derived Subgroup is Normal, $\sqbrk {G, G}$ is a normal subgroup of $G$.
Hence the above definition is valid.
By definition of derived subgroup:
:$\forall x, y \in G: \sqbrk {x, y} \in \sqbrk {G, G}$
where $\sqbrk {x, y}$ denotes the commut... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $G^{\mathrm {ab} }$ denote the [[Definition:Abelianization of Group|abelianization]] of $G$.
Then $G^{\mathrm {ab} }$ is an [[Definition:Abelian Group|abelian group]]. | {{Recall|Abelianization of Group}}
{{:Definition:Abelianization of Group}}
From [[Derived Subgroup is Normal]], $\sqbrk {G, G}$ is a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Hence the above definition is valid.
By definition of [[Definition:Derived Subgroup|derived subgroup]]:
:$\forall x, y \in G: ... | Abelianization of Group is Abelian | https://proofwiki.org/wiki/Abelianization_of_Group_is_Abelian | https://proofwiki.org/wiki/Abelianization_of_Group_is_Abelian | [
"Abelianizations of Groups",
"Abelian Groups"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Abelianization of Group",
"Definition:Abelian Group"
] | [
"Derived Subgroup is Normal",
"Definition:Normal Subgroup",
"Definition:Derived Subgroup",
"Definition:Commutator/Group",
"Quotient Group is Abelian iff All Commutators in Divisor",
"Definition:Abelian Group"
] |
proofwiki-21145 | Matrix Multiplication on Diagonal Matrices is Commutative | Let $\mathbf A$ and $\mathbf B$ be diagonal matrices.
Then:
:$\mathbf A \mathbf B = \mathbf B \mathbf A$
where $\mathbf A \mathbf B$ denotes (conventional) matrix product. | {{begin-eqn}}
{{eqn | l = \mathbf A
| o = :=
| r = \sqbrk {a_{ij} }_n
| c =
}}
{{eqn | l = \mathbf B
| o = :=
| r = \sqbrk {b_{ij} }_n
| c =
}}
{{end-eqn}}
Note that the orders of $\mathbf A$ and $\mathbf B$ must be equal in order for matrix product to be defined.
Then we have:
{{b... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Diagonal Matrix|diagonal matrices]].
Then:
:$\mathbf A \mathbf B = \mathbf B \mathbf A$
where $\mathbf A \mathbf B$ denotes [[Definition:Matrix Product (Conventional)|(conventional) matrix product]]. | {{begin-eqn}}
{{eqn | l = \mathbf A
| o = :=
| r = \sqbrk {a_{ij} }_n
| c =
}}
{{eqn | l = \mathbf B
| o = :=
| r = \sqbrk {b_{ij} }_n
| c =
}}
{{end-eqn}}
Note that the [[Definition:Order of Square Matrix|orders]] of $\mathbf A$ and $\mathbf B$ must be equal in order for [[Defini... | Matrix Multiplication on Diagonal Matrices is Commutative | https://proofwiki.org/wiki/Matrix_Multiplication_on_Diagonal_Matrices_is_Commutative | https://proofwiki.org/wiki/Matrix_Multiplication_on_Diagonal_Matrices_is_Commutative | [
"Diagonal Matrices",
"Conventional Matrix Multiplication",
"Commutativity"
] | [
"Definition:Diagonal Matrix",
"Definition:Matrix Product (Conventional)"
] | [
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix Product (Conventional)",
"Definition:Matrix/Indices",
"Product of Diagonal Matrices is Diagonal",
"Commutative Law of Multiplication",
"Product of Diagonal Matrices is Diagonal"
] |
proofwiki-21146 | Group Commutators are Commuting Elements | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $\sqbrk {g, h}$ denote the commutator of $g$ and $h$.
Then $\sqbrk {g, h}$ commutes with $\sqbrk {h, g}$, in the sense that:
:$\sqbrk {g, h} \circ \sqbrk {h, g} = \sqbrk {h, g} \circ \sqbrk {g, h}$ | From Inverse of Group Commutator:
:$\forall g, h \in G: \sqbrk {g, h} = \sqbrk {h, g}^{-1}$
The result follows from Group Element Commutes with Inverse.
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\sqbrk {g, h}$ denote the [[Definition:Commutator of Group Elements|commutator]] of $g$ and $h$.
Then $\sqbrk {g, h}$ [[Definition:Commuting Elements|commutes]] with $\sqbrk {h, g}$, in the sense that... | From [[Inverse of Group Commutator]]:
:$\forall g, h \in G: \sqbrk {g, h} = \sqbrk {h, g}^{-1}$
The result follows from [[Group Element Commutes with Inverse]].
{{qed}} | Group Commutators are Commuting Elements | https://proofwiki.org/wiki/Group_Commutators_are_Commuting_Elements | https://proofwiki.org/wiki/Group_Commutators_are_Commuting_Elements | [
"Group Commutators",
"Commutativity"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutator/Group",
"Definition:Commutative/Elements"
] | [
"Inverse of Group Commutator",
"Group Element Commutes with Inverse"
] |
proofwiki-21147 | Inverse of Group Commutator | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $\sqbrk {g, h}$ denote the commutator of $g$ and $h$.
Then $\sqbrk {g, h}$ is the inverse of $\sqbrk {h, g}$:
:$\sqbrk {g, h} = \sqbrk {h, g}^{-1}$ | {{begin-eqn}}
{{eqn | l = \sqbrk {g, h} \circ \sqbrk {h, g}
| r = \paren {g^{-1} \circ h^{-1} \circ g \circ h} \circ \paren {h^{-1} \circ g^{-1} \circ h \circ g}
| c = {{Defof|Commutator of Group Elements}}
}}
{{eqn | r = \paren {g^{-1} \circ h^{-1} \circ g \circ h} \circ \paren {g^{-1} \circ h^{-1} \circ g... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\sqbrk {g, h}$ denote the [[Definition:Commutator of Group Elements|commutator]] of $g$ and $h$.
Then $\sqbrk {g, h}$ is the [[Definition:Inverse Element|inverse]] of $\sqbrk {h, g}$:
:$\sqbrk {g, h} ... | {{begin-eqn}}
{{eqn | l = \sqbrk {g, h} \circ \sqbrk {h, g}
| r = \paren {g^{-1} \circ h^{-1} \circ g \circ h} \circ \paren {h^{-1} \circ g^{-1} \circ h \circ g}
| c = {{Defof|Commutator of Group Elements}}
}}
{{eqn | r = \paren {g^{-1} \circ h^{-1} \circ g \circ h} \circ \paren {g^{-1} \circ h^{-1} \circ g... | Inverse of Group Commutator | https://proofwiki.org/wiki/Inverse_of_Group_Commutator | https://proofwiki.org/wiki/Inverse_of_Group_Commutator | [
"Group Commutators",
"Inverse Elements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutator/Group",
"Definition:Inverse (Abstract Algebra)/Inverse"
] | [
"Inverse of Group Product/General Result",
"Inverse of Group Product/General Result",
"Category:Group Commutators",
"Category:Inverse Elements"
] |
proofwiki-21148 | Commutator of Group Element with Identity is Identity | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $\sqbrk {g, h}$ denote the commutator of $g$ and $h$.
Then:
:$\sqbrk {g, e} = e = \sqbrk {e, g}$ | {{begin-eqn}}
{{eqn | q = \forall g \in G
| l = \sqbrk {g, e}
| r = g^{-1} \circ e^{-1} \circ g \circ e
| c = {{Defof|Commutator of Group Elements}}
}}
{{eqn | r = g^{-1} \circ e \circ g \circ e
| c = Identity is Self-Inverse
}}
{{eqn | r = g^{-1} \circ g
| c = {{Defof|Identity Element}}
}... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\sqbrk {g, h}$ denote the [[Definition:Commutator of Group Elements|commutator]] of $g$ and $h$.
Then:
:$\sqbrk {g, e} = e = \sqbrk {e, g}$ | {{begin-eqn}}
{{eqn | q = \forall g \in G
| l = \sqbrk {g, e}
| r = g^{-1} \circ e^{-1} \circ g \circ e
| c = {{Defof|Commutator of Group Elements}}
}}
{{eqn | r = g^{-1} \circ e \circ g \circ e
| c = [[Identity is Self-Inverse]]
}}
{{eqn | r = g^{-1} \circ g
| c = {{Defof|Identity Element... | Commutator of Group Element with Identity is Identity | https://proofwiki.org/wiki/Commutator_of_Group_Element_with_Identity_is_Identity | https://proofwiki.org/wiki/Commutator_of_Group_Element_with_Identity_is_Identity | [
"Group Commutators",
"Inverse Elements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutator/Group"
] | [
"Inverse of Identity Element is Itself",
"Inverse of Identity Element is Itself",
"Category:Group Commutators",
"Category:Inverse Elements"
] |
proofwiki-21149 | Derived Subgroup is Subgroup | Let $G$ be a group whose identity is $e$.
Let $\sqbrk {G, G}$ denote the derived subgroup of $G$.
Then $\sqbrk {G, G}$ is indeed a subgroup of $G$. | Recall the definition of $\sqbrk {G, G}$:
:$\sqbrk {G, G}$ is the subgroup of $G$ generated by all its commutators.
Recall also the definition of the commutator of $g, h \in G$:
:$\sqbrk {g, h} = g^{-1} h^{-1} g h$
We note that from Commutator of Group Element with Identity is Identity:
:$\sqbrk {e, e} = e$
and so $e \... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\sqbrk {G, G}$ denote the [[Definition:Derived Subgroup|derived subgroup]] of $G$.
Then $\sqbrk {G, G}$ is indeed a [[Definition:Subgroup|subgroup]] of $G$. | Recall the definition of $\sqbrk {G, G}$:
:$\sqbrk {G, G}$ is the [[Definition:Generated Subgroup|subgroup of $G$ generated]] by all its [[Definition:Commutator of Group Elements|commutators]].
Recall also the definition of the [[Definition:Commutator of Group Elements|commutator]] of $g, h \in G$:
:$\sqbrk {g, h} = g... | Derived Subgroup is Subgroup | https://proofwiki.org/wiki/Derived_Subgroup_is_Subgroup | https://proofwiki.org/wiki/Derived_Subgroup_is_Subgroup | [
"Derived Subgroups"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Derived Subgroup",
"Definition:Subgroup"
] | [
"Definition:Generated Subgroup",
"Definition:Commutator/Group",
"Definition:Commutator/Group",
"Commutator of Group Element with Identity is Identity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Generated Subgroup",
"Two-Step Subgroup Test",
"Definition:Subgroup",
"Category:Derived... |
proofwiki-21150 | Prime Sum of Squares of 3 Prime Numbers | Let $a$, $b$ and $c$ be prime numbers with the property that:
:$a^2 + b^2 + c^2 = p$
where $p$ is a prime number.
Then either $1$ or $2$ of $a$, $b$ and $c$ is equal to $3$. | First we note that sets of $3$ prime numbers with this property are plentiful.
For example, these are all such sets for $a, b, c < 20$:
{{begin-eqn}}
{{eqn | l = 2^2 + 2^2 + 3^2
| r = 17
| c =
}}
{{eqn | l = 3^2 + 3^2 + 5^2
| r = 43
| c =
}}
{{eqn | l = 3^2 + 3^2 + 7^2
| r = 67
| c... | Let $a$, $b$ and $c$ be [[Definition:Prime Number|prime numbers]] with the property that:
:$a^2 + b^2 + c^2 = p$
where $p$ is a [[Definition:Prime Number|prime number]].
Then either $1$ or $2$ of $a$, $b$ and $c$ is equal to $3$. | First we note that [[Definition:Set|sets]] of $3$ [[Definition:Prime Number|prime numbers]] with this property are plentiful.
For example, these are all such [[Definition:Set|sets]] for $a, b, c < 20$:
{{begin-eqn}}
{{eqn | l = 2^2 + 2^2 + 3^2
| r = 17
| c =
}}
{{eqn | l = 3^2 + 3^2 + 5^2
| r = 43
... | Prime Sum of Squares of 3 Prime Numbers | https://proofwiki.org/wiki/Prime_Sum_of_Squares_of_3_Prime_Numbers | https://proofwiki.org/wiki/Prime_Sum_of_Squares_of_3_Prime_Numbers | [
"Prime Numbers",
"3"
] | [
"Definition:Prime Number",
"Definition:Prime Number"
] | [
"Definition:Set",
"Definition:Prime Number",
"Definition:Set",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Square Modulo 3",
"Modulo Addition is Well-Defined",
"Definition:Divisor (Algebra)/I... |
proofwiki-21151 | Topological Space is Connected iff any Proper Non-Empty Subset has Non-Empty Boundary | Let $\struct {X, \tau}$ be a topological space.
Then $\struct {X, \tau}$ is connected {{iff}} for each proper non-empty subset $S \subseteq X$, we have $\partial S \ne \O$. | From Connected iff no Proper Clopen Sets, we have that:
:$\struct {X, \tau}$ is connected {{iff}} there exists no proper non-empty clopen set $S \subseteq X$.
From Set is Clopen iff Boundary is Empty, we have that:
:$S \subseteq X$ is clopen {{iff}} $\partial S = \O$.
Hence we have:
:$\struct {X, \tau}$ is connected {... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Then $\struct {X, \tau}$ is [[Definition:Connected Topological Space|connected]] {{iff}} for each [[Definition:Proper Subset|proper]] [[Definition:Non-Empty Set|non-empty subset]] $S \subseteq X$, we have $\partial S \ne \O$. | From [[Connected iff no Proper Clopen Sets]], we have that:
:$\struct {X, \tau}$ is [[Definition:Connected Topological Space|connected]] {{iff}} there exists no [[Definition:Proper Subset|proper]] [[Definition:Non-Empty Set|non-empty]] [[Definition:Clopen Set|clopen set]] $S \subseteq X$.
From [[Set is Clopen iff Boun... | Topological Space is Connected iff any Proper Non-Empty Subset has Non-Empty Boundary | https://proofwiki.org/wiki/Topological_Space_is_Connected_iff_any_Proper_Non-Empty_Subset_has_Non-Empty_Boundary | https://proofwiki.org/wiki/Topological_Space_is_Connected_iff_any_Proper_Non-Empty_Subset_has_Non-Empty_Boundary | [
"Connected Topological Spaces",
"Set Boundaries"
] | [
"Definition:Topological Space",
"Definition:Connected Topological Space",
"Definition:Proper Subset",
"Definition:Non-Empty Set"
] | [
"Connected iff no Proper Clopen Sets",
"Definition:Connected Topological Space",
"Definition:Proper Subset",
"Definition:Non-Empty Set",
"Definition:Clopen Set",
"Set is Clopen iff Boundary is Empty",
"Definition:Clopen Set",
"Definition:Connected Topological Space",
"Definition:Proper Subset",
"D... |
proofwiki-21152 | Finite Direct Sum of Noetherian Module is Noetherian | Let $A$ be a commutative ring with unity.
Let $n \in \N_{>0}$.
Let $M_1, \ldots, M_n$ be $A$-Noetherian modules.
Then the direct sum:
:$\ds \bigoplus_{i \mathop = 1}^n M_i$
is an $A$-Noetherian module. | By Direct Sum of Modules is Module, it is an $A$-module.
Thus we only need to show that it is Noetherian.
We prove it by induction.
For $n = 1$, there is nothing to prove.
Let $n \ge 2$.
Suppose that the claim is true for $n-1$, i.e.:
:$\ds \bigoplus_{i \mathop = 1}^{n-1} M_i$
is Noetherian.
Then, consider the short ex... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $n \in \N_{>0}$.
Let $M_1, \ldots, M_n$ be [[Definition:Noetherian Module|$A$-Noetherian modules]].
Then the [[Definition:Module Direct Sum|direct sum]]:
:$\ds \bigoplus_{i \mathop = 1}^n M_i$
is an [[Definition:Noetherian Mod... | By [[Direct Sum of Modules is Module]], it is an [[Definition:Module over Ring|$A$-module]].
Thus we only need to show that it is [[Definition:Noetherian Module|Noetherian]].
We prove it by [[Principle of Mathematical Induction/One-Based|induction]].
For $n = 1$, there is nothing to prove.
Let $n \ge 2$.
Suppose... | Finite Direct Sum of Noetherian Module is Noetherian | https://proofwiki.org/wiki/Finite_Direct_Sum_of_Noetherian_Module_is_Noetherian | https://proofwiki.org/wiki/Finite_Direct_Sum_of_Noetherian_Module_is_Noetherian | [
"Noetherian Modules"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Noetherian Module",
"Definition:Direct Sum of Modules",
"Definition:Noetherian Module"
] | [
"Direct Sum of Modules is Module",
"Definition:Module over Ring",
"Definition:Noetherian Module",
"Principle of Mathematical Induction/One-Based",
"Definition:Noetherian Module",
"Definition:Short Exact Sequence of Modules",
"Definition:Noetherian Module",
"Short Exact Sequence Condition of Noetherian... |
proofwiki-21153 | Complement of Closed Disk in Complex Plane is Path-Connected | Let $R > 0$.
Let:
:$S_R = \set {z \in \C : \cmod z > R}$
Then $S_R$ is path-connected. | {{ProofWanted|Someone else can take this, we should probably have a diagram}}
Category:Path-Connected Sets
942u636v6iw8dgrnuolq9kbar324so6 | Let $R > 0$.
Let:
:$S_R = \set {z \in \C : \cmod z > R}$
Then $S_R$ is [[Definition:Path-Connected Set|path-connected]]. | {{ProofWanted|Someone else can take this, we should probably have a diagram}}
[[Category:Path-Connected Sets]]
942u636v6iw8dgrnuolq9kbar324so6 | Complement of Closed Disk in Complex Plane is Path-Connected | https://proofwiki.org/wiki/Complement_of_Closed_Disk_in_Complex_Plane_is_Path-Connected | https://proofwiki.org/wiki/Complement_of_Closed_Disk_in_Complex_Plane_is_Path-Connected | [
"Path-Connected Sets"
] | [
"Definition:Path-Connected/Set"
] | [
"Category:Path-Connected Sets"
] |
proofwiki-21154 | Complement of Bounded Set in Complex Plane has at most One Unbounded Component | Let $S \subseteq \C$ be bounded.
Then $\C \setminus S$ has at most one unbounded component. | If $\C \setminus S$ has no unbounded components, we are done.
Suppose that $\C \setminus S$ has at least one unbounded component.
We must show that it has at most one.
Since $S$ is bounded, there exists $R > 0$ such that:
:$\cmod z \le R$ for all $z \in S$.
Let $C$ be an unbounded component of $\C \setminus S$.
Since ... | Let $S \subseteq \C$ be [[Definition:Bounded Subset of Complex Plane|bounded]].
Then $\C \setminus S$ has at most one [[Definition:Unbounded Subset of Complex Plane|unbounded]] [[Definition:Component (Topology)|component]]. | If $\C \setminus S$ has no [[Definition:Unbounded Subset of Complex Plane|unbounded]] [[Definition:Component (Topology)|components]], we are done.
Suppose that $\C \setminus S$ has at least one [[Definition:Unbounded Subset of Complex Plane|unbounded]] [[Definition:Component (Topology)|component]].
We must show that ... | Complement of Bounded Set in Complex Plane has at most One Unbounded Component | https://proofwiki.org/wiki/Complement_of_Bounded_Set_in_Complex_Plane_has_at_most_One_Unbounded_Component | https://proofwiki.org/wiki/Complement_of_Bounded_Set_in_Complex_Plane_has_at_most_One_Unbounded_Component | [
"Components (Topology)"
] | [
"Definition:Bounded Metric Space/Complex",
"Definition:Bounded Metric Space/Complex/Unbounded",
"Definition:Component (Topology)"
] | [
"Definition:Bounded Metric Space/Complex/Unbounded",
"Definition:Component (Topology)",
"Definition:Bounded Metric Space/Complex/Unbounded",
"Definition:Component (Topology)",
"Definition:Bounded Metric Space/Complex",
"Definition:Bounded Metric Space/Complex/Unbounded",
"Definition:Component (Topology)... |
proofwiki-21155 | Unitization of Algebra over Field is Unital Algebra over Field | Let $K$ be a field.
Let $A$ be an algebra over $K$.
Let $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }_K$ be the unitization of $A$.
Then $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }_K$ is a unital algebra over $K$. | From Direct Product of Vector Spaces is Vector Space, $\struct {A_+, +_{A_+}, \cdot_{A_+} }_K = \struct {A \times K, +_{A \times K}, \cdot_{A \times K} }_K$ is a vector space over $K$.
We show that $\circ_{A_+} : A_+ \times A_+ \to A_+$ is a bilinear mapping.
Let $\tuple {u, \alpha} \in A_+$, $\tuple {v, \beta} \in A_+... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra over $K$]].
Let $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }_K$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Then $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }... | From [[Direct Product of Vector Spaces is Vector Space]], $\struct {A_+, +_{A_+}, \cdot_{A_+} }_K = \struct {A \times K, +_{A \times K}, \cdot_{A \times K} }_K$ is a [[Definition:Vector Space|vector space]] over $K$.
We show that $\circ_{A_+} : A_+ \times A_+ \to A_+$ is a [[Definition:Bilinear Mapping|bilinear mappin... | Unitization of Algebra over Field is Unital Algebra over Field | https://proofwiki.org/wiki/Unitization_of_Algebra_over_Field_is_Unital_Algebra_over_Field | https://proofwiki.org/wiki/Unitization_of_Algebra_over_Field_is_Unital_Algebra_over_Field | [
"Algebras over Fields",
"Unitizations of Algebras over Fields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Unitization of Algebra over Field",
"Definition:Unital Algebra",
"Definition:Algebra over Field"
] | [
"Direct Product of Vector Spaces is Vector Space",
"Definition:Vector Space",
"Definition:Bilinear Mapping",
"Definition:Bilinear Mapping",
"Definition:Algebra over Field",
"Definition:Unital Algebra",
"Definition:Unital Algebra",
"Definition:Algebra over Field",
"Category:Algebras over Fields",
"... |
proofwiki-21156 | Algebra over Field Embeds into Unitization as Ideal | Let $K$ be a field.
Let $A$ be an algebra over $K$.
Let $A_+$ be the unitization of $A$.
Let:
:$A_0 = \set {\tuple {x, 0_K} : x \in A} \subseteq A_+$.
Then $A_0$ is an ideal in $A_+$. | This follows from Ideal of Algebra over Field Embeds into Unitization as Ideal, since $A$ is an ideal of itself.
{{qed}}
Category:Unitizations of Algebras over Fields
rk50tujy4ssp9br568eau9doxpkyu32 | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra over $K$]].
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Let:
:$A_0 = \set {\tuple {x, 0_K} : x \in A} \subseteq A_+$.
Then $A_0$ is an [[Definition:Ideal of A... | This follows from [[Ideal of Algebra over Field Embeds into Unitization as Ideal]], since $A$ is an [[Definition:Ideal of Algebra|ideal]] of itself.
{{qed}}
[[Category:Unitizations of Algebras over Fields]]
rk50tujy4ssp9br568eau9doxpkyu32 | Algebra over Field Embeds into Unitization as Ideal | https://proofwiki.org/wiki/Algebra_over_Field_Embeds_into_Unitization_as_Ideal | https://proofwiki.org/wiki/Algebra_over_Field_Embeds_into_Unitization_as_Ideal | [
"Unitizations of Algebras over Fields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Unitization of Algebra over Field",
"Definition:Ideal of Algebra"
] | [
"Ideal of Algebra over Field Embeds into Unitization as Ideal",
"Definition:Ideal of Algebra",
"Category:Unitizations of Algebras over Fields"
] |
proofwiki-21157 | Algebra over Field Embeds into Unitization as Vector Subspace | Let $K$ be a field.
Let $A$ be an algebra over $K$ that is not unital.
Let $A_+$ be the unitization of $A$.
Let:
:$A_0 = \set {\tuple {x, 0_K} : x \in A} \subseteq A_+$.
Then $A_0$ is a vector subspace of $A_+$. | This follows from Vector Subspace of Algebra over Field Embeds into Unitization as Vector Subspace.
{{qed}}
Category:Unitizations of Algebras over Fields
442yj1xu778eqhj0v737phnaoxaucby | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra over $K$]] that is not [[Definition:Unital Algebra|unital]].
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Let:
:$A_0 = \set {\tuple {x, 0_K} : x \in A} \subseteq... | This follows from [[Vector Subspace of Algebra over Field Embeds into Unitization as Vector Subspace]].
{{qed}}
[[Category:Unitizations of Algebras over Fields]]
442yj1xu778eqhj0v737phnaoxaucby | Algebra over Field Embeds into Unitization as Vector Subspace | https://proofwiki.org/wiki/Algebra_over_Field_Embeds_into_Unitization_as_Vector_Subspace | https://proofwiki.org/wiki/Algebra_over_Field_Embeds_into_Unitization_as_Vector_Subspace | [
"Unitizations of Algebras over Fields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Unital Algebra",
"Definition:Unitization of Algebra over Field",
"Definition:Vector Subspace"
] | [
"Vector Subspace of Algebra over Field Embeds into Unitization as Vector Subspace",
"Category:Unitizations of Algebras over Fields"
] |
proofwiki-21158 | Unitization of Normed Algebra is Unital Normed Algebra | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra that is not unital as an algebra.
Let $A_+$ be the unitization of $A$.
Define $\norm {\, \cdot \,}_{A_+} : A_+ \to \hointr 0 \infty$ by:
:$\norm {\tuple {x, \lambda} }_{A_+} = \norm x + \cmod \lambda$
for each $\tuple {x, \lambd... | From Unitization of Algebra over Field is Unital Algebra over Field, $A_+$ is a unital algebra.
We show that $\norm {\, \cdot \,}_{A_+}$ is an algebra norm and that $\norm { {\mathbf 1}_{A_+} } = 1$. | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] that is not [[Definition:Unital Algebra|unital as an algebra]].
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Define $\norm {\, \cdot \,}_{A_+} : A_+ \to \h... | From [[Unitization of Algebra over Field is Unital Algebra over Field]], $A_+$ is a [[Definition:Unital Algebra|unital algebra]].
We show that $\norm {\, \cdot \,}_{A_+}$ is an [[Definition:Norm on Algebra|algebra norm]] and that $\norm { {\mathbf 1}_{A_+} } = 1$. | Unitization of Normed Algebra is Unital Normed Algebra | https://proofwiki.org/wiki/Unitization_of_Normed_Algebra_is_Unital_Normed_Algebra | https://proofwiki.org/wiki/Unitization_of_Normed_Algebra_is_Unital_Normed_Algebra | [
"Unitizations of Normed Algebras"
] | [
"Definition:Normed Algebra",
"Definition:Unital Algebra",
"Definition:Unitization of Algebra over Field",
"Definition:Unital Normed Algebra"
] | [
"Unitization of Algebra over Field is Unital Algebra over Field",
"Definition:Unital Algebra",
"Definition:Norm/Algebra",
"Definition:Norm/Algebra"
] |
proofwiki-21159 | Finitely Satisfiable Set of Sentences has Maximal Finitely Satisfiable Extension | Let $\LL$ be a language of predicate logic.
Let $\FF$ be a finitely satisfiable set of $\LL$-sentences.
Then there exists a finitely satisfiable set of $\LL$-sentences $\FF' \supseteq \FF$ such that:
:For all $\LL$-sentences $\phi$, either $\phi \in \FF'$ or $\sqbrk {\neg \phi} \in \FF'$ | Let $S$ be the set of all $\LL$-sentences.
For every finite subset $P \subseteq S$, define:
:$M_P = \set {t \in \Bbb B^P : \exists \MM : \paren {\MM \models_{\mathrm {PL}} \FF \cap P} \land \paren {\forall \phi \in P : \map t \phi = T \iff \MM \models_{\mathrm {PL}} \phi}}$
Define:
:$M = \bigcup_{P \mathop \in I} M_K$
... | Let $\LL$ be a [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $\FF$ be a [[Definition:Finitely Satisfiable|finitely satisfiable]] [[Definition:Set|set]] of $\LL$-[[Definition:Sentence|sentences]].
Then there exists a [[Definition:Finitely Satisfiable|finitely satisfiable]] [[Definition:Se... | Let $S$ be the [[Definition:Set|set]] of all $\LL$-[[Definition:Sentence|sentences]].
For every [[Definition:Finite Subset|finite subset]] $P \subseteq S$, define:
:$M_P = \set {t \in \Bbb B^P : \exists \MM : \paren {\MM \models_{\mathrm {PL}} \FF \cap P} \land \paren {\forall \phi \in P : \map t \phi = T \iff \MM \mo... | Finitely Satisfiable Set of Sentences has Maximal Finitely Satisfiable Extension | https://proofwiki.org/wiki/Finitely_Satisfiable_Set_of_Sentences_has_Maximal_Finitely_Satisfiable_Extension | https://proofwiki.org/wiki/Finitely_Satisfiable_Set_of_Sentences_has_Maximal_Finitely_Satisfiable_Extension | [
"Model Theory for Predicate Logic"
] | [
"Definition:Language of Predicate Logic",
"Definition:Finitely Satisfiable",
"Definition:Set",
"Definition:Classes of WFFs/Sentence",
"Definition:Finitely Satisfiable",
"Definition:Set",
"Definition:Classes of WFFs/Sentence",
"Definition:Classes of WFFs/Sentence"
] | [
"Definition:Set",
"Definition:Classes of WFFs/Sentence",
"Definition:Finite Subset",
"Definition:Set",
"Definition:Finite Subset",
"Definition:Binary Mess",
"Axiom:Binary Mess Axioms",
"Intersection is Subset",
"Subset of Finite Set is Finite",
"Definition:Finite Subset",
"Definition:Finitely Sa... |
proofwiki-21160 | Equation for Perpendicular Bisector of Two Points in Complex Plane/Standard Form | $L$ can be expressed by the equation:
:$\map \Re {z_2 - z_1} x + \map \Im {z_2 - z_1} y = \dfrac {\cmod {z_2}^2 - \cmod {z_1}^2} 2$ | Let $z_1$ and $z_2$ be represented by the points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the complex plane.
By Equation for Perpendicular Bisector of Two Points, the equation of their perpendicular bisector can be expressed as:
{{begin-eqn}}
{{eqn | l = y - \frac {y_1 + y_2} 2
| r = \f... | $L$ can be expressed by the equation:
:$\map \Re {z_2 - z_1} x + \map \Im {z_2 - z_1} y = \dfrac {\cmod {z_2}^2 - \cmod {z_1}^2} 2$ | Let $z_1$ and $z_2$ be represented by the [[Definition:Point|points]] $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the [[Definition:Complex Plane|complex plane]].
By [[Equation for Perpendicular Bisector of Two Points]], the equation of their [[Definition:Perpendicular Bisector|perpendicular bis... | Equation for Perpendicular Bisector of Two Points in Complex Plane/Standard Form | https://proofwiki.org/wiki/Equation_for_Perpendicular_Bisector_of_Two_Points_in_Complex_Plane/Standard_Form | https://proofwiki.org/wiki/Equation_for_Perpendicular_Bisector_of_Two_Points_in_Complex_Plane/Standard_Form | [
"Equation for Perpendicular Bisector of Two Points in Complex Plane"
] | [] | [
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Equation for Perpendicular Bisector of Two Points",
"Definition:Perpendicular Bisector",
"Difference of Two Squares",
"Category:Equation for Perpendicular Bisector of Two Points in Complex Plane"
] |
proofwiki-21161 | Extend Theory to Satisfy Witness Property | Let $\LL$ be a language of predicate logic.
Let $T$ be a set of $\LL$-sentences.
Then there exists a language $\LL^*$ and a set of $\LL^*$-sentences $T^*$ satisfying the following properties:
* $T^*$ is finitely satisfiable {{iff}} $T$ is finitely satisfiable.
* If $T^*$ is satisfiable, then $T$ is also.
* For every $\... | {{Proofread}}
We will recursively define a sequence of languages.
Define:
:$\LL_0 := \LL$
Let the set of predicates in $\LL$ be $\PP$.
Let the set of functions in $\LL$ be $\FF = \set {\KK, \FF_1, \FF_2, \dotsc}$.
For every $i \in \N$, define:
:$\Phi_i$ as the set of $\LL_i$-WFFs of $1$ free variable.
:$\KK_i$ as the s... | Let $\LL$ be a [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $T$ be a [[Definition:Set|set]] of $\LL$-[[Definition:Sentence|sentences]].
Then there exists a [[Definition:Language of Predicate Logic|language]] $\LL^*$ and a [[Definition:Set|set]] of $\LL^*$-[[Definition:Sentence|sentence... | {{Proofread}}
We will [[Principle of Recursive Definition|recursively define]] a [[Definition:Infinite Sequence|sequence]] of [[Definition:Language of Predicate Logic|languages]].
Define:
:$\LL_0 := \LL$
Let the [[Definition:Set|set]] of [[Definition:Predicate Symbol|predicates]] in $\LL$ be $\PP$.
Let the [[Defini... | Extend Theory to Satisfy Witness Property | https://proofwiki.org/wiki/Extend_Theory_to_Satisfy_Witness_Property | https://proofwiki.org/wiki/Extend_Theory_to_Satisfy_Witness_Property | [
"Model Theory for Predicate Logic"
] | [
"Definition:Language of Predicate Logic",
"Definition:Set",
"Definition:Classes of WFFs/Sentence",
"Definition:Language of Predicate Logic",
"Definition:Set",
"Definition:Classes of WFFs/Sentence",
"Definition:Finitely Satisfiable",
"Definition:Finitely Satisfiable",
"Definition:Satisfiable",
"Def... | [
"Principle of Recursive Definition",
"Definition:Sequence/Infinite Sequence",
"Definition:Language of Predicate Logic",
"Definition:Set",
"Definition:Predicate Symbol",
"Definition:Set",
"Definition:Function Symbol",
"Definition:Set",
"Definition:Language of Predicate Logic/Formal Grammar",
"Defin... |
proofwiki-21162 | Maximal Finitely Satisfiable Theory with Witness Property is Satisfiable | {{Tidy|If we are using the $\mathrm{PL}$ semantics, then only sentences exist in a theory. We can then show that $T$ is a theory in Finitely Satisfiable Set of Sentences has Maximal Finitely Satisfiable Extension, and make some of this reasoning a little simpler.}}
Let $\LL$ be a language of predicate logic.
Let $T$ be... | Let $M$ be the set of all $\LL$-terms that contain no variables.
For each $n$-ary function $f$ in $\LL$, define:
:$\map {F_f} {t_1, \dotsc, t_n} = \sqbrk {\map f {t_1, \dotsc, t_n}}$
For each $n$-ary predicate $p$ in $\LL$, define:
:$\map {P_p} {t_1, \dotsc, t_n} = \begin{cases}
\top & : \sqbrk {\map p {t_1, \dotsc, t_... | {{Tidy|If we are using the $\mathrm{PL}$ semantics, then only sentences exist in a theory. We can then show that $T$ is a theory in [[Finitely Satisfiable Set of Sentences has Maximal Finitely Satisfiable Extension]], and make some of this reasoning a little simpler.}}
Let $\LL$ be a [[Definition:Language of Predicate... | Let $M$ be the [[Definition:Set|set]] of all $\LL$-[[Definition:Term (Predicate Logic)|terms]] that contain no [[Definition:Variable|variables]].
For each $n$-[[Definition:Arity|ary]] [[Definition:Function Symbol|function]] $f$ in $\LL$, define:
:$\map {F_f} {t_1, \dotsc, t_n} = \sqbrk {\map f {t_1, \dotsc, t_n}}$
Fo... | Maximal Finitely Satisfiable Theory with Witness Property is Satisfiable | https://proofwiki.org/wiki/Maximal_Finitely_Satisfiable_Theory_with_Witness_Property_is_Satisfiable | https://proofwiki.org/wiki/Maximal_Finitely_Satisfiable_Theory_with_Witness_Property_is_Satisfiable | [
"Model Theory for Predicate Logic"
] | [
"Finitely Satisfiable Set of Sentences has Maximal Finitely Satisfiable Extension",
"Definition:Language of Predicate Logic",
"Definition:Set",
"Definition:Classes of WFFs/Sentence",
"Definition:Finitely Satisfiable",
"Definition:Classes of WFFs/Sentence",
"Definition:Witness Property"
] | [
"Definition:Set",
"Definition:Language of Predicate Logic/Formal Grammar/Term",
"Definition:Variable",
"Definition:Operation/Arity",
"Definition:Function Symbol",
"Definition:Operation/Arity",
"Definition:Predicate Symbol",
"Definition:Classes of WFFs/Sentence",
"Definition:Language of Predicate Log... |
proofwiki-21163 | Spectrum of Idempotent in Algebra over Complex Numbers | Let $A$ be an algebra over $\C$.
Let $p \in A$ be idempotent, that is:
:$p^2 = p$
Let $\map {\sigma_A} p$ be the spectrum of $p$ in $A$.
Then:
:$\map {\sigma_A} p \subseteq \set {0, 1}$ | Suppose first that $A$ is not unital and let $p \in A$ be idempotent.
Let $A_+$ be the unitization of $A$.
Then we have:
{{begin-eqn}}
{{eqn | l = \tuple {p, 0} \tuple {p, 0}
| r = \tuple {p^2 + 0p + 0 p, 0}
}}
{{eqn | r = \tuple {p, 0}
}}
{{end-eqn}}
So $\tuple {p, 0}$ is an idempotent in $A_+$.
Since, by definitio... | Let $A$ be an [[Definition:Algebra over Field|algebra]] over $\C$.
Let $p \in A$ be [[Definition:Idempotent Element|idempotent]], that is:
:$p^2 = p$
Let $\map {\sigma_A} p$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $p$ in $A$.
Then:
:$\map {\sigma_A} p \subseteq \set {0, 1}$ | Suppose first that $A$ is not [[Definition:Unital Algebra|unital]] and let $p \in A$ be [[Definition:Idempotent Element|idempotent]].
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Then we have:
{{begin-eqn}}
{{eqn | l = \tuple {p, 0} \tuple {p, 0}
| r = \tuple {p^2 + 0p + 0 ... | Spectrum of Idempotent in Algebra over Complex Numbers | https://proofwiki.org/wiki/Spectrum_of_Idempotent_in_Algebra_over_Complex_Numbers | https://proofwiki.org/wiki/Spectrum_of_Idempotent_in_Algebra_over_Complex_Numbers | [
"Spectra (Spectral Theory)"
] | [
"Definition:Algebra over Field",
"Definition:Idempotence/Element",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Unital Algebra",
"Definition:Idempotence/Element",
"Definition:Unitization of Algebra over Field",
"Definition:Idempotence/Element",
"Definition:Spectrum (Spectral Theory)/Non-Unital Algebra",
"Definition:Unital Algebra",
"Definition:Idempotence/Element",
"Definition:Unital Algebra",
"De... |
proofwiki-21164 | Commutativity of Spectrum in Algebra over Complex Numbers | Let $A$ be an algebra over $\C$.
Let $x, y \in A$.
Let $\map {\sigma_A} {x y}$ and $\map {\sigma_A} {y x}$ be the spectrum of $x y$ and $y x$ respectively in $A$.
Then:
:$\map {\sigma_A} {x y} \cup \set 0 = \map {\sigma_A} {y x} \cup \set 0$ | Let $\map G A$ be the group of units of $A$.
{{WLOG}}, by replacing $A$ by its unitization if necessary, that $A$ is unital.
Let $x, y \in A$.
We show that ${\mathbf 1}_A - x y \in \map G A$ {{iff}} ${\mathbf 1}_A - y x \in \map G A$.
By swapping $x$ and $y$, it suffices to show that if ${\mathbf 1}_A - x y \in \map G ... | Let $A$ be an [[Definition:Algebra over Field|algebra]] over $\C$.
Let $x, y \in A$.
Let $\map {\sigma_A} {x y}$ and $\map {\sigma_A} {y x}$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $x y$ and $y x$ respectively in $A$.
Then:
:$\map {\sigma_A} {x y} \cup \set 0 = \map {\sigma_A} {y x} \cup \set 0... | Let $\map G A$ be the [[Definition:Group of Units|group of units]] of $A$.
{{WLOG}}, by replacing $A$ by its [[Definition:Unitization of Algebra over Field|unitization]] if necessary, that $A$ is [[Definition:Unital Algebra|unital]].
Let $x, y \in A$.
We show that ${\mathbf 1}_A - x y \in \map G A$ {{iff}} ${\mathbf... | Commutativity of Spectrum in Algebra over Complex Numbers | https://proofwiki.org/wiki/Commutativity_of_Spectrum_in_Algebra_over_Complex_Numbers | https://proofwiki.org/wiki/Commutativity_of_Spectrum_in_Algebra_over_Complex_Numbers | [
"Spectra (Spectral Theory)"
] | [
"Definition:Algebra over Field",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Group of Units",
"Definition:Unitization of Algebra over Field",
"Definition:Unital Algebra"
] |
proofwiki-21165 | Norm on Unitization of Normed Algebra is Equivalent to Direct Product Norm | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$ that is not unital as an algebra.
Let $\struct {A_+, \norm {\, \cdot \,}_1 }$ be the normed unitization of $\struct {A, \norm {\, \cdot \,} }$.
Let $\norm {\, \cdot \,}_2$ be the direct product norm on $A \times \GF$.
... | Let $u, v \in \R$.
We have:
:$\ds \frac 1 2 \paren {\size u + \size v} \le \frac 1 2 \paren {\max \set {\size u, \size v} + \max \set {\size u, \size v} } = \max \set {\size u, \size v}$
and:
:$\ds \max \set {\size u, \size v} \le \max \set {\size u, \size v} + \min \set {\size u, \size v} = \size u + \size v$
Let $x ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $\GF$ that is not [[Definition:Unital Algebra|unital as an algebra]].
Let $\struct {A_+, \norm {\, \cdot \,}_1 }$ be the [[Definition:Unitization of Normed Algebra|normed unitization]] of $\st... | Let $u, v \in \R$.
We have:
:$\ds \frac 1 2 \paren {\size u + \size v} \le \frac 1 2 \paren {\max \set {\size u, \size v} + \max \set {\size u, \size v} } = \max \set {\size u, \size v}$
and:
:$\ds \max \set {\size u, \size v} \le \max \set {\size u, \size v} + \min \set {\size u, \size v} = \size u + \size v$
Let $... | Norm on Unitization of Normed Algebra is Equivalent to Direct Product Norm | https://proofwiki.org/wiki/Norm_on_Unitization_of_Normed_Algebra_is_Equivalent_to_Direct_Product_Norm | https://proofwiki.org/wiki/Norm_on_Unitization_of_Normed_Algebra_is_Equivalent_to_Direct_Product_Norm | [
"Unitizations of Normed Algebras"
] | [
"Definition:Normed Algebra",
"Definition:Unital Algebra",
"Definition:Unitization of Normed Algebra",
"Definition:Direct Product Norm",
"Definition:Equivalence of Norms"
] | [
"Definition:Equivalence of Norms",
"Category:Unitizations of Normed Algebras"
] |
proofwiki-21166 | Normed Algebra Embeds into Unitization as Closed Ideal | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$ that is not unital as an algebra.
Let $\struct {A_+, \norm {\, \cdot \,}_{A_+} }$ be the normed unitization of $\struct {A, \norm {\, \cdot \,} }$.
Let:
:$A_0 = \set {\tuple {x, 0} : x \in A} \subseteq A_+$
Then $A_0$ ... | From Algebra over Field Embeds into Unitization as Ideal, $A_0$ is an ideal of $A$.
From Norm on Unitization of Normed Algebra is Equivalent to Direct Product Norm, $\norm {\, \cdot \,}_{A_+}$ is equivalent to the direct product norm $\norm {\, \cdot \,}_{A \times \GF}$.
Let $\sequence {\tuple {x_n, 0} }_{n \in \N}$ be... | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $\GF$ that is not [[Definition:Unital Algebra|unital as an algebra]].
Let $\struct {A_+, \norm {\, \cdot \,}_{A_+} }$ be the [[Definition:Unitization of Normed Algebra|normed unitization]] of ... | From [[Algebra over Field Embeds into Unitization as Ideal]], $A_0$ is an [[Definition:Ideal of Algebra over Field|ideal]] of $A$.
From [[Norm on Unitization of Normed Algebra is Equivalent to Direct Product Norm]], $\norm {\, \cdot \,}_{A_+}$ is [[Definition:Equivalence of Norms|equivalent]] to the [[Definition:Direc... | Normed Algebra Embeds into Unitization as Closed Ideal | https://proofwiki.org/wiki/Normed_Algebra_Embeds_into_Unitization_as_Closed_Ideal | https://proofwiki.org/wiki/Normed_Algebra_Embeds_into_Unitization_as_Closed_Ideal | [
"Unitizations of Normed Algebras"
] | [
"Definition:Normed Algebra",
"Definition:Unital Algebra",
"Definition:Unitization of Normed Algebra",
"Definition:Closed Set",
"Definition:Ideal of Algebra"
] | [
"Algebra over Field Embeds into Unitization as Ideal",
"Definition:Ideal of Algebra over Field",
"Norm on Unitization of Normed Algebra is Equivalent to Direct Product Norm",
"Definition:Equivalence of Norms",
"Definition:Direct Product Norm",
"Definition:Convergent Sequence",
"Definition:Limit of Seque... |
proofwiki-21167 | Resolvent Set of Element of Banach Algebra is Open | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $x \in A$.
Let $\map {\rho_A} x$ be the resolvent set of $x$ in $A$.
{{mistake|The resolvent set is only defined for unital algebras}}
Then $\map {\rho_A} x$ is open. | {{WLOG}} suppose that $A$ is unital, swapping $A$ for its unitization if necessary.
Let $\map G A$ be the group of units of $A$.
Define $S : \C \to A$ by:
:$\map S \lambda = \lambda {\mathbf 1}_A - x$
From Resolvent Mapping is Continuous: Continuous, $S$ is continuous.
From Group of Units in Unital Banach Algebra is O... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $x \in A$.
Let $\map {\rho_A} x$ be the [[Definition:Resolvent Set|resolvent set]] of $x$ in $A$.
{{mistake|The resolvent set is only defined for unital algebras}}
Then $\map {\rho_A} x$ is [[Definition:Open Se... | {{WLOG}} suppose that $A$ is [[Definition:Unital Banach Algebra|unital]], swapping $A$ for its [[Definition:Unitization of Normed Algebra|unitization]] if necessary.
Let $\map G A$ be the [[Definition:Group of Units|group of units]] of $A$.
Define $S : \C \to A$ by:
:$\map S \lambda = \lambda {\mathbf 1}_A - x$
Fro... | Resolvent Set of Element of Banach Algebra is Open | https://proofwiki.org/wiki/Resolvent_Set_of_Element_of_Banach_Algebra_is_Open | https://proofwiki.org/wiki/Resolvent_Set_of_Element_of_Banach_Algebra_is_Open | [
"Resolvent Sets",
"Spectral Theory of Banach Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Resolvent Set",
"Definition:Open Set"
] | [
"Definition:Unital Banach Algebra",
"Definition:Unitization of Normed Algebra",
"Definition:Group of Units",
"Resolvent Mapping is Continuous/Banach Algebra",
"Definition:Continuous Mapping",
"Group of Units in Unital Banach Algebra is Open",
"Definition:Open Set",
"Definition:Resolvent Set",
"Defin... |
proofwiki-21168 | Spectrum of Element of Banach Algebra is Closed | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$.
Then $\map {\sigma_A} x$ is closed. | {{WLOG}} suppose that $A$ is unital, swapping $A$ for its unitization if necessary.
Let $\map {\rho_A} x$ be the resolvent set of $x$ in $A$.
From Resolvent Set of Element of Banach Algebra is Open, $\map {\rho_A} x$ is open.
From the definition of the spectrum, we have $\map {\sigma_A} x = \C \setminus \map {\rho_A} x... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $x$ in $A$.
Then $\map {\sigma_A} x$ is [[Definition:Closed Set|closed]]. | {{WLOG}} suppose that $A$ is [[Definition:Unital Banach Algebra|unital]], swapping $A$ for its [[Definition:Unitization of Normed Algebra|unitization]] if necessary.
Let $\map {\rho_A} x$ be the [[Definition:Resolvent Set|resolvent set]] of $x$ in $A$.
From [[Resolvent Set of Element of Banach Algebra is Open]], $\ma... | Spectrum of Element of Banach Algebra is Closed | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Closed | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Closed | [
"Spectra (Spectral Theory)",
"Spectral Theory of Banach Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Spectrum (Spectral Theory)",
"Definition:Closed Set"
] | [
"Definition:Unital Banach Algebra",
"Definition:Unitization of Normed Algebra",
"Definition:Resolvent Set",
"Resolvent Set of Element of Banach Algebra is Open",
"Definition:Open Set",
"Definition:Spectrum (Spectral Theory)",
"Definition:Closed Set",
"Definition:Closed Set",
"Category:Spectra (Spect... |
proofwiki-21169 | Spectrum of Element of Banach Algebra is Bounded | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$.
Then $\map {\sigma_A} x$ is bounded, and in particular:
:$\cmod \lambda \le \norm x$ for all $\lambda \in \map {\sigma_A} x$ | Suppose first that $\struct {A, \norm {\, \cdot \,} }$ is unital.
Let $\map G A$ be the group of units.
Let $\lambda \in \C$ be such that $\cmod \lambda > \norm x$.
Then from {{NormAxiomVector|2}}, we have:
:$\ds \norm {\frac x \lambda} < 1$
From Element of Unital Banach Algebra Close to Identity is Invertible:
:$\ds {... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $x$ in $A$.
Then $\map {\sigma_A} x$ is [[Definition:Bounded Subset of Complex Plane|bounded]], and in particular:... | Suppose first that $\struct {A, \norm {\, \cdot \,} }$ is [[Definition:Unital Banach Algebra|unital]].
Let $\map G A$ be the [[Definition:Group of Units|group of units]].
Let $\lambda \in \C$ be such that $\cmod \lambda > \norm x$.
Then from {{NormAxiomVector|2}}, we have:
:$\ds \norm {\frac x \lambda} < 1$
From [[... | Spectrum of Element of Banach Algebra is Bounded | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Bounded | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Bounded | [
"Spectra (Spectral Theory)",
"Banach Algebras",
"Spectrum of Element of Banach Algebra is Bounded"
] | [
"Definition:Banach Algebra",
"Definition:Spectrum (Spectral Theory)",
"Definition:Bounded Metric Space/Complex"
] | [
"Definition:Unital Banach Algebra",
"Definition:Group of Units",
"Element of Unital Banach Algebra Close to Identity is Invertible",
"Definition:Unital Banach Algebra",
"Definition:Unital Banach Algebra",
"Definition:Unitization of Normed Algebra",
"Definition:Spectrum (Spectral Theory)/Non-Unital Algeb... |
proofwiki-21170 | Spectrum of Element of Banach Algebra is Compact | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$.
Then $\map {\sigma_A} x$ is compact. | The result follows immediately from:
:Spectrum of Element of Banach Algebra is Bounded
:Spectrum of Element of Banach Algebra is Closed
{{qed}}
Category:Spectra (Spectral Theory)
Category:Banach Algebras
ky0zt3jh1z769hy75uudpejradwrek7 | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $x$ in $A$.
Then $\map {\sigma_A} x$ is [[Definition:Compact Subset of Complex Plane|compact]]. | The result follows immediately from:
:[[Spectrum of Element of Banach Algebra is Bounded]]
:[[Spectrum of Element of Banach Algebra is Closed]]
{{qed}}
[[Category:Spectra (Spectral Theory)]]
[[Category:Banach Algebras]]
ky0zt3jh1z769hy75uudpejradwrek7 | Spectrum of Element of Banach Algebra is Compact | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Compact | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Compact | [
"Spectra (Spectral Theory)",
"Banach Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Spectrum (Spectral Theory)",
"Definition:Compact Space/Metric Space/Complex"
] | [
"Spectrum of Element of Banach Algebra is Bounded",
"Spectrum of Element of Banach Algebra is Closed",
"Category:Spectra (Spectral Theory)",
"Category:Banach Algebras"
] |
proofwiki-21171 | Real Number Line is not Topological Continuum | The real number line is not a '''continuum''' in the topological sense. | {{Recall|Continuum (Topology)|continuum}}
{{:Definition:Continuum (Topology)}}
However, we have the result Real Number Line is not Compact.
Hence the result.
{{qed}} | The [[Definition:Real Number Line|real number line]] is not a '''[[Definition:Continuum (Topology)|continuum]]''' in the [[Definition:Topology (Mathematical Branch)|topological]] sense. | {{Recall|Continuum (Topology)|continuum}}
{{:Definition:Continuum (Topology)}}
However, we have the result [[Real Number Line is not Compact]].
Hence the result.
{{qed}} | Real Number Line is not Topological Continuum | https://proofwiki.org/wiki/Real_Number_Line_is_not_Topological_Continuum | https://proofwiki.org/wiki/Real_Number_Line_is_not_Topological_Continuum | [
"Real Numbers",
"Examples of Continua (Topology)"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Continuum/Topology",
"Definition:Topology (Mathematical Branch)"
] | [
"Real Number Line is not Compact"
] |
proofwiki-21172 | Real Number Line is not Compact | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is not compact. | We have:
:Compact Space is Countably Compact
:Real Number Line is not Countably Compact
Hence, as $\struct {\R, \tau_d}$ is not countably compact, it follows that it is not compact.
{{qed}}
Category:Real Number Line with Euclidean Topology
Category:Examples of Compact Topological Spaces
kspi9j5em2ftdhm328t16x9051tgq57 | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is not [[Definition:Compact Topological Space|compact]]. | We have:
:[[Compact Space is Countably Compact]]
:[[Real Number Line is not Countably Compact]]
Hence, as $\struct {\R, \tau_d}$ is not [[Definition:Countably Compact Space|countably compact]], it follows that it is not [[Definition:Compact Topological Space|compact]].
{{qed}}
[[Category:Real Number Line with Euclide... | Real Number Line is not Compact | https://proofwiki.org/wiki/Real_Number_Line_is_not_Compact | https://proofwiki.org/wiki/Real_Number_Line_is_not_Compact | [
"Real Number Line with Euclidean Topology",
"Examples of Compact Topological Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Compact Topological Space"
] | [
"Compact Space is Countably Compact",
"Real Number Line is not Countably Compact",
"Definition:Countably Compact Space",
"Definition:Compact Topological Space",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Compact Topological Spaces"
] |
proofwiki-21173 | Equivalence of Definitions of Inconsistent (Logic) | {{TFAE|def = Inconsistent (Logic)|view = Inconsistent|context = Logic}}
Let $\LL$ be a logical language.
Let $\mathscr P$ be a proof system for $\LL$. | === Definition $(1)$ implies Definition $(2)$ ===
Let $\FF$ be an inconsistent set of logical formulas by definition $1$.
Let $\phi$ be an arbitrary logical formula in $\FF$.
Then {{hypothesis}}:
:$\phi \land \lnot \phi$
is a logical formula in $\FF$.
Thus $\FF$ is an inconsistent set of logical formulas by definition ... | {{TFAE|def = Inconsistent (Logic)|view = Inconsistent|context = Logic}}
Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr P$ be a [[Definition:Proof System|proof system]] for $\LL$. | === Definition $(1)$ implies Definition $(2)$ ===
Let $\FF$ be an [[Definition:Inconsistent (Logic)/Definition 1|inconsistent]] [[Definition:Set|set]] of [[Definition:Logical Formula|logical formulas]] by [[Definition:Inconsistent (Logic)/Definition 1|definition $1$]].
Let $\phi$ be an arbitrary [[Definition:Logical ... | Equivalence of Definitions of Inconsistent (Logic) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Inconsistent_(Logic) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Inconsistent_(Logic) | [
"Inconsistent (Logic)"
] | [
"Definition:Logical Language",
"Definition:Proof System"
] | [
"Definition:Inconsistent (Logic)/Definition 1",
"Definition:Set",
"Definition:Logical Formula",
"Definition:Inconsistent (Logic)/Definition 1",
"Definition:Logical Formula",
"Definition:Logical Formula",
"Definition:Inconsistent (Logic)/Definition 2",
"Definition:Set",
"Definition:Logical Formula",
... |
proofwiki-21174 | Three Points are Coplanar | Let $P_1$, $P_2$ and $P_3$ be points in Euclidean $3$-space.
Then there exists a plane $\PP$ such that $P_1$, $P_2$ and $P_3$ all lie in $\PP$.
That is, $P_1$, $P_2$ and $P_3$ are coplanar. | Let the straight line $P_1 P_2$ be constructed according to Euclid's first postulate.
Let the straight line $P_2 P_3$ be constructed according to Euclid's first postulate.
Thus $P_1 P_2$ and $P_2 P_3$ intersect at $P_2$.
From Two Intersecting Straight Lines are in One Plane, $P_1 P_2$ and $P_2 P_3$ are coplanar.
Hence ... | Let $P_1$, $P_2$ and $P_3$ be [[Definition:Point|points]] in [[Definition:Euclidean Space|Euclidean $3$-space]].
Then there exists a [[Definition:Plane|plane]] $\PP$ such that $P_1$, $P_2$ and $P_3$ all lie in $\PP$.
That is, $P_1$, $P_2$ and $P_3$ are [[Definition:Coplanar Points|coplanar]]. | Let the [[Definition:Straight Line|straight line]] $P_1 P_2$ be constructed according to [[Axiom:Euclid's First Postulate|Euclid's first postulate]].
Let the [[Definition:Straight Line|straight line]] $P_2 P_3$ be constructed according to [[Axiom:Euclid's First Postulate|Euclid's first postulate]].
Thus $P_1 P_2$ and... | Three Points are Coplanar | https://proofwiki.org/wiki/Three_Points_are_Coplanar | https://proofwiki.org/wiki/Three_Points_are_Coplanar | [
"Coplanar Points"
] | [
"Definition:Point",
"Definition:Euclidean Space",
"Definition:Plane Surface",
"Definition:Coplanar/Points"
] | [
"Definition:Line/Straight Line",
"Axiom:Euclid's First Postulate",
"Definition:Line/Straight Line",
"Axiom:Euclid's First Postulate",
"Definition:Intersection (Geometry)",
"Two Intersecting Straight Lines are in One Plane",
"Definition:Coplanar/Lines",
"Definition:Coplanar/Points"
] |
proofwiki-21175 | Condition for 4 Points to be Coplanar | Let:
{{begin-eqn}}
{{eqn | l = p_1
| r = \tuple {x_1, y_1, z_1}
}}
{{eqn | l = p_2
| r = \tuple {x_2, y_2, z_2}
}}
{{eqn | l = p_3
| r = \tuple {x_3, y_3, z_3}
}}
{{eqn | l = p_4
| r = \tuple {x_4, y_4, z_4}
}}
{{end-eqn}}
be distinct points in Cartesian $3$-space.
Then $p_1$, $p_2$, $p_3$ and $... | === Sufficient Condition ===
Let $p_1$, $p_2$, $p_3$ and $p_4$ be on a plane $P$.
By the equation of plane, determinant form on the points $p_2$, $p_3$ and $p_4$, the equation of $P$ is:
{{begin-eqn}}
{{eqn | n = 1
| l = \begin {vmatrix} x & y & z & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z... | Let:
{{begin-eqn}}
{{eqn | l = p_1
| r = \tuple {x_1, y_1, z_1}
}}
{{eqn | l = p_2
| r = \tuple {x_2, y_2, z_2}
}}
{{eqn | l = p_3
| r = \tuple {x_3, y_3, z_3}
}}
{{eqn | l = p_4
| r = \tuple {x_4, y_4, z_4}
}}
{{end-eqn}}
be [[Definition:Distinct Elements|distinct]] [[Definition:Point|points]... | === Sufficient Condition ===
Let $p_1$, $p_2$, $p_3$ and $p_4$ be on a plane $P$.
By the [[Three-Point Form of Equation of Plane/Determinant Form|equation of plane, determinant form]] on the points $p_2$, $p_3$ and $p_4$, the equation of $P$ is:
{{begin-eqn}}
{{eqn | n = 1
| l = \begin {vmatrix} x & y & z & 1 ... | Condition for 4 Points to be Coplanar | https://proofwiki.org/wiki/Condition_for_4_Points_to_be_Coplanar | https://proofwiki.org/wiki/Condition_for_4_Points_to_be_Coplanar | [
"Coplanar Points"
] | [
"Definition:Distinct/Plural",
"Definition:Point",
"Definition:Cartesian 3-Space",
"Definition:Coplanar/Points",
"Definition:Determinant"
] | [
"Three-Point Form of Equation of Plane/Determinant Form",
"Three-Point Form of Equation of Plane/Determinant Form"
] |
proofwiki-21176 | Coordinate Axes are Copunctal | Let $\CC$ be a coordinate system in ordinary $3$-dimensional space.
Then the coordinate axes of $S$ are copunctal. | Implicit by definition of coordinate system.
{{qed}} | Let $\CC$ be a [[Definition:Coordinate System|coordinate system]] in [[Definition:Ordinary Space|ordinary $3$-dimensional space]].
Then the [[Definition:Coordinate Axis|coordinate axes]] of $S$ are [[Definition:Copunctal Lines|copunctal]]. | Implicit by definition of [[Definition:Coordinate System|coordinate system]].
{{qed}} | Coordinate Axes are Copunctal | https://proofwiki.org/wiki/Coordinate_Axes_are_Copunctal | https://proofwiki.org/wiki/Coordinate_Axes_are_Copunctal | [
"Copunctal Lines",
"Coordinate Systems"
] | [
"Definition:Coordinate System",
"Definition:Ordinary Space",
"Definition:Axis/Coordinate Axes",
"Definition:Copunctal/Lines"
] | [
"Definition:Coordinate System"
] |
proofwiki-21177 | Character on Unital Banach Algebra is Unital Algebra Homomorphism | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.
Let $\phi : A \to \C$ be a character on $A$.
Then $\phi$ is a unital algebra homomorphism. | By the definition of a character, $\phi$ is a non-zero algebra homomorphism.
We only need to verify that:
:$\map \phi { {\mathbf 1}_A} = 1$
We have:
:$\map \phi { {\mathbf 1}_A} = \map \phi { {\mathbf 1}_A^2} = \paren {\map \phi { {\mathbf 1}_A} }^2$
So, we have:
:$\map \phi { {\mathbf 1}_A} \in \set {0, 1}$
Note that ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $\phi : A \to \C$ be a [[Definition:Character (Banach Algebra)|character]] on $A$.
Then $\phi$ is a [[Definition:Unital Algebra Homomorphism|unital algebra homomorphism]]. | By the definition of a [[Definition:Character (Banach Algebra)|character]], $\phi$ is a non-zero [[Definition:Algebra Homomorphism|algebra homomorphism]].
We only need to verify that:
:$\map \phi { {\mathbf 1}_A} = 1$
We have:
:$\map \phi { {\mathbf 1}_A} = \map \phi { {\mathbf 1}_A^2} = \paren {\map \phi { {\mathbf ... | Character on Unital Banach Algebra is Unital Algebra Homomorphism | https://proofwiki.org/wiki/Character_on_Unital_Banach_Algebra_is_Unital_Algebra_Homomorphism | https://proofwiki.org/wiki/Character_on_Unital_Banach_Algebra_is_Unital_Algebra_Homomorphism | [
"Characters (Banach Algebras)"
] | [
"Definition:Unital Banach Algebra",
"Definition:Character (Banach Algebra)",
"Definition:Unital Algebra Homomorphism"
] | [
"Definition:Character (Banach Algebra)",
"Definition:Algebra Homomorphism",
"Definition:Unital Algebra Homomorphism"
] |
proofwiki-21178 | Character on Banach Algebra is Surjective | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $\phi : A \to \C$ be a character on $A$.
Then $\phi$ is surjective. | From Image of Submodule under Linear Transformation is Submodule, $\phi \sqbrk A$ is a vector subspace of $\C$.
From Dimension of Proper Subspace is Less Than its Superspace, we have:
:$\dim \phi \sqbrk A \le \dim \C = 1$
and so we either have $\phi \sqbrk A = \set 0$ or $\phi \sqbrk A = \C$.
Since $\phi \ne 0$ by the... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $\phi : A \to \C$ be a [[Definition:Character (Banach Algebra)|character]] on $A$.
Then $\phi$ is [[Definition:Surjection|surjective]]. | From [[Image of Submodule under Linear Transformation is Submodule]], $\phi \sqbrk A$ is a [[Definition:Vector Subspace|vector subspace]] of $\C$.
From [[Dimension of Proper Subspace is Less Than its Superspace]], we have:
:$\dim \phi \sqbrk A \le \dim \C = 1$
and so we either have $\phi \sqbrk A = \set 0$ or $\phi \... | Character on Banach Algebra is Surjective/Proof 1 | https://proofwiki.org/wiki/Character_on_Banach_Algebra_is_Surjective | https://proofwiki.org/wiki/Character_on_Banach_Algebra_is_Surjective/Proof_1 | [
"Character on Banach Algebra is Surjective",
"Characters (Banach Algebras)",
"Character on Banach Algebra is Surjective"
] | [
"Definition:Banach Algebra",
"Definition:Character (Banach Algebra)",
"Definition:Surjection"
] | [
"Image of Submodule under Linear Transformation is Submodule",
"Definition:Vector Subspace",
"Dimension of Proper Subspace is Less Than its Superspace",
"Definition:Character (Banach Algebra)"
] |
proofwiki-21179 | Character on Banach Algebra is Surjective | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $\phi : A \to \C$ be a character on $A$.
Then $\phi$ is surjective. | As $\phi$ is non-zero, there exists an $x_0 \in A$ such that:
:$\map \phi {x_0} \in \C \setminus \set 0$
Thus, for each $a \in \C$:
{{begin-eqn}}
{{eqn | l = \frac a {\map \phi {x_0} } x_0
| o = \in
| r = A
| c = as $A$ is a $\C$-algebra
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \map \phi {\frac a... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $\phi : A \to \C$ be a [[Definition:Character (Banach Algebra)|character]] on $A$.
Then $\phi$ is [[Definition:Surjection|surjective]]. | As $\phi$ is non-zero, there exists an $x_0 \in A$ such that:
:$\map \phi {x_0} \in \C \setminus \set 0$
Thus, for each $a \in \C$:
{{begin-eqn}}
{{eqn | l = \frac a {\map \phi {x_0} } x_0
| o = \in
| r = A
| c = as $A$ is a $\C$-[[Definition:Algebra over Ring|algebra]]
}}
{{end-eqn}}
and:
{{begin-eq... | Character on Banach Algebra is Surjective/Proof 2 | https://proofwiki.org/wiki/Character_on_Banach_Algebra_is_Surjective | https://proofwiki.org/wiki/Character_on_Banach_Algebra_is_Surjective/Proof_2 | [
"Character on Banach Algebra is Surjective",
"Characters (Banach Algebras)",
"Character on Banach Algebra is Surjective"
] | [
"Definition:Banach Algebra",
"Definition:Character (Banach Algebra)",
"Definition:Surjection"
] | [
"Definition:Algebra over Ring",
"Definition:Algebra Homomorphism"
] |
proofwiki-21180 | Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital commutative Banach algebra over $\C$.
Let $\phi : A \to \C$ be a character on $A$.
Then $\ker \phi$ is a maximal ideal of $A$. | From Kernel of Ring Homomorphism is Ideal, $\ker \phi$ is a ring ideal of $A$.
From the First Ring Isomorphism Theorem, we have:
:$\phi \sqbrk A$ and $\dfrac A {\ker \phi}$ are isomorphic as rings.
From Character on Banach Algebra is Surjective, we have that $\phi \sqbrk A = \C$.
Hence:
:$\dfrac A {\ker \phi} \cong \C... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Algebra|unital]] [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $\phi : A \to \C$ be a [[Definition:Character (Banach Algebra)|character]] on $A$.
Then $\ker \phi$ is a [[... | From [[Kernel of Ring Homomorphism is Ideal]], $\ker \phi$ is a [[Definition:Ideal of Ring|ring ideal]] of $A$.
From the [[First Ring Isomorphism Theorem]], we have:
:$\phi \sqbrk A$ and $\dfrac A {\ker \phi}$ are [[Definition:Ring Isomorphism|isomorphic as rings]].
From [[Character on Banach Algebra is Surjective]]... | Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal/Proof 1 | https://proofwiki.org/wiki/Kernel_of_Character_on_Unital_Commutative_Banach_Algebra_is_Maximal_Ideal | https://proofwiki.org/wiki/Kernel_of_Character_on_Unital_Commutative_Banach_Algebra_is_Maximal_Ideal/Proof_1 | [
"Characters (Banach Algebras)",
"Commutative Banach Algebras"
] | [
"Definition:Unital Algebra",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Banach Algebra",
"Definition:Character (Banach Algebra)",
"Definition:Maximal Ideal of Algebra"
] | [
"Kernel of Ring Homomorphism is Ideal",
"Definition:Ideal of Ring",
"First Isomorphism Theorem/Rings",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Character on Banach Algebra is Surjective",
"Definition:Field (Abstract Algebra)",
"Maximal Ideal iff Quotient Ring is Field",
"Definiti... |
proofwiki-21181 | Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital commutative Banach algebra over $\C$.
Let $\phi : A \to \C$ be a character on $A$.
Then $\ker \phi$ is a maximal ideal of $A$. | Let $I$ be an ideal of $A$ such that:
:$\ker \phi \subsetneq I$
We need to show $I = A$.
That is, we need to show:
:${\mathbf 1}_A \in I$
Let:
:$x \in I \setminus \ker \phi$
Then:
:$\map \phi x \ne 0$
Thus we can define:
:$\ds \tilde x := {\map \phi x}^{-1} x$
Then:
{{begin-eqn}}
{{eqn | l = \map \phi { {\mathbf 1}_A -... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Algebra|unital]] [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $\phi : A \to \C$ be a [[Definition:Character (Banach Algebra)|character]] on $A$.
Then $\ker \phi$ is a [[... | Let $I$ be an [[Definition:Ideal of Algebra|ideal]] of $A$ such that:
:$\ker \phi \subsetneq I$
We need to show $I = A$.
That is, we need to show:
:${\mathbf 1}_A \in I$
Let:
:$x \in I \setminus \ker \phi$
Then:
:$\map \phi x \ne 0$
Thus we can define:
:$\ds \tilde x := {\map \phi x}^{-1} x$
Then:
{{begin-eqn}}
... | Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal/Proof 2 | https://proofwiki.org/wiki/Kernel_of_Character_on_Unital_Commutative_Banach_Algebra_is_Maximal_Ideal | https://proofwiki.org/wiki/Kernel_of_Character_on_Unital_Commutative_Banach_Algebra_is_Maximal_Ideal/Proof_2 | [
"Characters (Banach Algebras)",
"Commutative Banach Algebras"
] | [
"Definition:Unital Algebra",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Banach Algebra",
"Definition:Character (Banach Algebra)",
"Definition:Maximal Ideal of Algebra"
] | [
"Definition:Ideal of Algebra"
] |
proofwiki-21182 | Character on Unital Banach Algebra is Uniquely Identified by Kernel | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.
Let $\phi, \psi : A \to \C$ be characters on $A$ such that:
:$\ker \phi = \ker \psi$
Then $\phi = \psi$. | From Character on Unital Banach Algebra is Unital Algebra Homomorphism, we have $\map \phi { {\mathbf 1}_A} = 1$ and $\map \psi { {\mathbf 1}_A} = 1$.
Let $x \in A$.
We have:
{{begin-eqn}}
{{eqn | l = \map \phi {x - \map \phi x {\mathbf 1}_A}
| r = \map \phi x - \map \phi {\map \phi x {\mathbf 1}_A}
}}
{{eqn | r = \... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $\phi, \psi : A \to \C$ be [[Definition:Character (Banach Algebra)|characters]] on $A$ such that:
:$\ker \phi = \ker \psi$
Then $\phi = \psi$. | From [[Character on Unital Banach Algebra is Unital Algebra Homomorphism]], we have $\map \phi { {\mathbf 1}_A} = 1$ and $\map \psi { {\mathbf 1}_A} = 1$.
Let $x \in A$.
We have:
{{begin-eqn}}
{{eqn | l = \map \phi {x - \map \phi x {\mathbf 1}_A}
| r = \map \phi x - \map \phi {\map \phi x {\mathbf 1}_A}
}}
{{eqn |... | Character on Unital Banach Algebra is Uniquely Identified by Kernel | https://proofwiki.org/wiki/Character_on_Unital_Banach_Algebra_is_Uniquely_Identified_by_Kernel | https://proofwiki.org/wiki/Character_on_Unital_Banach_Algebra_is_Uniquely_Identified_by_Kernel | [
"Characters (Banach Algebras)"
] | [
"Definition:Unital Banach Algebra",
"Definition:Character (Banach Algebra)"
] | [
"Character on Unital Banach Algebra is Unital Algebra Homomorphism",
"Category:Characters (Banach Algebras)"
] |
proofwiki-21183 | Character on Banach Algebra is Continuous | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $\phi : A \to \C$ be a character on $A$.
Then $\phi$ is continuous and further:
:$\norm \phi_{A^\ast} \le 1$ | Let $\map G A$ be the group of units of $A$.
Suppose first that $\struct {A, \norm {\, \cdot \,} }$ is unital.
We show that:
:$\cmod {\map \phi x} \le \norm x$ for each $x \in A$.
From Continuity of Linear Functionals, we will then have that $\phi$ is continuous with $\norm \phi_{A^\ast} \le 1$.
{{AimForCont}} that $x ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $\phi : A \to \C$ be a [[Definition:Character (Banach Algebra)|character]] on $A$.
Then $\phi$ is [[Definition:Continuous Mapping|continuous]] and further:
:$\norm \phi_{A^\ast} \le 1$ | Let $\map G A$ be the [[Definition:Group of Units|group of units]] of $A$.
Suppose first that $\struct {A, \norm {\, \cdot \,} }$ is [[Definition:Unital Banach Algebra|unital]].
We show that:
:$\cmod {\map \phi x} \le \norm x$ for each $x \in A$.
From [[Continuity of Linear Functionals]], we will then have that $\ph... | Character on Banach Algebra is Continuous | https://proofwiki.org/wiki/Character_on_Banach_Algebra_is_Continuous | https://proofwiki.org/wiki/Character_on_Banach_Algebra_is_Continuous | [
"Characters (Banach Algebras)",
"Character on Banach Algebra is Continuous"
] | [
"Definition:Banach Algebra",
"Definition:Character (Banach Algebra)",
"Definition:Continuous Mapping"
] | [
"Definition:Group of Units",
"Definition:Unital Banach Algebra",
"Continuity of Linear Functionals",
"Definition:Continuous Mapping",
"Element of Unital Banach Algebra Close to Identity is Invertible",
"Character on Unital Banach Algebra is Unital Algebra Homomorphism",
"Definition:Continuous Mapping",
... |
proofwiki-21184 | Closure of Proper Ideal in Unital Banach Algebra is Proper Ideal | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $I$ be a proper ideal of $A$.
Then $I^-$ is a proper ideal of $A$. | From Closure of Subspace of Normed Vector Space is Subspace, $I^-$ is a vector subspace of $A$.
Let $x \in I^-$ and $y \in A$.
We need to show that $x y \in I^-$.
From the definition of a closed set in a normed vector space, there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ valued in $I$ such that:
:$x_n \... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $I$ be a [[Definition:Proper Subset|proper]] [[Definition:Ideal of Algebra over Field|ideal]] of $A$.
Then $I^-$ is a [[Definition:Proper Ideal of Algebra over Field|proper ideal]] of $A$. | From [[Closure of Subspace of Normed Vector Space is Subspace]], $I^-$ is a [[Definition:Vector Subspace|vector subspace]] of $A$.
Let $x \in I^-$ and $y \in A$.
We need to show that $x y \in I^-$.
From the definition of a [[Definition:Closed Set of Normed Vector Space|closed set in a normed vector space]], there ... | Closure of Proper Ideal in Unital Banach Algebra is Proper Ideal | https://proofwiki.org/wiki/Closure_of_Proper_Ideal_in_Unital_Banach_Algebra_is_Proper_Ideal | https://proofwiki.org/wiki/Closure_of_Proper_Ideal_in_Unital_Banach_Algebra_is_Proper_Ideal | [
"Banach Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Proper Subset",
"Definition:Ideal of Algebra over Field",
"Definition:Proper Ideal of Algebra over Field"
] | [
"Closure of Subspace of Normed Vector Space is Subspace",
"Definition:Vector Subspace",
"Definition:Closed Set/Normed Vector Space",
"Definition:Sequence",
"Definition:Ideal of Algebra over Field",
"Product Rule for Sequence in Normed Algebra",
"Definition:Closure (Topology)",
"Definition:Ideal of Alg... |
proofwiki-21185 | Maximal Ideal in Unital Banach Algebra is Closed | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $I$ be a maximal ideal of $A$.
Then $I$ is closed. | From Closure of Proper Ideal in Unital Banach Algebra is Proper Ideal, the closure $I^-$ is a proper ideal of $A$ with $I \subseteq I^-$.
Since $I$ is a maximal ideal, we have $I = I^-$.
From Set is Closed iff Equals Topological Closure, we conclude that $I$ is closed.
{{qed}}
Category:Banach Algebras
d2kvhsrycz8cqxcg7... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $I$ be a [[Definition:Maximal Ideal|maximal ideal]] of $A$.
Then $I$ is [[Definition:Closed Set|closed]]. | From [[Closure of Proper Ideal in Unital Banach Algebra is Proper Ideal]], the [[Definition:Topological Closure|closure]] $I^-$ is a [[Definition:Proper Ideal|proper ideal]] of $A$ with $I \subseteq I^-$.
Since $I$ is a [[Definition:Maximal Ideal|maximal ideal]], we have $I = I^-$.
From [[Set is Closed iff Equals Top... | Maximal Ideal in Unital Banach Algebra is Closed | https://proofwiki.org/wiki/Maximal_Ideal_in_Unital_Banach_Algebra_is_Closed | https://proofwiki.org/wiki/Maximal_Ideal_in_Unital_Banach_Algebra_is_Closed | [
"Banach Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Maximal Ideal",
"Definition:Closed Set"
] | [
"Closure of Proper Ideal in Unital Banach Algebra is Proper Ideal",
"Definition:Closure (Topology)",
"Definition:Ideal of Ring/Proper Ideal",
"Definition:Maximal Ideal",
"Set is Closed iff Equals Topological Closure",
"Definition:Closed Set",
"Category:Banach Algebras"
] |
proofwiki-21186 | Product of Commuting Elements in Monoid is Unit iff Each Element is Unit | Let $A$ be a monoid.
Let $\map G A$ be the group of units of $A$.
Let $n \ge 2$ be an integer.
Let $x_1, \ldots, x_n$ be commuting elements in $A$.
Let:
:$\ds x = \prod_{i \mathop = 1}^n x_i$
Then:
:$x \in \map G A$ {{iff}} $x_i \in \map G A$ for each $1 \le i \le n$. | If $x_1, \ldots, x_n \in \map G A$, then:
:$\ds \prod_{i \mathop = 1}^k x_i \in \map G A$
by Inverse of Product: Monoid: General Result.
Conversely, suppose:
:$\ds \prod_{i \mathop = 1}^k x_i \in \map G A$
That is, there is a $z \in A$ such that:
:$(1):\quad \ds z \paren {\prod_{i \mathop = 1}^k x_i} = \paren {\prod_{i... | Let $A$ be a [[Definition:Monoid|monoid]].
Let $\map G A$ be the [[Definition:Group of Units of Monoid|group of units]] of $A$.
Let $n \ge 2$ be an [[Definition:Integer|integer]].
Let $x_1, \ldots, x_n$ be [[Definition:Commuting Elements|commuting elements]] in $A$.
Let:
:$\ds x = \prod_{i \mathop = 1}^n x_i$
Th... | If $x_1, \ldots, x_n \in \map G A$, then:
:$\ds \prod_{i \mathop = 1}^k x_i \in \map G A$
by [[Inverse of Product/Monoid/General Result|Inverse of Product: Monoid: General Result]].
Conversely, suppose:
:$\ds \prod_{i \mathop = 1}^k x_i \in \map G A$
That is, there is a $z \in A$ such that:
:$(1):\quad \ds z \paren ... | Product of Commuting Elements in Monoid is Unit iff Each Element is Unit/Proof 2 | https://proofwiki.org/wiki/Product_of_Commuting_Elements_in_Monoid_is_Unit_iff_Each_Element_is_Unit | https://proofwiki.org/wiki/Product_of_Commuting_Elements_in_Monoid_is_Unit_iff_Each_Element_is_Unit/Proof_2 | [
"Monoids"
] | [
"Definition:Monoid",
"Definition:Group of Units/Monoid",
"Definition:Integer",
"Definition:Commutative/Elements"
] | [
"Inverse of Product/Monoid/General Result",
"Definition:Commutative/Elements",
"Definition:Commutative/Elements",
"Definition:Commutative/Elements"
] |
proofwiki-21187 | Resolvent Mapping is Continuous/Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.
Let ${\mathbf 1}_A$ be the identity element of $A$.
Let $x \in A$.
Let $\map {\rho_A} x$ be the resolvent set of $x$ in $A$.
Define $R : \map {\rho_A} x \to A$ by:
:$\map R \lambda = \paren {\lambda {\mathbf 1}_A - x}^{-1}$
Then $R$ is cont... | === Lemma ===
{{:Resolvent Mapping is Continuous/Banach Algebra/Lemma}}{{qed|lemma}}
From the Lemma, we have:
:the mapping $S : \C \to A$ defined by:
::$\map S \lambda = \lambda {\mathbf 1}_A - x$
:for each $\lambda \in \C$, is continuous.
From Restriction of Continuous Mapping is Continuous, $S \restriction_{\map {\rh... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let ${\mathbf 1}_A$ be the [[Definition:Identity Element|identity element]] of $A$.
Let $x \in A$.
Let $\map {\rho_A} x$ be the [[Definition:Resolvent Set|resolvent set]] of $x$ in $A$.
Define $R : \... | === [[Resolvent Mapping is Continuous/Banach Algebra/Lemma|Lemma]] ===
{{:Resolvent Mapping is Continuous/Banach Algebra/Lemma}}{{qed|lemma}}
From the [[Resolvent Mapping is Continuous/Banach Algebra/Lemma|Lemma]], we have:
:the [[Definition:Mapping|mapping]] $S : \C \to A$ defined by:
::$\map S \lambda = \lambda {\m... | Resolvent Mapping is Continuous/Banach Algebra | https://proofwiki.org/wiki/Resolvent_Mapping_is_Continuous/Banach_Algebra | https://proofwiki.org/wiki/Resolvent_Mapping_is_Continuous/Banach_Algebra | [
"Resolvent Mapping is Continuous",
"Spectral Theory of Banach Algebras",
"Resolvent Mapping is Continuous"
] | [
"Definition:Unital Banach Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Resolvent Set",
"Definition:Continuous Mapping"
] | [
"Resolvent Mapping is Continuous/Banach Algebra/Lemma",
"Resolvent Mapping is Continuous/Banach Algebra/Lemma",
"Definition:Mapping",
"Definition:Continuous Mapping",
"Restriction of Continuous Mapping is Continuous",
"Definition:Continuous Mapping",
"Definition:Resolvent Set/Unital Algebra",
"Inverse... |
proofwiki-21188 | Triangular Numbers are Primitive Recursive | Let $t : \N \to \N$ be defined as:
:$\map t k = T_k$
where $T_k$ is the $k$-th triangular number.
Then $t$ is a primitive recursive function. | Define $f : \N^0 \to \N$ as:
:$\map f {} = 0$.
Define $g : \N^2 \to \N$ as:
$\map g x y = x + y + 1$.
By Constant Function is Primitive Recursive and Addition is Primitive Recursive:
:$f$ and $g$ are primitive recursive.
By definition of primitive recursive, it suffices to show that $t$ is obtained by primitive recursi... | Let $t : \N \to \N$ be defined as:
:$\map t k = T_k$
where $T_k$ is the $k$-th [[Definition:Triangular Number|triangular number]].
Then $t$ is a [[Definition:Primitive Recursive Function|primitive recursive function]]. | Define $f : \N^0 \to \N$ as:
:$\map f {} = 0$.
Define $g : \N^2 \to \N$ as:
$\map g x y = x + y + 1$.
By [[Constant Function is Primitive Recursive]] and [[Addition is Primitive Recursive]]:
:$f$ and $g$ are [[Definition:Primitive Recursive Function|primitive recursive]].
By definition of [[Definition:Primitive Rec... | Triangular Numbers are Primitive Recursive | https://proofwiki.org/wiki/Triangular_Numbers_are_Primitive_Recursive | https://proofwiki.org/wiki/Triangular_Numbers_are_Primitive_Recursive | [
"Primitive Recursive Functions",
"Triangular Numbers"
] | [
"Definition:Triangular Number",
"Definition:Primitive Recursive/Function"
] | [
"Constant Function is Primitive Recursive",
"Addition is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Definition:Primitive Recursive/Function",
"Definition:Primitive Recursion",
"Definition:Triangular Number",
"Category:Primitive Recursive Functions",
"Category:Triangular Numbers"... |
proofwiki-21189 | Inverse of Cantor Pairing Function is Primitive Recursive | Define $k : \N \to \N$ as:
:$\map k z$ is the largest $k$ such that $T_k \le z$
where $T_k$ is the $k$-th triangular number.
Let $\pi_1 : \N \to \N$ be defined as:
:$\ds \map {\pi_1} z = z - T_{\map k z}$
Let $\pi_2 : \N \to \N$ be defined as:
:$\map {\pi_2} z = \map k z - \map {\pi_1} z$
Then, $\pi_1$ and $\pi_2$ are ... | As $n = \map k z$ is the largest $n$ such that:
:$T_n \le z$
holds, it follows that $n = \map k z + 1$ is the smallest $n$ such that:
:$T_n \le z$
fails.
Or, in other words, $n = \map k z$ the smallest $n$ such that:
:$T_{n + 1} > z$
holds.
It follows that we can define $k : \N \to \N$ as:
:$\map k z = \map {\mu n} {T_... | Define $k : \N \to \N$ as:
:$\map k z$ is the largest $k$ such that $T_k \le z$
where $T_k$ is the $k$-th [[Definition:Triangular Number|triangular number]].
Let $\pi_1 : \N \to \N$ be defined as:
:$\ds \map {\pi_1} z = z - T_{\map k z}$
Let $\pi_2 : \N \to \N$ be defined as:
:$\map {\pi_2} z = \map k z - \map {\pi_1... | As $n = \map k z$ is the largest $n$ such that:
:$T_n \le z$
holds, it follows that $n = \map k z + 1$ is the smallest $n$ such that:
:$T_n \le z$
fails.
Or, in other words, $n = \map k z$ the smallest $n$ such that:
:$T_{n + 1} > z$
holds.
It follows that we can define $k : \N \to \N$ as:
:$\map k z = \map {\mu n} {... | Inverse of Cantor Pairing Function is Primitive Recursive | https://proofwiki.org/wiki/Inverse_of_Cantor_Pairing_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Inverse_of_Cantor_Pairing_Function_is_Primitive_Recursive | [
"Cantor Pairing Function",
"Primitive Recursive Functions"
] | [
"Definition:Triangular Number",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursive/Function",
"Definition:Minimization",
"Definition:Bounded Minimization",
"Definition:Triangular Number",
"Definition:Contradiction",
"Definition:Primitive Recursive/Function",
"Triangular Numbers are Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Bo... |
proofwiki-21190 | Resolvent Mapping is Analytic/Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.
Let ${\mathbf 1}_A$ be the identity element of $A$.
Let $x \in A$.
Let $\map {\rho_A} x$ be the resolvent set of $x$ in $A$.
Define $R : \map {\rho_A} x \to A$ by:
:$\map R \lambda = \paren {\lambda {\mathbf 1}_A - x}^{-1}$
Then $R$ is anal... | Let $\lambda, \mu \in \map {\rho_A} x$ be such that $\lambda \ne \mu$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \frac {\paren {\mu {\mathbf 1}_A - x}^{-1} - \paren {\lambda {\mathbf 1}_A - x}^{-1} } {\mu - \lambda}
| r = \frac {\paren {\mu {\mathbf 1}_A - x}^{-1} \paren { {\mathbf 1}_A - \paren {\mu {\mathbf 1}_A... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let ${\mathbf 1}_A$ be the [[Definition:Identity Element|identity element]] of $A$.
Let $x \in A$.
Let $\map {\rho_A} x$ be the [[Definition:Resolvent Set|resolvent set]] of $x$ in $A$.
Define $R : \... | Let $\lambda, \mu \in \map {\rho_A} x$ be such that $\lambda \ne \mu$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \frac {\paren {\mu {\mathbf 1}_A - x}^{-1} - \paren {\lambda {\mathbf 1}_A - x}^{-1} } {\mu - \lambda}
| r = \frac {\paren {\mu {\mathbf 1}_A - x}^{-1} \paren { {\mathbf 1}_A - \paren {\mu {\mathbf 1}_... | Resolvent Mapping is Analytic/Banach Algebra | https://proofwiki.org/wiki/Resolvent_Mapping_is_Analytic/Banach_Algebra | https://proofwiki.org/wiki/Resolvent_Mapping_is_Analytic/Banach_Algebra | [
"Resolvent Mapping is Analytic",
"Banach Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Resolvent Set",
"Definition:Analytic Function/Banach Space Valued Function",
"Definition:Derivative/Function With Values in Normed Space"
] | [
"Resolvent Mapping is Continuous/Banach Algebra",
"Product Rule for Sequence in Normed Algebra",
"Category:Resolvent Mapping is Analytic",
"Category:Banach Algebras"
] |
proofwiki-21191 | Analytic Function on Banach Space is Continuous | Let $U$ be an open subset of $\C$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\C$.
Let $f : U \to X$ be an analytic function.
Then $f$ is continuous. | Let $x \in U$.
Since $f : U \to X$ is analytic function, the limit:
:$\ds \lim_{y \mathop \to x} \frac {\map f y - \map f x} {y - x} = \map {f'} x$ exists.
We have, from {{NormAxiomVector|2}}:
:$\ds \norm {\map f y - \map f x} = \cmod {y - x} \norm {\frac {\map f y - \map f x} {y - x} }$
From Modulus of Limit: Normed ... | Let $U$ be an [[Definition:Open Set (Complex Analysis)|open subset]] of $\C$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\C$.
Let $f : U \to X$ be an [[Definition:Analytic Function/Banach Space Valued Function|analytic function]].
Then $f$ is [[Definition:Continuous ... | Let $x \in U$.
Since $f : U \to X$ is [[Definition:Analytic Function/Banach Space Valued Function|analytic function]], the [[Definition:Limit of Mapping between Metric Spaces|limit]]:
:$\ds \lim_{y \mathop \to x} \frac {\map f y - \map f x} {y - x} = \map {f'} x$ exists.
We have, from {{NormAxiomVector|2}}:
:$\ds \n... | Analytic Function on Banach Space is Continuous | https://proofwiki.org/wiki/Analytic_Function_on_Banach_Space_is_Continuous | https://proofwiki.org/wiki/Analytic_Function_on_Banach_Space_is_Continuous | [
"Analytic Functions",
"Banach Spaces"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Banach Space",
"Definition:Analytic Function/Banach Space Valued Function",
"Definition:Continuous Function"
] | [
"Definition:Analytic Function/Banach Space Valued Function",
"Definition:Limit of Mapping between Metric Spaces",
"Modulus of Limit/Normed Vector Space",
"Combination Theorem for Sequences/Complex/Product Rule",
"Definition:Continuous Function",
"Definition:Continuous Function",
"Category:Analytic Funct... |
proofwiki-21192 | Spectrum of Element of Banach Algebra is Non-Empty | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$.
Then $\map {\sigma_A} x \ne \O$. | Suppose first that $\struct {A, \norm {\, \cdot \,} }$ is a unital Banach algebra.
Let $\map {\rho_A} x$ be the resolvent set of $x$ in $A$.
{{AimForCont}} that $\map {\sigma_A} x = \O$.
Then $\map {\rho_A} x = \C$.
Define $R : \C \to A$ by:
:$\map R \lambda = \paren {\lambda {\mathbf 1}_A - x}^{-1}$
for each $\lambda ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $x$ in $A$.
Then $\map {\sigma_A} x \ne \O$. | Suppose first that $\struct {A, \norm {\, \cdot \,} }$ is a [[Definition:Unital Banach Algebra|unital Banach algebra]].
Let $\map {\rho_A} x$ be the [[Definition:Resolvent Set|resolvent set]] of $x$ in $A$.
{{AimForCont}} that $\map {\sigma_A} x = \O$.
Then $\map {\rho_A} x = \C$.
Define $R : \C \to A$ by:
:$\map R... | Spectrum of Element of Banach Algebra is Non-Empty | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Non-Empty | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Non-Empty | [
"Spectral Theory of Banach Algebras",
"Spectrum of Element of Banach Algebra is Non-Empty"
] | [
"Definition:Banach Algebra",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Unital Banach Algebra",
"Definition:Resolvent Set",
"Resolvent Mapping is Analytic/Banach Algebra",
"Definition:Analytic Function/Banach Space Valued Function",
"Liouville's Theorem (Complex Analysis)/Banach Space",
"Definition:Bounded Mapping/Normed Vector Space",
"Element of Unital Banach ... |
proofwiki-21193 | Spectral Mapping Theorem for Polynomials | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.
Let $p : \C \to \C$ be a polynomial with:
:$\ds \map p z = \sum_{j \mathop = 0}^n a_j z^j$
for all $z \in \C$, for some $a_0, \ldots, a_n \in \C$.
Define:
:$\ds \map p y = a_0 {\mathbf 1}_A + \sum_{j \mathop = 1}^n a_j y^j$
for each $y \in ... | Let $\map G A$ be the group of units of $A$.
Suppose that $p$ is a constant polynomial, so that:
:$\map p z = \lambda$
for each $z \in \C$, and:
:$\map p x = \lambda {\mathbf 1}_A$
From Spectrum of Element of Banach Algebra is Non-Empty, we have:
:$\map {\sigma_A} x \ne \O$
so that:
:$p \sqbrk {\map {\sigma_A} x} = \s... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $p : \C \to \C$ be a [[Definition:Polynomial|polynomial]] with:
:$\ds \map p z = \sum_{j \mathop = 0}^n a_j z^j$
for all $z \in \C$, for some $a_0, \ldots, a_n \in \C$.
Define:
:$\ds \map p y = a_0... | Let $\map G A$ be the [[Definition:Group of Units|group of units]] of $A$.
Suppose that $p$ is a [[Definition:Constant Polynomial|constant polynomial]], so that:
:$\map p z = \lambda$
for each $z \in \C$, and:
:$\map p x = \lambda {\mathbf 1}_A$
From [[Spectrum of Element of Banach Algebra is Non-Empty]], we have:
:... | Spectral Mapping Theorem for Polynomials | https://proofwiki.org/wiki/Spectral_Mapping_Theorem_for_Polynomials | https://proofwiki.org/wiki/Spectral_Mapping_Theorem_for_Polynomials | [
"Spectra (Spectral Theory)"
] | [
"Definition:Unital Banach Algebra",
"Definition:Polynomial",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Group of Units",
"Definition:Constant Polynomial",
"Spectrum of Element of Banach Algebra is Non-Empty",
"Definition:Constant Polynomial",
"Definition:Constant Polynomial",
"Polynomial Factor Theorem/Corollary/Complex Numbers",
"Product of Commuting Elements in Monoid is Unit iff Each Elemen... |
proofwiki-21194 | Least Fixed Point of Enumeration Operator is Recursively Enumerable | Let $\psi : \powerset \N \to \powerset \N$ be an enumeration operator.
Then there exists a recursively enumerable set $A$ such that:
:$A$ is a fixed point of $\psi$
:Every fixed point of $\psi$ is a superset of $A$ | By Least Fixed Point of Enumeration Operator, such an $A$ can be defined as:
:$\ds \bigcup_{i \mathop \in \N} A_i$
where:
:$A_0 = \O$
:$A_{n + 1} = \map \psi {A_n}$
By definition of enumeration operator, there exists a recursively enumerable set $\phi \subseteq \N$ such that:
:$\map \psi A = \set {x \in \N : \exists \t... | Let $\psi : \powerset \N \to \powerset \N$ be an [[Definition:Enumeration Operator (Recursion Theory)|enumeration operator]].
Then there exists a [[Definition:Recursively Enumerable Set|recursively enumerable set]] $A$ such that:
:$A$ is a [[Definition:Fixed Point|fixed point]] of $\psi$
:Every [[Definition:Fixed Poin... | By [[Least Fixed Point of Enumeration Operator]], such an $A$ can be defined as:
:$\ds \bigcup_{i \mathop \in \N} A_i$
where:
:$A_0 = \O$
:$A_{n + 1} = \map \psi {A_n}$
By definition of [[Definition:Enumeration Operator (Recursion Theory)|enumeration operator]], there exists a [[Definition:Recursively Enumerable Set|... | Least Fixed Point of Enumeration Operator is Recursively Enumerable | https://proofwiki.org/wiki/Least_Fixed_Point_of_Enumeration_Operator_is_Recursively_Enumerable | https://proofwiki.org/wiki/Least_Fixed_Point_of_Enumeration_Operator_is_Recursively_Enumerable | [] | [
"Definition:Enumeration Operator (Recursion Theory)",
"Definition:Recursively Enumerable Set",
"Definition:Fixed Point",
"Definition:Fixed Point",
"Definition:Subset/Superset"
] | [
"Least Fixed Point of Enumeration Operator",
"Definition:Enumeration Operator (Recursion Theory)",
"Definition:Recursively Enumerable Set",
"Definition:Finite Set Coding",
"Definition:Cantor Pairing Function",
"Principle of Mathematical Induction",
"Subset of Empty Set iff Empty",
"Principle of Mathem... |
proofwiki-21195 | Generalized Sum Restricted to Non-zero Summands/Necessary Condition | Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\family{g }_{i \in I}$ be an indexed family of elements of $G$.
Let $J = \set{i \in I : g_i \ne 0_G}$
Let $h \in G$.
Let the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converge to $h$.
Then:
:the generalized sum $\ds \sum_{j \mathop \in J} g... | Let $U \subseteq G$ be an open subset of $G$ such that $h \in U$.
By definition of convergent net:
:$(1) \quad \exists F \subseteq I : F \ne \O : \forall E \subseteq I : E \supseteq F \implies \ds \sum_{i \mathop \in E} g_i \in U$
where $\ds \sum_{i \mathop \in E} g_i$ is the summation over $E$.
Let:
:$F'= F \cap J$
F... | Let $G$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Topological Semigroup|topological semigroup]] with [[Definition:Identity Element|identity]] $0_G$.
Let $\family{g }_{i \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] of $G$.
Let $J = \set{i \in ... | Let $U \subseteq G$ be an [[Definition:Open Set (Topology)|open subset]] of $G$ such that $h \in U$.
By definition of [[Definition:Convergent Net|convergent net]]:
:$(1) \quad \exists F \subseteq I : F \ne \O : \forall E \subseteq I : E \supseteq F \implies \ds \sum_{i \mathop \in E} g_i \in U$
where $\ds \sum_{i \ma... | Generalized Sum Restricted to Non-zero Summands/Necessary Condition | https://proofwiki.org/wiki/Generalized_Sum_Restricted_to_Non-zero_Summands/Necessary_Condition | https://proofwiki.org/wiki/Generalized_Sum_Restricted_to_Non-zero_Summands/Necessary_Condition | [
"Generalized Sum Restricted to Non-zero Summands"
] | [
"Definition:Commutative Semigroup",
"Definition:Topological Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Generalized Sum",
"Definition:Convergent... | [
"Definition:Open Set/Topology",
"Definition:Convergent Net",
"Definition:Summation over Finite Index",
"Set Difference and Intersection form Partition",
"Set Union Preserves Subsets",
"Union with Empty Set",
"Summation over Union of Disjoint Finite Index Sets",
"Definition:Convergent Net"
] |
proofwiki-21196 | Generalized Sum Restricted to Non-zero Summands/Sufficient Condition | Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\family{g }_{i \in I}$ be an indexed family of elements of $G$.
Let $J = \set{i \in I : g_i \ne 0_G}$
Let $h \in G$.
Let the generalized sum $\ds \sum_{j \mathop \in J} g_j$ converge to $h$.
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} g... | Let $U \subseteq G$ be an open subset of $G$ such that $h \in U$.
By definition of convergent net:
:$(2) \quad \exists F' \subseteq J : F' \ne \O : \forall E' \subseteq J : E' \supseteq F' \implies \ds \sum_{j \mathop \in E'} g_j \in U$
where $\ds \sum_{j \mathop \in E'} g_j$ is the summation over $E$.
We have:
:$F' \... | Let $G$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Topological Semigroup|topological semigroup]] with [[Definition:Identity Element|identity]] $0_G$.
Let $\family{g }_{i \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] of $G$.
Let $J = \set{i \in ... | Let $U \subseteq G$ be an [[Definition:Open Set (Topology)|open subset]] of $G$ such that $h \in U$.
By definition of [[Definition:Convergent Net|convergent net]]:
:$(2) \quad \exists F' \subseteq J : F' \ne \O : \forall E' \subseteq J : E' \supseteq F' \implies \ds \sum_{j \mathop \in E'} g_j \in U$
where $\ds \sum_... | Generalized Sum Restricted to Non-zero Summands/Sufficient Condition | https://proofwiki.org/wiki/Generalized_Sum_Restricted_to_Non-zero_Summands/Sufficient_Condition | https://proofwiki.org/wiki/Generalized_Sum_Restricted_to_Non-zero_Summands/Sufficient_Condition | [
"Generalized Sum Restricted to Non-zero Summands"
] | [
"Definition:Commutative Semigroup",
"Definition:Topological Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Generalized Sum",
"Definition:Convergent... | [
"Definition:Open Set/Topology",
"Definition:Convergent Net",
"Definition:Summation over Finite Index",
"Set Intersection Preserves Subsets",
"Intersection with Subset is Subset",
"Set Difference Union Intersection",
"Set Difference and Intersection are Disjoint",
"Union with Empty Set",
"Summation o... |
proofwiki-21197 | Group of Units of Submonoid is Subgroup | Let $T$ be a monoid.
Let $S$ be a submonoid of $T$.
Let $\map G T$ and $\map G S$ be the groups of units of $T$ and $S$ respectively.
Then $\map G S \subseteq \map G T$ and $\map G S$ is a subgroup of $\map G T$. | Let $x \in \map G S$.
Then $x \in S$ and there exists $y \in S$ such that $x y = y x = e$.
Since $S \subseteq T$, we have $y \in T$.
So $x \in \map G T$.
So we have $\map G S \subseteq \map G T$.
From Group of Units is Group, $\map G S$ is a group.
So $\map G S$ is a subgroup of $\map G T$.
{{qed}}
Category:Groups of... | Let $T$ be a [[Definition:Monoid|monoid]].
Let $S$ be a [[Definition:Submonoid|submonoid]] of $T$.
Let $\map G T$ and $\map G S$ be the [[Definition:Group of Units|groups of units]] of $T$ and $S$ respectively.
Then $\map G S \subseteq \map G T$ and $\map G S$ is a [[Definition:Subgroup|subgroup]] of $\map G T$. | Let $x \in \map G S$.
Then $x \in S$ and there exists $y \in S$ such that $x y = y x = e$.
Since $S \subseteq T$, we have $y \in T$.
So $x \in \map G T$.
So we have $\map G S \subseteq \map G T$.
From [[Group of Units is Group]], $\map G S$ is a [[Definition:Group|group]].
So $\map G S$ is a [[Definition:Subgro... | Group of Units of Submonoid is Subgroup | https://proofwiki.org/wiki/Group_of_Units_of_Submonoid_is_Subgroup | https://proofwiki.org/wiki/Group_of_Units_of_Submonoid_is_Subgroup | [
"Groups of Units"
] | [
"Definition:Monoid",
"Definition:Submonoid",
"Definition:Group of Units",
"Definition:Subgroup"
] | [
"Group of Units is Group",
"Definition:Group",
"Definition:Subgroup",
"Category:Groups of Units"
] |
proofwiki-21198 | Spectrum of Element in Unital Subalgebra | Let $A$ be a unital algebra over $\C$.
Let $B$ be a unital subalgebra of $A$.
Let $x \in B$.
Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the spectra of $x$ in $A$ and $B$ respectively.
Then:
:$\map {\sigma_A} x \subseteq \map {\sigma_B} x$ | Let $\map G A$ and $\map G B$ be the group of units of $A$ and $B$ respectively.
From Group of Units of Submonoid is Subgroup, we have:
:$\map G B \subseteq \map G A$
From Set Complement inverts Subsets, we have:
:$A \setminus \map G A \subseteq A \setminus \map G B$
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {\sigm... | Let $A$ be a [[Definition:Unital Algebra|unital algebra]] over $\C$.
Let $B$ be a [[Definition:Unital Subalgebra|unital subalgebra]] of $A$.
Let $x \in B$.
Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the [[Definition:Spectrum (Spectral Theory)|spectra]] of $x$ in $A$ and $B$ respectively.
Then:
:$\map {\s... | Let $\map G A$ and $\map G B$ be the [[Definition:Group of Units|group of units]] of $A$ and $B$ respectively.
From [[Group of Units of Submonoid is Subgroup]], we have:
:$\map G B \subseteq \map G A$
From [[Set Complement inverts Subsets]], we have:
:$A \setminus \map G A \subseteq A \setminus \map G B$
Then, we ha... | Spectrum of Element in Unital Subalgebra | https://proofwiki.org/wiki/Spectrum_of_Element_in_Unital_Subalgebra | https://proofwiki.org/wiki/Spectrum_of_Element_in_Unital_Subalgebra | [
"Spectra (Spectral Theory)",
"Unital Subalgebras",
"Spectrum of Element in Unital Subalgebra"
] | [
"Definition:Unital Algebra",
"Definition:Unital Subalgebra",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Group of Units",
"Group of Units of Submonoid is Subgroup",
"Set Complement inverts Subsets",
"Definition:Unital Subalgebra",
"Category:Spectra (Spectral Theory)",
"Category:Unital Subalgebras",
"Category:Spectrum of Element in Unital Subalgebra"
] |
proofwiki-21199 | Intersection of Subalgebras is Subalgebra | Let $K$ be a field.
Let $\struct {A, +, \cdot, \circ}_K$ be an algebra over $K$.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of subalgebras of $A$.
Let:
:$\ds B = \bigcap_{\alpha \mathop \in I} A_\alpha$
Then $B$ is a subalgebra of $A$. | From Set of Linear Subspaces is Closed under Intersection, $\struct {B, +, \cdot}_K$ is a vector subspace of $\struct {A, +, \cdot}_K$.
Now let $x, y \in B$.
That is, $x, y \in A_\alpha$ for each $\alpha \in I$.
Since $A_\alpha$ is a subalgebras of $A$ for each $\alpha \in I$, we have:
:$x y \in A_\alpha$ for each $\al... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\struct {A, +, \cdot, \circ}_K$ be an [[Definition:Algebra over Field|algebra]] over $K$.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|$I$-indexed family]] of [[Definition:Subalgebra|subalgebras]] of $A$.
Let:
:$\ds... | From [[Set of Linear Subspaces is Closed under Intersection]], $\struct {B, +, \cdot}_K$ is a [[Definition:Vector Subspace|vector subspace]] of $\struct {A, +, \cdot}_K$.
Now let $x, y \in B$.
That is, $x, y \in A_\alpha$ for each $\alpha \in I$.
Since $A_\alpha$ is a [[Definition:Subalgebra|subalgebras]] of $A$ for... | Intersection of Subalgebras is Subalgebra | https://proofwiki.org/wiki/Intersection_of_Subalgebras_is_Subalgebra | https://proofwiki.org/wiki/Intersection_of_Subalgebras_is_Subalgebra | [
"Algebras over Fields",
"Set Intersection"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Indexing Set/Family",
"Definition:Subalgebra",
"Definition:Subalgebra"
] | [
"Set of Linear Subspaces is Closed under Intersection",
"Definition:Vector Subspace",
"Definition:Subalgebra",
"Definition:Subalgebra",
"Category:Algebras over Fields",
"Category:Set Intersection"
] |
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