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 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-18500 | Inner Product is Continuous | Let $\struct {V, \innerprod \cdot \cdot}$ be a inner product space.
Let $\norm \cdot$ be the inner product norm on $V$.
Let $x, y \in V$.
Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be sequences converging in $\struct {V, \norm \cdot}$ to $x$ and $y$ respectively.
Then we have:
... | Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be sequences converging to $x$ and $y$ respectively.
From Modulus of Limit in Normed Vector Space, we have that:
:$\norm {x_n} \to \norm x$
and:
:$\norm {y_n} \to \norm y$
We have:
{{begin-eqn}}
{{eqn | l = \size {\innerprod {x_n} {y_n} ... | Let $\struct {V, \innerprod \cdot \cdot}$ be a [[Definition:Inner Product Space|inner product space]].
Let $\norm \cdot$ be the [[Definition:Inner Product Norm|inner product norm]] on $V$.
Let $x, y \in V$.
Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be [[Definition:Sequence|s... | Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be [[Definition:Sequence|sequences]] [[Definition:Convergent Sequence in Normed Vector Space|converging]] to $x$ and $y$ respectively.
From [[Modulus of Limit/Normed Vector Space|Modulus of Limit in Normed Vector Space]], we have that:
... | Inner Product is Continuous | https://proofwiki.org/wiki/Inner_Product_is_Continuous | https://proofwiki.org/wiki/Inner_Product_is_Continuous | [
"Inner Product Spaces"
] | [
"Definition:Inner Product Space",
"Definition:Inner Product Norm",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Modulus of Limit/Normed Vector Space",
"Triangle Inequality",
"Cauchy-Bunyakovsky-Schwarz Inequality",
"Convergent Real Sequence is Bounded",
"Definition:Bounded Sequence",
"Definition:Positive/Real Number",
"Definition:Co... |
proofwiki-18501 | Convergence of Sequence of Test Functions in Test Function Space implies Convergence in Schwartz Space | Let $\map \DD \R$ be the test function space.
Let $\map \SS \R$ be the Schwartz space.
Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence of test functions in $\map \DD \R$.
Let $\mathbf 0 : \R \to 0$ be the zero mapping.
Suppose $\sequence {\phi_n}$ converges to $\mathbf 0$ in $\map \DD \R$:
:$\phi_n \stackrel ... | For all $n \in \N$ let $\phi_n$ be a test function.
By definition, $\phi_n$ has a compact support $I_n \subset \R$:
:$\forall x \notin I_n \implies \map {\phi_n} x = 0$
Let:
:$a \in \R : a > 0 : \forall n \in \N : I_n \subseteq \closedint {-a} a$
Then:
{{begin-eqn}}
{{eqn | l = \forall m,k \in \N : \sup_{x \mathop \in ... | Let $\map \DD \R$ be the [[Definition:Test Function Space|test function space]].
Let $\map \SS \R$ be the [[Definition:Schwartz Space|Schwartz space]].
Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Test Function|test functions]] in $\map \DD \R$.
Let $\mathbf 0 : \... | For all $n \in \N$ let $\phi_n$ be a [[Definition:Test Function|test function]].
By definition, $\phi_n$ has a [[Definition:Support of Schwartz Distribution|compact support]] $I_n \subset \R$:
:$\forall x \notin I_n \implies \map {\phi_n} x = 0$
Let:
:$a \in \R : a > 0 : \forall n \in \N : I_n \subseteq \closedint ... | Convergence of Sequence of Test Functions in Test Function Space implies Convergence in Schwartz Space | https://proofwiki.org/wiki/Convergence_of_Sequence_of_Test_Functions_in_Test_Function_Space_implies_Convergence_in_Schwartz_Space | https://proofwiki.org/wiki/Convergence_of_Sequence_of_Test_Functions_in_Test_Function_Space_implies_Convergence_in_Schwartz_Space | [
"Test Functions",
"Uniform Convergence",
"Convergence"
] | [
"Definition:Test Function Space",
"Definition:Schwartz Space",
"Definition:Sequence",
"Definition:Test Function",
"Definition:Zero Mapping",
"Definition:Convergent Sequence/Test Function Space",
"Definition:Zero-Limit Sequence in Schwartz Space"
] | [
"Definition:Test Function",
"Definition:Support of Schwartz Distribution",
"Absolute Value Function is Completely Multiplicative",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Uniform Convergence/Real Sequence",
"Definition:Zero-Limit Sequence in Schwartz Space"
] |
proofwiki-18502 | Null Ring is Commutative Ring | Let $R$ be the null ring.
That is, let:
:$R := \struct {\set {0_R}, +, \circ}$
where ring addition and ring product are defined as:
{{begin-eqn}}
{{eqn | l = 0_R + 0_R
| r = 0_R
| c =
}}
{{eqn | l = 0_R \circ 0_R
| r = 0_R
| c =
}}
{{end-eqn}}
Then $R$ is a commutative ring. | From Null Ring is Trivial Ring, we have that $R$ is a trivial ring.
The result follows from Trivial Ring is Commutative Ring.
{{qed}} | Let $R$ be the [[Definition:Null Ring|null ring]].
That is, let:
:$R := \struct {\set {0_R}, +, \circ}$
where [[Definition:Ring Addition|ring addition]] and [[Definition:Ring Product|ring product]] are defined as:
{{begin-eqn}}
{{eqn | l = 0_R + 0_R
| r = 0_R
| c =
}}
{{eqn | l = 0_R \circ 0_R
| r ... | From [[Null Ring is Trivial Ring]], we have that $R$ is a [[Definition:Trivial Ring|trivial ring]].
The result follows from [[Trivial Ring is Commutative Ring]].
{{qed}} | Null Ring is Commutative Ring | https://proofwiki.org/wiki/Null_Ring_is_Commutative_Ring | https://proofwiki.org/wiki/Null_Ring_is_Commutative_Ring | [
"Null Ring",
"Commutative Rings"
] | [
"Definition:Null Ring",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Commutative Ring"
] | [
"Null Ring is Trivial Ring",
"Definition:Trivial Ring",
"Trivial Ring is Commutative Ring"
] |
proofwiki-18503 | Orthogonal Projection is Linear Transformation | Let $\GF \in \set {\R, \C}$.
{{explain|Does this hold for all subfields of $\C$?}}
Let $H$ be a Hilbert space over $\mathbb F$ with inner product $\innerprod \cdot \cdot$.
Let $K$ be a closed linear subspace of $H$.
Let $P_K$ denote the orthogonal projection on $K$.
Then $P_K$ is a linear transformation on $H$. | Let $x, y \in H$.
Let $\alpha, \beta \in \GF$.
Let $k \in \KK$.
Since the inner product is linear in its first argument, we have:
:$\innerprod {\paren {\alpha x + \beta y} - \paren {\alpha \map {P_K} x + \beta \map {P_K} y} } k = \alpha \innerprod {x - \map {P_K} x} k + \beta \innerprod {y - \map {P_K} y} k$
From U... | Let $\GF \in \set {\R, \C}$.
{{explain|Does this hold for all subfields of $\C$?}}
Let $H$ be a [[Definition:Hilbert Space|Hilbert space]] over $\mathbb F$ with [[Definition:Inner Product|inner product]] $\innerprod \cdot \cdot$.
Let $K$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $H$.
Let ... | Let $x, y \in H$.
Let $\alpha, \beta \in \GF$.
Let $k \in \KK$.
Since the [[Definition:Inner Product|inner product]] is [[Definition:Linear Mapping|linear]] in its first argument, we have:
:$\innerprod {\paren {\alpha x + \beta y} - \paren {\alpha \map {P_K} x + \beta \map {P_K} y} } k = \alpha \innerprod {x -... | Orthogonal Projection is Linear Transformation | https://proofwiki.org/wiki/Orthogonal_Projection_is_Linear_Transformation | https://proofwiki.org/wiki/Orthogonal_Projection_is_Linear_Transformation | [
"Orthogonal Projections"
] | [
"Definition:Hilbert Space",
"Definition:Inner Product",
"Definition:Closed Linear Subspace",
"Definition:Orthogonal Projection",
"Definition:Linear Transformation"
] | [
"Definition:Inner Product",
"Definition:Linear Transformation",
"Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space",
"Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space",
"Definition:Unique",
"Definition:Linear Transformation"
] |
proofwiki-18504 | Orthogonal Projection is Bounded | Let $H$ be a Hilbert space with inner product $\innerprod \cdot \cdot$ and inner product norm $\norm \cdot$.
Let $K$ be a closed linear subspace of $H$.
Let $P_K$ denote the orthogonal projection on $K$.
Then $P_K$ is bounded.
That is:
:$\norm {\map {P_K} h} \le \norm h$
for each $h \in H$. | Let $h \in H$.
Note that we can write:
:$h = \paren {h - \map {P_K} h} + \map {P_K} h$
We have, by the definition of orthogonal projection:
:$\map {P_K} h \in K$
From Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space, we have:
:$h - \map {P_K} h \in K^\bot$
so that:
:$\innerprod {\map {P_... | Let $H$ be a [[Definition:Hilbert Space|Hilbert space]] with [[Definition:Inner Product|inner product]] $\innerprod \cdot \cdot$ and [[Definition:Inner Product Norm|inner product norm]] $\norm \cdot$.
Let $K$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $H$.
Let $P_K$ denote the [[Definition:O... | Let $h \in H$.
Note that we can write:
:$h = \paren {h - \map {P_K} h} + \map {P_K} h$
We have, by the definition of [[Definition:Orthogonal Projection|orthogonal projection]]:
:$\map {P_K} h \in K$
From [[Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space]], we have:
:$h - \map {P_K} ... | Orthogonal Projection is Bounded | https://proofwiki.org/wiki/Orthogonal_Projection_is_Bounded | https://proofwiki.org/wiki/Orthogonal_Projection_is_Bounded | [
"Orthogonal Projections"
] | [
"Definition:Hilbert Space",
"Definition:Inner Product",
"Definition:Inner Product Norm",
"Definition:Closed Linear Subspace",
"Definition:Orthogonal Projection",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Orthogonal Projection",
"Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space",
"Pythagoras's Theorem (Inner Product Space)"
] |
proofwiki-18505 | Orthogonal Projection is Projection | Let $\HH$ be a Hilbert space.
Let $K$ be a closed linear subspace of $H$.
Let $P_K$ denote the orthogonal projection onto $K$.
Then $P_K$ is a projection. | Let $h \in H$.
From the definition of the orthogonal projection, we have:
:$\map {P_K} h \in K$
So, from Fixed Points of Orthogonal Projection, we have:
:$\map {P_K^2} h = \map {P_K} {\map {P_K} h} = \map {P_K} h$
Since $h$ was arbitrary, we have:
:$P_K^2 = P_K$
So $P_K$ is an idempotent.
Further, from Kernel of Ortho... | Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $K$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $H$.
Let $P_K$ denote the [[Definition:Orthogonal Projection|orthogonal projection]] onto $K$.
Then $P_K$ is a [[Definition:Projection (Hilbert Spaces)|projection]]. | Let $h \in H$.
From the definition of the [[Definition:Orthogonal Projection|orthogonal projection]], we have:
:$\map {P_K} h \in K$
So, from [[Fixed Points of Orthogonal Projection]], we have:
:$\map {P_K^2} h = \map {P_K} {\map {P_K} h} = \map {P_K} h$
Since $h$ was arbitrary, we have:
:$P_K^2 = P_K$
So $P_K$... | Orthogonal Projection is Projection | https://proofwiki.org/wiki/Orthogonal_Projection_is_Projection | https://proofwiki.org/wiki/Orthogonal_Projection_is_Projection | [
"Orthogonal Projections"
] | [
"Definition:Hilbert Space",
"Definition:Closed Linear Subspace",
"Definition:Orthogonal Projection",
"Definition:Projection (Hilbert Spaces)"
] | [
"Definition:Orthogonal Projection",
"Fixed Points of Orthogonal Projection",
"Definition:Idempotent Operator",
"Kernel of Orthogonal Projection",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement",
"Range of Orthogonal Projection",
"Definition:Projection (Hilbert Spaces)"
] |
proofwiki-18506 | Kernel of Orthogonal Projection | Let $H$ be a Hilbert space.
Let $K$ be a closed linear subspace of $H$.
Let $P_K$ denote the orthogonal projection on $K$.
Then:
:$\ker P_K = K^\bot$
where:
:$\ker P_K$ denotes the kernel of $P_K$
:$K^\bot$ denotes the orthocomplement of $K$. | We first prove that:
:$\ker P_K \subseteq K^\bot$
Let $h \in \ker P_K$.
Then:
:$\map {P_K} h = 0$
From Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space, we have:
:$h - \map {P_K} h \in K^\bot$
That is:
:$h \in K^\bot$
So:
:$\ker P_K \subseteq K^\bot$
We now prove that:
:$K^\bot \subseteq ... | Let $H$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $K$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $H$.
Let $P_K$ denote the [[Definition:Orthogonal Projection|orthogonal projection]] on $K$.
Then:
:$\ker P_K = K^\bot$
where:
:$\ker P_K$ denotes the [[Definition:Kernel|kernel]]... | We first prove that:
:$\ker P_K \subseteq K^\bot$
Let $h \in \ker P_K$.
Then:
:$\map {P_K} h = 0$
From [[Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space]], we have:
:$h - \map {P_K} h \in K^\bot$
That is:
:$h \in K^\bot$
So:
:$\ker P_K \subseteq K^\bot$
We now prove that:
:$... | Kernel of Orthogonal Projection | https://proofwiki.org/wiki/Kernel_of_Orthogonal_Projection | https://proofwiki.org/wiki/Kernel_of_Orthogonal_Projection | [
"Orthogonal Projections"
] | [
"Definition:Hilbert Space",
"Definition:Closed Linear Subspace",
"Definition:Orthogonal Projection",
"Definition:Kernel",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement"
] | [
"Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space",
"Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space"
] |
proofwiki-18507 | Range of Orthogonal Projection | Let $H$ be a Hilbert space.
Let $K$ be a closed linear subspace of $H$.
Let $P_K$ denote the orthogonal projection on $K$.
Then:
:$P_K \sqbrk H = K$
where $P_K \sqbrk H$ denotes the image of $H$ under $P_K$. | We first show that $P_K \sqbrk H \subseteq K$.
Let $k \in P_K \sqbrk H$.
Then there exists $h \in H$ such that:
:$\map {P_K} h = k$
From the definition of the orthogonal projection, we have:
:$\map {P_K} h \in K$
so:
:$h \in K$
giving:
:$P_K \sqbrk H \subseteq K$
{{qed|lemma}}
We now show that:
:$K \subseteq P_K \... | Let $H$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $K$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $H$.
Let $P_K$ denote the [[Definition:Orthogonal Projection|orthogonal projection]] on $K$.
Then:
:$P_K \sqbrk H = K$
where $P_K \sqbrk H$ denotes the [[Definition:Image of Mappin... | We first show that $P_K \sqbrk H \subseteq K$.
Let $k \in P_K \sqbrk H$.
Then there exists $h \in H$ such that:
:$\map {P_K} h = k$
From the definition of the [[Definition:Orthogonal Projection|orthogonal projection]], we have:
:$\map {P_K} h \in K$
so:
:$h \in K$
giving:
:$P_K \sqbrk H \subseteq K$
{{qed|... | Range of Orthogonal Projection | https://proofwiki.org/wiki/Range_of_Orthogonal_Projection | https://proofwiki.org/wiki/Range_of_Orthogonal_Projection | [
"Orthogonal Projections"
] | [
"Definition:Hilbert Space",
"Definition:Closed Linear Subspace",
"Definition:Orthogonal Projection",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Orthogonal Projection",
"Fixed Points of Orthogonal Projection",
"Definition:Set Equality"
] |
proofwiki-18508 | Fixed Points of Orthogonal Projection | Let $\struct {H, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $\norm \cdot$ be the inner product norm of $H$.
Let $K$ be a closed linear subspace of $H$.
Let $P_K$ denote the orthogonal projection on $K$.
Let $h \in H$.
Then:
:$\map {P_K} h = h$
{{iff}} $h \in K$. | Let $d$ be the metric induced by $\norm \cdot$.
Let $h \in H$.
By the definition of orthogonal projection, we have:
:$\map d {h, \map {P_K} h} = \map d {h, K}$
Note that by the definition of a metric, we have that:
:$\map d {h, \map {P_K} h} = 0$
{{iff}}:
:$h = \map {P_K} h$
So, we have:
:$h = \map {P_K} h$
{{iff}}... | Let $\struct {H, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $\norm \cdot$ be the [[Definition:Inner Product|inner product norm]] of $H$.
Let $K$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $H$.
Let $P_K$ denote the [[Definition:Orthogonal Projection|orthogo... | Let $d$ be the [[Definition:Metric|metric]] [[Definition:Metric Induced by Norm|induced]] by $\norm \cdot$.
Let $h \in H$.
By the definition of [[Definition:Orthogonal Projection|orthogonal projection]], we have:
:$\map d {h, \map {P_K} h} = \map d {h, K}$
Note that by the definition of a [[Definition:Metric Spac... | Fixed Points of Orthogonal Projection | https://proofwiki.org/wiki/Fixed_Points_of_Orthogonal_Projection | https://proofwiki.org/wiki/Fixed_Points_of_Orthogonal_Projection | [
"Orthogonal Projections"
] | [
"Definition:Hilbert Space",
"Definition:Inner Product",
"Definition:Closed Linear Subspace",
"Definition:Orthogonal Projection"
] | [
"Definition:Metric Space/Metric",
"Definition:Metric Induced by Norm",
"Definition:Orthogonal Projection",
"Definition:Metric Space",
"Definition:Closed Set/Topology",
"Subset of Metric Space is Closed iff contains all Zero Distance Points",
"Category:Orthogonal Projections"
] |
proofwiki-18509 | Sum of Cosets of Ideals is Sum in Quotient Ring | The sum $X +_\PP Y$ in $\powerset R$ is also their sum in the quotient ring $R / J$. | As $\struct {R, +, \circ}$ is a ring, it follows that $\struct {R, +}$ is an abelian group.
Thus by Subgroup of Abelian Group is Normal, all subgroups of $\struct {R, +, \circ}$ are normal.
So from the definition of quotient group, it follows directly that $X +_\PP Y$ in $\powerset R$ is also the sum in the quotient ri... | The [[Definition:Subset Product|sum]] $X +_\PP Y$ in $\powerset R$ is also their [[Definition:Subset Product|sum]] in the [[Definition:Quotient Ring|quotient ring]] $R / J$. | As $\struct {R, +, \circ}$ is a [[Definition:Ring (Abstract Algebra)|ring]], it follows that $\struct {R, +}$ is an [[Definition:Abelian Group|abelian group]].
Thus by [[Subgroup of Abelian Group is Normal]], all [[Definition:Subgroup|subgroups]] of $\struct {R, +, \circ}$ are [[Definition:Normal Subgroup|normal]].
S... | Sum of Cosets of Ideals is Sum in Quotient Ring | https://proofwiki.org/wiki/Sum_of_Cosets_of_Ideals_is_Sum_in_Quotient_Ring | https://proofwiki.org/wiki/Sum_of_Cosets_of_Ideals_is_Sum_in_Quotient_Ring | [
"Ring Operations on Coset Space of Ideal"
] | [
"Definition:Subset Product",
"Definition:Subset Product",
"Definition:Quotient Ring"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Abelian Group",
"Subgroup of Abelian Group is Normal",
"Definition:Subgroup",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Quotient Ring"
] |
proofwiki-18510 | Principal Ideal is Smallest Ideal | Let $\struct {R, +, \circ}$ be a ring with unity.
Let $a \in R$.
Let $\ideal a$ be the principal ideal of $R$ generated by $a$.
Let $J$ be an ideal of $R$ such that $a \in J$.
Then $\ideal a \subseteq J$.
That is, $\ideal a$ is the smallest ideal of $R$ to which $a$ belongs. | Let $J$ be an ideal of $R$ such that $a \in J$.
By the definition of an ideal:
:$\forall r, s \in R: r \circ a \circ s \in J$
Also, $J$ is a group under $+$.
So every element of $\ideal a$ is in $J$.
Thus $\ideal a \subseteq J$.
{{qed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $a \in R$.
Let $\ideal a$ be the [[Definition:Principal Ideal of Ring|principal ideal]] of $R$ generated by $a$.
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$ such that $a \in J$.
Then $\ideal a \subseteq J$.
That is, $... | Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$ such that $a \in J$.
By the definition of an [[Definition:Ideal of Ring|ideal]]:
:$\forall r, s \in R: r \circ a \circ s \in J$
Also, $J$ is a [[Definition:Group|group]] under $+$.
So every [[Definition:Element|element]] of $\ideal a$ is in $J$.
Thus $\ideal a... | Principal Ideal is Smallest Ideal | https://proofwiki.org/wiki/Principal_Ideal_is_Smallest_Ideal | https://proofwiki.org/wiki/Principal_Ideal_is_Smallest_Ideal | [
"Principal Ideals of Rings"
] | [
"Definition:Ring with Unity",
"Definition:Principal Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Smallest Set by Set Inclusion"
] | [
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Group",
"Definition:Element"
] |
proofwiki-18511 | Prime Element of Integral Domain is Irreducible | Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$.
Let $p$ be a prime element of $\struct {D, +, \circ}$.
Then $p$ is an irreducible element of $\struct {D, +, \circ}$. | By definition of prime element, $p$ is neither zero nor a unit of $\struct {D, +, \circ}$.
{{AimForCont}}:
:$p = a \circ b$
for some non-units $a, b \in D$.
From Element of Integral Domain is Divisor of Itself:
:$p \divides a \circ b$
By definition of prime element:
:$p \divides a$ or $p \divides b$
{{WLOG}}, suppose $... | Let $\struct {D, +, \circ}$ be an [[Definition:Integral Domain|integral domain]] whose [[Definition:Unity of Ring|unity]] is $1_D$.
Let $p$ be a [[Definition:Prime Element of Ring|prime element]] of $\struct {D, +, \circ}$.
Then $p$ is an [[Definition:Irreducible Element of Ring|irreducible element]] of $\struct {D, ... | By definition of [[Definition:Prime Element of Ring|prime element]], $p$ is neither [[Definition:Ring Zero|zero]] nor a [[Definition:Unit of Ring|unit]] of $\struct {D, +, \circ}$.
{{AimForCont}}:
:$p = a \circ b$
for some non-[[Definition:Unit of Ring|units]] $a, b \in D$.
From [[Element of Integral Domain is Diviso... | Prime Element of Integral Domain is Irreducible | https://proofwiki.org/wiki/Prime_Element_of_Integral_Domain_is_Irreducible | https://proofwiki.org/wiki/Prime_Element_of_Integral_Domain_is_Irreducible | [
"Integral Domains",
"Irreducible Elements of Rings",
"Prime Elements of Rings"
] | [
"Definition:Integral Domain",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Prime Element of Ring",
"Definition:Irreducible Element of Ring"
] | [
"Definition:Prime Element of Ring",
"Definition:Ring Zero",
"Definition:Unit of Ring",
"Definition:Unit of Ring",
"Element of Integral Domain is Divisor of Itself",
"Definition:Prime Element of Ring",
"Cancellation Law for Ring Product of Integral Domain",
"Definition:Unit of Ring",
"Definition:Cont... |
proofwiki-18512 | Functor Category is Category | Let $\mathbf C$ and $\mathbf D$ be categories.
Then the functor category $\map {\operatorname {Funct} } {\mathbf C, \mathbf D}$ is a metacategory. | We check the metacategory axioms. | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Then the [[Definition:Functor Category|functor category]] $\map {\operatorname {Funct} } {\mathbf C, \mathbf D}$ is a [[Definition:Metacategory|metacategory]]. | We check the [[Definition:Metacategory|metacategory axioms]]. | Functor Category is Category | https://proofwiki.org/wiki/Functor_Category_is_Category | https://proofwiki.org/wiki/Functor_Category_is_Category | [
"Functor Categories"
] | [
"Definition:Category",
"Definition:Functor Category",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Metacategory"
] |
proofwiki-18513 | Equivalence of Definitions of Projective Module | Let $A$ be a ring.
Let $M$ be an $A$-module.
The following are equivalent:
:$(1): \quad$ $M$ is a projective module, that is, $M$ is a projective object in the category of left $A$-modules.
:$(2): \quad$ $M$ is a direct summand of a free module.
:$(3): \quad$ Every short exact sequence of the form:
::<nowiki>$\xymatrix... | === $(1)$ implies $(2)$ ===
By Surjection by Free Module there is a free module $Y$ and a surjection $g : Y \to M$.
By Epimorphism of modules iff surjective $g$ is an epimorphism.
By Definition:Projective Object applied to $\operatorname {id}_M$, there is a homomorphism $s : M \to Y$ with $g \circ s = \operatorname {id... | Let $A$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $M$ be an [[Definition:Module over Ring|$A$-module]].
The following are equivalent:
:$(1): \quad$ $M$ is a [[Definition:Projective Module|projective module]], that is, $M$ is a [[Definition:Projective Object|projective object]] in the category of left $A$... | === $(1)$ implies $(2)$ ===
By [[Surjection by Free Module]] there is a free [[Definition:Module over Ring|module]] $Y$ and a [[Definition:Surjection|surjection]] $g : Y \to M$.
By [[Epimorphism of modules iff surjective]] $g$ is an [[Definition:Epimorphism (Category Theory)|epimorphism]].
By [[Definition:Projective... | Equivalence of Definitions of Projective Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Projective_Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Projective_Module | [
"Projective Modules"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Module over Ring",
"Definition:Projective Module",
"Definition:Projective Object",
"Definition:Internal Direct Sum of Rings/Direct Summand",
"Definition:Free Module over Ring",
"Definition:Short Exact Sequence of Modules",
"Definition:Split Short Exact... | [
"Surjection by Free Module",
"Definition:Module over Ring",
"Definition:Surjection",
"Epimorphism of modules iff surjective",
"Definition:Epimorphism (Category Theory)",
"Definition:Projective Object",
"Definition:Module over Ring",
"Epimorphism of modules iff surjective",
"Definition:Module over Ri... |
proofwiki-18514 | Tensor Product of Projective Modules is Projective | Let $A$ be a commutative ring with unity.
Let $P$ and $Q$ be projective $A$-modules.
Then the tensor product $P \otimes_A Q$ is a projective $A$-module. | By Projective iff Direct Summand of Free Module, there exist $A$-modules $P'$ and $Q'$, such that $P \oplus P'$ and $Q \oplus Q'$ are free.
By Tensor Product Distributes over Direct Sum, there is an isomorphism:
:$\paren {P \oplus P'} \otimes_A \paren {Q \oplus Q'} \cong \paren {P \otimes_A Q} \oplus \paren {P' \otimes... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $P$ and $Q$ be [[Definition:Projective Module|projective]] [[Definition:Module over Ring|$A$-modules]].
Then the [[Definition:Tensor Product of Modules|tensor product]] $P \otimes_A Q$ is a [[Definition:Projective Module|project... | By [[Projective iff Direct Summand of Free Module]], there [[Definition:Existential Quantifier|exist]] [[Definition:Module over Ring|$A$-modules]] $P'$ and $Q'$, such that $P \oplus P'$ and $Q \oplus Q'$ are [[Definition:Free Module over Ring|free]].
By [[Tensor Product Distributes over Direct Sum]], there is an [[Def... | Tensor Product of Projective Modules is Projective | https://proofwiki.org/wiki/Tensor_Product_of_Projective_Modules_is_Projective | https://proofwiki.org/wiki/Tensor_Product_of_Projective_Modules_is_Projective | [
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Projective Module",
"Definition:Module over Ring",
"Definition:Tensor Product of Modules",
"Definition:Projective Module",
"Definition:Module over Ring"
] | [
"Projective iff Direct Summand of Free Module",
"Definition:Existential Quantifier",
"Definition:Module over Ring",
"Definition:Free Module over Ring",
"Tensor Product Distributes over Direct Sum",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Module Isomorphism",
"Tensor ... |
proofwiki-18515 | Surjection by Free Module | Let $A$ be a ring.
Let $M$ be a left $A$-module.
Then there exists a free $A$-module $F$ and a surjective $A$-module homomorphism $f : F \to M$. | Let $F = A^{\paren M}$ be the free $A$-module on the set $M$.
Let $c : M \to A^{\paren M}$ be the canonical mapping on $F$.
Let $f : F \to M$ be the $A$-module homomorphism induced the by the Universal Property of Free Modules applied to the identity $\operatorname {id}_M$ of $M$.
We have:
:$f \circ c = \operatorname {... | Let $A$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $M$ be a [[Definition:Left Module over Ring|left $A$-module]].
Then there [[Definition:Existential Quantifier|exists]] a [[Definition:Free Module over Ring|free]] [[Definition:Left Module over Ring|$A$-module]] $F$ and a [[Definition:Surjection|surjectiv... | Let $F = A^{\paren M}$ be the [[Definition:Free Module on Set|free $A$-module on the set]] $M$.
Let $c : M \to A^{\paren M}$ be the [[Definition:Canonical Mapping on Free Module on Set|canonical mapping]] on $F$.
Let $f : F \to M$ be the $A$-[[Definition:Module Homomorphism|module homomorphism]] induced the by the [[... | Surjection by Free Module | https://proofwiki.org/wiki/Surjection_by_Free_Module | https://proofwiki.org/wiki/Surjection_by_Free_Module | [
"Homological Algebra"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Left Module over Ring",
"Definition:Existential Quantifier",
"Definition:Free Module over Ring",
"Definition:Left Module over Ring",
"Definition:Surjection",
"Definition:Linear Transformation"
] | [
"Definition:Free Module on Set",
"Definition:Canonical Mapping on Free Module on Set",
"Definition:Linear Transformation",
"Universal Property of Free Modules",
"Definition:Identity Morphism",
"Definition:Split Epimorphism",
"Definition:Category of Sets",
"Split Epimorphism is Epic",
"Surjection iff... |
proofwiki-18516 | Category of Modules has Enough Projectives | Let $A$ be a ring.
Then the category of left $A$-modules has enough projectives. | Let $M$ be an $A$-module.
By Surjection by Free Module there is a free $A$-module $F$ and a surjection $f : F \to M$.
By Epimorphism of Modules Iff Surjection $f$ is an epimorphism.
By Free Module is Projective $F$ is projective.
{{qed}}
Category:Homological Algebra
orzs0y7ub10gsga3bmaiubjagtcf5f0 | Let $A$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Then the [[Definition:Category of Modules|category of left $A$-modules]] has [[Definition:Enough Projectives|enough projectives]]. | Let $M$ be an [[Definition:Module over Ring|$A$-module]].
By [[Surjection by Free Module]] there is a [[Definition:Free Module over Ring|free $A$-module]] $F$ and a [[Definition:Surjection|surjection]] $f : F \to M$.
By [[Epimorphism of Modules Iff Surjection]] $f$ is an [[Definition:Epimorphism (Category Theory)|epi... | Category of Modules has Enough Projectives | https://proofwiki.org/wiki/Category_of_Modules_has_Enough_Projectives | https://proofwiki.org/wiki/Category_of_Modules_has_Enough_Projectives | [
"Homological Algebra"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Category of Modules",
"Definition:Enough Projectives"
] | [
"Definition:Module over Ring",
"Surjection by Free Module",
"Definition:Free Module over Ring",
"Definition:Surjection",
"Epimorphism of Modules Iff Surjection",
"Definition:Epimorphism (Category Theory)",
"Free Module is Projective",
"Definition:Projective Module",
"Category:Homological Algebra"
] |
proofwiki-18517 | Tensor Product of Free Modules is Free | Let $A$ be a commutative ring with unity.
Let $F$ and $F'$ be free $A$-modules.
Then the tensor product $F \otimes_A F'$ is a free $A$-module. | By Free Module is Isomorphic to Free Module on Set there are sets $I$ and $I'$ and isomorphisms $\Psi : A^{\paren I} \to F$ and $\Psi' : A^{\paren {I'} } \to F'$.
By Tensor Product Distributes over Direct Sum, there is an isomorphism:
:$\ds A^{\paren I} \otimes_A A^{\paren {I'} } \cong \bigoplus_{i \mathop \in I} \bigo... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $F$ and $F'$ be [[Definition:Free Module over Ring|free $A$-modules]].
Then the [[Definition:Tensor Product of Modules|tensor product]] $F \otimes_A F'$ is a [[Definition:Free Module over Ring|free $A$-module]]. | By [[Free Module is Isomorphic to Free Module on Set]] there are sets $I$ and $I'$ and isomorphisms $\Psi : A^{\paren I} \to F$ and $\Psi' : A^{\paren {I'} } \to F'$.
By [[Tensor Product Distributes over Direct Sum]], there is an [[Definition:Module Isomorphism|isomorphism]]:
:$\ds A^{\paren I} \otimes_A A^{\paren {I'... | Tensor Product of Free Modules is Free | https://proofwiki.org/wiki/Tensor_Product_of_Free_Modules_is_Free | https://proofwiki.org/wiki/Tensor_Product_of_Free_Modules_is_Free | [
"Commutative Algebra",
"Free Modules",
"Homological Algebra",
"Module Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Free Module over Ring",
"Definition:Tensor Product of Modules",
"Definition:Free Module over Ring"
] | [
"Free Module is Isomorphic to Free Module on Set",
"Tensor Product Distributes over Direct Sum",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Module Isomorphism",
"Direct Sum of Direct Sums is Direct Sum",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomo... |
proofwiki-18518 | Equivalence of Definitions of Kernel of Morphism | Let $\mathbf C$ be a category with zero object $0$.
Let $f : A \to B$ be a morphism in $\mathbf C$.
Then the following definitions of kernel of $f$ are equivalent. | === Definition 1 implies Definition 2 ===
By definition of Zero Object, $0$ is an initial object, so Definition 1 is possible.
Let $f : A \to B$ be a morphism in $\mathbf C$.
Let $k : K \to A$ be a pullback of $f$ along the zero morphism $0 : 0 \to B$.
We check the universal property of the equalizer of $f$ and the zer... | Let $\mathbf C$ be a [[Definition:Category|category]] with [[Definition:Zero Object|zero object]] $0$.
Let $f : A \to B$ be a [[Definition:Morphism|morphism]] in $\mathbf C$.
Then the following definitions of [[Definition:Kernel (Category Theory)|kernel]] of $f$ are equivalent. | === [[Definition:Kernel (Category Theory)/Definition 1|Definition 1]] implies [[Definition:Kernel (Category Theory)/Definition 2|Definition 2]] ===
By definition of [[Definition:Zero Object|Zero Object]], $0$ is an [[Definition:Initial Object|initial object]], so Definition 1 is possible.
Let $f : A \to B$ be a [[Def... | Equivalence of Definitions of Kernel of Morphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Kernel_of_Morphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Kernel_of_Morphism | [
"Morphisms"
] | [
"Definition:Category",
"Definition:Zero Object",
"Definition:Morphism",
"Definition:Kernel (Category Theory)"
] | [
"Definition:Kernel (Category Theory)/Definition 1",
"Definition:Kernel (Category Theory)/Definition 2",
"Definition:Zero Object",
"Definition:Initial Object",
"Definition:Morphism",
"Definition:Pullback (Category Theory)",
"Definition:Zero Morphism via Zero Object",
"Definition:Equalizer",
"Definiti... |
proofwiki-18519 | Zero Morphism does not Depend on Zero Object | Let $\mathbf C$ be a category.
Let $A$ and $B$ be objects of $\mathbf C$.
Let $0_1$ and $0_2$ be zero objects of $\mathbf C$.
Then the morphism defined as the composition
:$\beta \circ \alpha : A \to 0_1 \to B$
of the unique morphism $\alpha : A \to 0_1$ and the unique morphism $\beta : 0_1 \to B$ is equal to the mor... | There are unique morphisms $\epsilon : 0_1 \to 0_2$ and $\zeta : 0_2 \to 0_1$.
Since $0_1$ is terminal, we have
: $\zeta \circ \epsilon = \operatorname{id}_{0_1}$
: $\beta \circ \zeta = \delta$
Since $0_2$ is terminal, we have
: $\epsilon \circ \alpha = \gamma$
Hence
{{begin-eqn}}
{{eqn | l = \beta \circ \alpha
|... | Let $\mathbf C$ be a [[Definition:Category|category]].
Let $A$ and $B$ be [[Definition:Object (Category Theory)|objects]] of $\mathbf C$.
Let $0_1$ and $0_2$ be [[Definition:Zero Object|zero objects]] of $\mathbf C$.
Then the [[Definition:Morphism|morphism]] defined as the composition
:$\beta \circ \alpha : A \to... | There are [[Definition:Unique|unique]] [[Definition:Morphism|morphisms]] $\epsilon : 0_1 \to 0_2$ and $\zeta : 0_2 \to 0_1$.
Since $0_1$ is [[Definition:Terminal Object|terminal]], we have
: $\zeta \circ \epsilon = \operatorname{id}_{0_1}$
: $\beta \circ \zeta = \delta$
Since $0_2$ is [[Definition:Terminal Object|ter... | Zero Morphism does not Depend on Zero Object | https://proofwiki.org/wiki/Zero_Morphism_does_not_Depend_on_Zero_Object | https://proofwiki.org/wiki/Zero_Morphism_does_not_Depend_on_Zero_Object | [] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Zero Object",
"Definition:Morphism",
"Definition:Unique",
"Definition:Morphism",
"Definition:Unique",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Unique",
"Definition:Morphism",
"Definition:Unique",
"Defi... | [
"Definition:Unique",
"Definition:Morphism",
"Definition:Terminal Object",
"Definition:Terminal Object"
] |
proofwiki-18520 | Freyd-Mitchell Embedding Theorem | Let $\AA$ be a small abelian category.
Then there exists a ring with unity $R$ and a fully faithful and exact functor $F : \AA \to R \text{-} \mathbf{Mod}$ to the category of left $R$-modules. | {{ProofWanted}}
{{Namedfor|Peter John Freyd|name2 = Barry M. Mitchell|cat = Mitchell|cat2 = Freyd}} | Let $\AA$ be a [[Definition:Small Category|small]] [[Definition:Abelian Category|abelian category]].
Then there [[Definition:Existential Quantifier|exists]] a [[Definition:Ring with Unity|ring with unity]] $R$ and a [[Definition:Full Functor|fully]] [[Definition:Faithful Functor|faithful]] and [[Definition:Exact Fun... | {{ProofWanted}}
{{Namedfor|Peter John Freyd|name2 = Barry M. Mitchell|cat = Mitchell|cat2 = Freyd}} | Freyd-Mitchell Embedding Theorem | https://proofwiki.org/wiki/Freyd-Mitchell_Embedding_Theorem | https://proofwiki.org/wiki/Freyd-Mitchell_Embedding_Theorem | [
"Category Theory",
"Homological Algebra"
] | [
"Definition:Small Category",
"Definition:Abelian Category",
"Definition:Existential Quantifier",
"Definition:Ring with Unity",
"Definition:Full Functor",
"Definition:Faithful Functor",
"Definition:Exact Functor",
"Definition:Category of Modules"
] | [] |
proofwiki-18521 | Eigenvalues of Hermitian Operator have Orthogonal Eigenspaces | The eigenvectors of a Hermitian operator have eigenspaces which are orthogonal. | Directly follows from Hermitian Operator is Normal and Eigenvalues of Normal Operator have Orthogonal Eigenspaces.
{{qed}}
Category:Linear Algebra
Category:Linear Operators
Category:Linear Transformations on Hilbert Spaces
2m73pteuz0snmf9xolvtxhj5n6jm72n | The [[Definition:Eigenvector of Linear Operator|eigenvectors]] of a [[Definition:Hermitian Operator|Hermitian operator]] have [[Definition:Eigenspace of Linear Operator|eigenspaces]] which are [[Definition:Orthogonal Sets|orthogonal]]. | Directly follows from [[Hermitian Operator is Normal]] and [[Eigenvalues of Normal Operator have Orthogonal Eigenspaces]].
{{qed}}
[[Category:Linear Algebra]]
[[Category:Linear Operators]]
[[Category:Linear Transformations on Hilbert Spaces]]
2m73pteuz0snmf9xolvtxhj5n6jm72n | Eigenvalues of Hermitian Operator have Orthogonal Eigenspaces | https://proofwiki.org/wiki/Eigenvalues_of_Hermitian_Operator_have_Orthogonal_Eigenspaces | https://proofwiki.org/wiki/Eigenvalues_of_Hermitian_Operator_have_Orthogonal_Eigenspaces | [
"Linear Algebra",
"Linear Operators",
"Linear Transformations on Hilbert Spaces"
] | [
"Definition:Eigenvector/Linear Operator",
"Definition:Hermitian Operator",
"Definition:Eigenspace/Linear Operator",
"Definition:Orthogonal (Linear Algebra)/Sets"
] | [
"Hermitian Operator is Normal",
"Eigenvalues of Normal Operator have Orthogonal Eigenspaces",
"Category:Linear Algebra",
"Category:Linear Operators",
"Category:Linear Transformations on Hilbert Spaces"
] |
proofwiki-18522 | Top is Complete | The category of topological spaces is complete. | Let $\II$ be a small category.
Let $D : \II \to \mathbf {Top}$ be a diagram in the category of topological spaces $\mathbf {Top}$.
Let $\family {\lim D, \family {\pi_i}_{i \mathop \in \II}}$ be the limit of topological spaces of $D$.
By Limit of Topological Spaces is Limit, $\family {\lim D, \family {\pi_i}_{i \mathop ... | The [[Definition:Category of Topological Spaces|category of topological spaces]] is [[Definition:Complete Category|complete]]. | Let $\II$ be a [[Definition:Small Category|small category]].
Let $D : \II \to \mathbf {Top}$ be a [[Definition:Diagram (Category Theory)|diagram]] in the [[Definition:Category of Topological Spaces|category of topological spaces]] $\mathbf {Top}$.
Let $\family {\lim D, \family {\pi_i}_{i \mathop \in \II}}$ be the [[D... | Top is Complete | https://proofwiki.org/wiki/Top_is_Complete | https://proofwiki.org/wiki/Top_is_Complete | [
"Topological Spaces",
"Category Theory"
] | [
"Definition:Category of Topological Spaces",
"Definition:Complete Category"
] | [
"Definition:Small Category",
"Definition:Diagram (Category Theory)",
"Definition:Category of Topological Spaces",
"Definition:Limit of Topological Spaces",
"Limit of Topological Spaces is Limit",
"Definition:Limit (Category Theory)",
"Category:Topological Spaces",
"Category:Category Theory"
] |
proofwiki-18523 | Projective Resolution Exists Iff Enough Projectives | Let $\AA$ be an abelian category.
Then $\AA$ has enough projectives {{iff}} any object in $\AA$ has a projective resolution. | Suppose $\AA$ has enough projectives.
Let $X$ be an object in $\AA$.
Then there is an epimorphism $\varepsilon : P_0 \to X$ for some projective object $P_0$.
In particular
: $P_0 \to X \to 0$
is exact at $X$.
Since $\AA$ is abelian it has kernels.
Thus $\varepsilon$ has a kernel $K \to P_0$.
Since $\AA$ has enough proj... | Let $\AA$ be an [[Definition:Abelian Category|abelian category]].
Then $\AA$ has [[Definition:Enough Projectives|enough projectives]] {{iff}} any [[Definition:Object (Category Theory)|object]] in $\AA$ has a [[Definition:Projective Resolution|projective resolution]]. | Suppose $\AA$ has [[Definition:Enough Projectives|enough projectives]].
Let $X$ be an [[Definition:Object (Category Theory)|object]] in $\AA$.
Then there is an [[Definition:Epimorphism (Category Theory)|epimorphism]] $\varepsilon : P_0 \to X$ for some [[Definition:Projective Object|projective object]] $P_0$.
In part... | Projective Resolution Exists Iff Enough Projectives | https://proofwiki.org/wiki/Projective_Resolution_Exists_Iff_Enough_Projectives | https://proofwiki.org/wiki/Projective_Resolution_Exists_Iff_Enough_Projectives | [
"Homological Algebra"
] | [
"Definition:Abelian Category",
"Definition:Enough Projectives",
"Definition:Object (Category Theory)",
"Definition:Projective Resolution"
] | [
"Definition:Enough Projectives",
"Definition:Object (Category Theory)",
"Definition:Epimorphism (Category Theory)",
"Definition:Projective Object",
"Definition:Exactness of Chain Complex at Object",
"Definition:Abelian Category",
"Definition:Kernel (Category Theory)",
"Definition:Kernel (Category Theo... |
proofwiki-18524 | Inner Product with Zero Vector | Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.
Let $0_V$ be the zero vector of $V$.
Then:
:$\innerprod {0_V} x = \innerprod x {0_V} = 0$
for all $x \in V$. | We have:
{{begin-eqn}}
{{eqn | l = \innerprod {0_V} x
| r = \innerprod {0_V + 0_V} x
| c = {{Defof|Zero Vector}}
}}
{{eqn | r = \innerprod {0_V} x + \innerprod {0_V} x
| c = linearity of inner product in first argument
}}
{{end-eqn}}
so:
:$\innerprod {0_V} x = 0$
From conjugate symmetry, we have:
:$\innerprod x {0... | Let $\struct {V, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]].
Let $0_V$ be the [[Definition:Zero Vector|zero vector]] of $V$.
Then:
:$\innerprod {0_V} x = \innerprod x {0_V} = 0$
for all $x \in V$. | We have:
{{begin-eqn}}
{{eqn | l = \innerprod {0_V} x
| r = \innerprod {0_V + 0_V} x
| c = {{Defof|Zero Vector}}
}}
{{eqn | r = \innerprod {0_V} x + \innerprod {0_V} x
| c = [[Definition:Linear Mapping|linearity]] of [[Definition:Inner Product|inner product]] in first argument
}}
{{end-eqn}}
so:
:$\innerprod {0_... | Inner Product with Zero Vector | https://proofwiki.org/wiki/Inner_Product_with_Zero_Vector | https://proofwiki.org/wiki/Inner_Product_with_Zero_Vector | [
"Inner Product Spaces"
] | [
"Definition:Inner Product Space",
"Definition:Zero Vector"
] | [
"Definition:Linear Transformation",
"Definition:Inner Product",
"Definition:Conjugate Symmetric Mapping",
"Category:Inner Product Spaces"
] |
proofwiki-18525 | Distributional Solution to x T = 0 | Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Let $\delta$ be the Dirac delta distribution.
Let $\mathbf 0$ be the zero distribution.
Suppose $T$ satisfies the following equation in the distributional sense:
:$x T = \mathbf 0$
Then $T = \alpha \delta$ where $c \in \C$. | Let $\phi \in \map \DD \R$ be a test function.
Let $c \in \C$.
Suppose:
:$T = c \delta$
Then:
{{begin-eqn}}
{{eqn | l = x \map T {\map \phi x}
| r = x \paren{c \map \delta {\map \phi x} }
}}
{{eqn | r = \map \delta {c x \map \phi x}
| c = {{Defof|Multiplication of Schwartz Distribution by Smooth Function}... | Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Let $\delta$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Let $\mathbf 0$ be the [[Definition:Zero Distribution|zero distribution]].
Suppose $T$ satisfies the following [[Definition:Equation|equati... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
Let $c \in \C$.
Suppose:
:$T = c \delta$
Then:
{{begin-eqn}}
{{eqn | l = x \map T {\map \phi x}
| r = x \paren{c \map \delta {\map \phi x} }
}}
{{eqn | r = \map \delta {c x \map \phi x}
| c = {{Defof|Multiplication of Schwart... | Distributional Solution to x T = 0 | https://proofwiki.org/wiki/Distributional_Solution_to_x_T_=_0 | https://proofwiki.org/wiki/Distributional_Solution_to_x_T_=_0 | [
"Examples of Distributional Solutions"
] | [
"Definition:Schwartz Distribution",
"Definition:Dirac Delta Distribution",
"Definition:Zero Mapping/Schwartz Distribution",
"Definition:Equation",
"Definition:Schwartz Distribution"
] | [
"Definition:Test Function",
"Definition:Test Function",
"Definition:Test Function",
"Definition:Test Function"
] |
proofwiki-18526 | Restriction of Sheaf to Open Set is Sheaf | Let $X$ be a topological space.
Let $\FF$ be a sheaf of sets on $X$.
Let $U \subset X$ be an open subset.
Then the restriction $\FF {\restriction_U}$ of $\FF$ to $U$ is a sheaf. | {{ProofWanted}}
Category:Algebraic Geometry
Category:Sheaf Theory
8zp0igs959sgsu9ajsja8o89lifbtxa | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\FF$ be a [[Definition:Sheaf on Topological Space|sheaf of sets]] on $X$.
Let $U \subset X$ be an [[Definition:Open Set (Topology)|open subset]].
Then the [[Definition:Restriction of Presheaf to Open Set|restriction]] $\FF {\restriction_U}$ of $\... | {{ProofWanted}}
[[Category:Algebraic Geometry]]
[[Category:Sheaf Theory]]
8zp0igs959sgsu9ajsja8o89lifbtxa | Restriction of Sheaf to Open Set is Sheaf | https://proofwiki.org/wiki/Restriction_of_Sheaf_to_Open_Set_is_Sheaf | https://proofwiki.org/wiki/Restriction_of_Sheaf_to_Open_Set_is_Sheaf | [
"Algebraic Geometry",
"Sheaf Theory"
] | [
"Definition:Topological Space",
"Definition:Sheaf on Topological Space",
"Definition:Open Set/Topology",
"Definition:Restriction of Presheaf to Open Set",
"Definition:Sheaf on Topological Space"
] | [
"Category:Algebraic Geometry",
"Category:Sheaf Theory"
] |
proofwiki-18527 | Open Neighborhood contains Affine Open Neighborhood | Let $\struct {X, \OO_X}$ be a scheme.
Let $U \subset X$ be an open subset.
Let $x \in U$.
Then there exists an open subset $V \subset U$ with $x \in V$, such that the restriction $\struct {V, \OO_X {\restriction_V}}$ of $\struct {X, \OO_X}$ to $V$ is an affine scheme. | {{ProofWanted}}
Category:Algebraic Geometry
Category:Schemes
gd4gmnyt0n0yjt571srduhp8c0n2psa | Let $\struct {X, \OO_X}$ be a [[Definition:Scheme|scheme]].
Let $U \subset X$ be an [[Definition:Open Set (Topology)|open subset]].
Let $x \in U$.
Then there [[Definition:Existential Quantifier|exists]] an [[Definition:Open Set (Topology)|open subset]] $V \subset U$ with $x \in V$, such that the [[Definition:Restri... | {{ProofWanted}}
[[Category:Algebraic Geometry]]
[[Category:Schemes]]
gd4gmnyt0n0yjt571srduhp8c0n2psa | Open Neighborhood contains Affine Open Neighborhood | https://proofwiki.org/wiki/Open_Neighborhood_contains_Affine_Open_Neighborhood | https://proofwiki.org/wiki/Open_Neighborhood_contains_Affine_Open_Neighborhood | [
"Algebraic Geometry",
"Schemes"
] | [
"Definition:Scheme",
"Definition:Open Set/Topology",
"Definition:Existential Quantifier",
"Definition:Open Set/Topology",
"Definition:Restriction of Ringed Space to Open Set",
"Definition:Affine Scheme"
] | [
"Category:Algebraic Geometry",
"Category:Schemes"
] |
proofwiki-18528 | Open Subscheme is Scheme | Let $\struct {X, \OO_X}$ be a scheme.
Let $U \subset X$ be an open subset.
Then the open subscheme $\struct {U, \OO_X {\restriction U}}$ defined by $U$ is a scheme. | Let $x \in U$.
By Open Neighborhood contains Affine Open Neighborhood, there is an open subset $V \subset U$ with $x \in V$, such that $\struct {V, \OO_X {\restriction V}}$ is an affine scheme.
By Restriction of Restriction of Functor is Restriction $\OO_X {\restriction U} {\restriction V} = \OO_X {\restriction V}$.
By... | Let $\struct {X, \OO_X}$ be a [[Definition:Scheme|scheme]].
Let $U \subset X$ be an [[Definition:Open Set (Topology)|open subset]].
Then the [[Definition:Open Subscheme|open subscheme]] $\struct {U, \OO_X {\restriction U}}$ defined by $U$ is a [[Definition:Scheme|scheme]]. | Let $x \in U$.
By [[Open Neighborhood contains Affine Open Neighborhood]], there is an [[Definition:Open Set (Topology)|open subset]] $V \subset U$ with $x \in V$, such that $\struct {V, \OO_X {\restriction V}}$ is an [[Definition:Affine Scheme|affine scheme]].
By [[Restriction of Restriction of Functor is Restrictio... | Open Subscheme is Scheme | https://proofwiki.org/wiki/Open_Subscheme_is_Scheme | https://proofwiki.org/wiki/Open_Subscheme_is_Scheme | [
"Algebraic Geometry",
"Schemes"
] | [
"Definition:Scheme",
"Definition:Open Set/Topology",
"Definition:Open Subscheme",
"Definition:Scheme"
] | [
"Open Neighborhood contains Affine Open Neighborhood",
"Definition:Open Set/Topology",
"Definition:Affine Scheme",
"Restriction of Restriction of Functor is Restriction",
"Definition:Scheme",
"Definition:Scheme",
"Category:Algebraic Geometry",
"Category:Schemes"
] |
proofwiki-18529 | Vanishing of Quasi-Coherent Sheaf Cohomology of Affine Scheme | Let $X = \Spec A$ be the spectrum of a commutative ring $A$.
Let $\FF$ be a quasi-coherent sheaf on $X$.
Then for all $i \in \Z$ with $i > 0$ the $i$-th sheaf cohomology $\map {H^i} {X, \FF} = 0$. | {{ProofWanted|Proof following EGA III (1.3.1)}} | Let $X = \Spec A$ be the [[Definition:Prime Spectrum of Ring|spectrum]] of a [[Definition:Commutative Ring with Unity|commutative ring]] $A$.
Let $\FF$ be a [[Definition:Quasi-Coherent Sheaf of Modules|quasi-coherent sheaf]] on $X$.
Then for all $i \in \Z$ with $i > 0$ the $i$-th [[Definition:Sheaf Cohomology|sheaf ... | {{ProofWanted|Proof following EGA III (1.3.1)}} | Vanishing of Quasi-Coherent Sheaf Cohomology of Affine Scheme | https://proofwiki.org/wiki/Vanishing_of_Quasi-Coherent_Sheaf_Cohomology_of_Affine_Scheme | https://proofwiki.org/wiki/Vanishing_of_Quasi-Coherent_Sheaf_Cohomology_of_Affine_Scheme | [
"Algebraic Geometry",
"Schemes",
"Sheaf Cohomologies"
] | [
"Definition:Prime Spectrum of Ring",
"Definition:Commutative and Unitary Ring",
"Definition:Quasi-Coherent Sheaf of Modules",
"Definition:Sheaf Cohomology"
] | [] |
proofwiki-18530 | Separated Morphism is Quasi-Separated | Let $f$ be a separated morphism of schemes.
Then $f$ is quasi-separated. | Let $f$ be a separated morphism of schemes.
By definition, the diagonal morphism $\Delta_f$ is a closed immersion.
By Closed Immersion is Quasi-Compact $\Delta_f$ is quasi-compact.
Thus, by definition, $f$ is quasi-separated.
{{qed}}
Category:Algebraic Geometry
Category:Schemes
tj2m4qmcx0na8v74jggb5uj5np4p8rp | Let $f$ be a [[Definition:Separated Morphism of Schemes|separated morphism of schemes]].
Then $f$ is [[Definition:Quasi-Separated Morphism of Schemes|quasi-separated]]. | Let $f$ be a [[Definition:Separated Morphism of Schemes|separated morphism of schemes]].
By [[Definition:Separated Morphism of Schemes|definition]], the [[Definition:Diagonal Morphism|diagonal morphism]] $\Delta_f$ is a [[Definition:Closed Immersion of Schemes|closed immersion]].
By [[Closed Immersion is Quasi-Compac... | Separated Morphism is Quasi-Separated | https://proofwiki.org/wiki/Separated_Morphism_is_Quasi-Separated | https://proofwiki.org/wiki/Separated_Morphism_is_Quasi-Separated | [
"Algebraic Geometry",
"Schemes"
] | [
"Definition:Separated Morphism of Schemes",
"Definition:Quasi-Separated Morphism of Schemes"
] | [
"Definition:Separated Morphism of Schemes",
"Definition:Separated Morphism of Schemes",
"Definition:Diagonal Morphism",
"Definition:Closed Immersion of Schemes",
"Closed Immersion is Quasi-Compact",
"Definition:Quasi-Compact Morphism of Schemes",
"Definition:Quasi-Separated Morphism of Schemes",
"Defi... |
proofwiki-18531 | Restriction of Ringed Space to Open Set is Ringed Space | Let $\struct {X, \OO_X}$ be a ringed space.
Let $U \subset X$ be an open subset of $X$.
Let $\struct {U, \OO_X {\restriction_U}}$ denote the restriction of $\struct {X, \OO_X}$ to $U$.
Then $\struct {U, \OO_X {\restriction_U}}$ is a ringed space. | By Restriction of Sheaf to Open Set is Sheaf $\OO_X {\restriction_U}$ is a sheaf of commutative rings on $U$.
It follows, that $\struct {U, \OO_X {\restriction_U}}$ is a ringed space.
{{qed}}
Category:Ringed Spaces
d0mxvtekfjiupgwl52cpnwug2f8ugkl | Let $\struct {X, \OO_X}$ be a [[Definition:Ringed Space|ringed space]].
Let $U \subset X$ be an [[Definition:Open Set (Topology)|open subset]] of $X$.
Let $\struct {U, \OO_X {\restriction_U}}$ denote the [[Definition:Restriction of Ringed Space to Open Set|restriction]] of $\struct {X, \OO_X}$ to $U$.
Then $\struct... | By [[Restriction of Sheaf to Open Set is Sheaf]] $\OO_X {\restriction_U}$ is a [[Definition:Sheaf on Topological Space|sheaf of commutative rings]] on $U$.
It follows, that $\struct {U, \OO_X {\restriction_U}}$ is a [[Definition:Ringed Space|ringed space]].
{{qed}}
[[Category:Ringed Spaces]]
d0mxvtekfjiupgwl52cpnwug... | Restriction of Ringed Space to Open Set is Ringed Space | https://proofwiki.org/wiki/Restriction_of_Ringed_Space_to_Open_Set_is_Ringed_Space | https://proofwiki.org/wiki/Restriction_of_Ringed_Space_to_Open_Set_is_Ringed_Space | [
"Ringed Spaces"
] | [
"Definition:Ringed Space",
"Definition:Open Set/Topology",
"Definition:Restriction of Ringed Space to Open Set",
"Definition:Ringed Space"
] | [
"Restriction of Sheaf to Open Set is Sheaf",
"Definition:Sheaf on Topological Space",
"Definition:Ringed Space",
"Category:Ringed Spaces"
] |
proofwiki-18532 | Regular Representation on Subgroup is Bijection to Coset/Left | Let $y H$ denote the left coset of $H$ by $y$.
The mapping $\lambda_x: H \to x H$, where $\lambda_x$ is the left regular representation of $H$ with respect to $x$, is a bijection from $H$ to $x H$. | Let $h \in H$.
Then, by definition of left regular representation:
:$\map {\lambda_x} h = x h \in x H$
Thus:
:$\forall h \in H: \map {\lambda_x} h \in x H$
So $\lambda_x: H \to x H$ is a mapping.
A permutation is {{afortiori}} a bijection.
As Regular Representations in Group are Permutations, it follows that $\lambda_x... | Let $y H$ denote the [[Definition:Left Coset|left coset]] of $H$ by $y$.
The mapping $\lambda_x: H \to x H$, where $\lambda_x$ is the [[Definition:Left Regular Representation|left regular representation]] of $H$ with respect to $x$, is a [[Definition:Bijection|bijection]] from $H$ to $x H$. | Let $h \in H$.
Then, by definition of [[Definition:Left Regular Representation|left regular representation]]:
:$\map {\lambda_x} h = x h \in x H$
Thus:
:$\forall h \in H: \map {\lambda_x} h \in x H$
So $\lambda_x: H \to x H$ is a [[Definition:Mapping|mapping]].
A [[Definition:Permutation|permutation]] is {{afortior... | Regular Representation on Subgroup is Bijection to Coset/Left | https://proofwiki.org/wiki/Regular_Representation_on_Subgroup_is_Bijection_to_Coset/Left | https://proofwiki.org/wiki/Regular_Representation_on_Subgroup_is_Bijection_to_Coset/Left | [
"Regular Representation on Subgroup is Bijection to Coset"
] | [
"Definition:Coset/Left Coset",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Bijection"
] | [
"Definition:Regular Representations/Left Regular Representation",
"Definition:Mapping",
"Definition:Permutation",
"Definition:Bijection",
"Regular Representations in Group are Permutations",
"Definition:Bijection"
] |
proofwiki-18533 | Sheaf Associated to Injective Module over Noetherian Ring is Flasque | Let $A$ be a Noetherian commutative ring.
Let $I$ be an injective $A$-module.
Then the sheaf $\tilde I$ associated to $I$ on $\Spec A$ is flasque. | {{ProofWanted|Hartshorne III.3.4}} | Let $A$ be a [[Definition:Noetherian Ring|Noetherian]] [[Definition:Commutative Ring with Unity|commutative ring]].
Let $I$ be an [[Definition:Injective Module|injective]] $A$-[[Definition:Module over Ring|module]].
Then the [[Definition:Sheaf of Modules Associated to Module|sheaf $\tilde I$ associated to]] $I$ on $... | {{ProofWanted|Hartshorne III.3.4}} | Sheaf Associated to Injective Module over Noetherian Ring is Flasque | https://proofwiki.org/wiki/Sheaf_Associated_to_Injective_Module_over_Noetherian_Ring_is_Flasque | https://proofwiki.org/wiki/Sheaf_Associated_to_Injective_Module_over_Noetherian_Ring_is_Flasque | [
"Schemes"
] | [
"Definition:Noetherian Ring",
"Definition:Commutative and Unitary Ring",
"Definition:Injective Module",
"Definition:Module over Ring",
"Definition:Sheaf of Modules Associated to Module",
"Definition:Flasque Sheaf of Sets on Topological Space"
] | [] |
proofwiki-18534 | Injective iff Projective in Dual Category | Let $\mathbf A$ be an abelian category.
Let $X$ be an object in $\mathbf A$.
Then:
:$X$ is injective in $\mathbf A$
{{iff}}:
:$X$ is projective in the dual category $\mathbf A^{\mathrm{op}}$ of $\mathbf A$. | === Sufficient Condition ===
Let $X$ be injective in $\mathbf A$.
Let $f : B \to A$ be an epimorphism in $\mathbf A^{\mathrm {op} }$.
By Monomorphism iff Epimorphism in Dual Category:
:$f : A \to B$ is a monomorphism in $\mathbf A$.
By definition of injective object, the mapping:
:$\alpha: \map {\mathrm {Hom}_{\mathbf ... | Let $\mathbf A$ be an [[Definition:Abelian Category|abelian category]].
Let $X$ be an [[Definition:Object (Category Theory)|object]] in $\mathbf A$.
Then:
:$X$ is [[Definition:Injective Object|injective]] in $\mathbf A$
{{iff}}:
:$X$ is [[Definition:Projective Object|projective]] in the [[Definition:Dual Category|du... | === Sufficient Condition ===
Let $X$ be [[Definition:Injective Object|injective]] in $\mathbf A$.
Let $f : B \to A$ be an [[Definition:Epimorphism (Category Theory)|epimorphism]] in $\mathbf A^{\mathrm {op} }$.
By [[Monomorphism iff Epimorphism in Dual Category]]:
:$f : A \to B$ is a [[Definition:Monomorphism (Categ... | Injective iff Projective in Dual Category | https://proofwiki.org/wiki/Injective_iff_Projective_in_Dual_Category | https://proofwiki.org/wiki/Injective_iff_Projective_in_Dual_Category | [
"Category Theory",
"Homological Algebra"
] | [
"Definition:Abelian Category",
"Definition:Object (Category Theory)",
"Definition:Injective Object",
"Definition:Projective Object",
"Definition:Dual Category"
] | [
"Definition:Injective Object",
"Definition:Epimorphism (Category Theory)",
"Monomorphism iff Epimorphism in Dual Category",
"Definition:Monomorphism (Category Theory)",
"Definition:Injective Object",
"Definition:Mapping",
"Definition:Surjection",
"User:Wandynsky",
"User talk:Wandynsky",
"Definitio... |
proofwiki-18535 | Composition with Zero Morphism is Zero Morphism | Let $\mathbf C$ be a category.
Let $0$ be a zero object in $\mathbf C$.
Let $A, B, C, D$ be objects in $\mathbf C$.
Let $f : A \to B$, $g : B \to C$ and $h : C \to D$ be morphisms in $\mathbf C$.
Let $g$ be a zero morphism.
Then $g \circ f$ and $h \circ g$ are zero morphisms. | By definition of zero morphism, $g$ is the composition of the unique morphism $\alpha : B \to 0$ with the unique morphism $\beta : 0 \to C$.
Since $\alpha \circ f : A \to 0$ is a morphism with codomain $0$ and $0$ is a terminal object, $\alpha \circ f$ is the unique morphism $A \to 0$.
It follows that $g \circ f = \bet... | Let $\mathbf C$ be a [[Definition:Category|category]].
Let $0$ be a [[Definition:Zero Object|zero object]] in $\mathbf C$.
Let $A, B, C, D$ be [[Definition:Object (Category Theory)|objects]] in $\mathbf C$.
Let $f : A \to B$, $g : B \to C$ and $h : C \to D$ be [[Definition:Morphism|morphisms]] in $\mathbf C$.
Let $... | By definition of [[Definition:Zero Morphism|zero morphism]], $g$ is the [[Definition:Composition of Morphisms|composition]] of the [[Definition:Unique|unique]] [[Definition:Morphism|morphism]] $\alpha : B \to 0$ with the [[Definition:Unique|unique]] [[Definition:Morphism|morphism]] $\beta : 0 \to C$.
Since $\alpha \ci... | Composition with Zero Morphism is Zero Morphism | https://proofwiki.org/wiki/Composition_with_Zero_Morphism_is_Zero_Morphism | https://proofwiki.org/wiki/Composition_with_Zero_Morphism_is_Zero_Morphism | [
"Morphisms"
] | [
"Definition:Category",
"Definition:Zero Object",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Zero Morphism via Zero Object",
"Definition:Zero Morphism via Zero Object"
] | [
"Definition:Zero Morphism via Zero Object",
"Definition:Composition of Morphisms",
"Definition:Unique",
"Definition:Morphism",
"Definition:Unique",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Codomain (Category Theory)",
"Definition:Terminal Object",
"Definition:Unique",
"Definitio... |
proofwiki-18536 | Differentiability of Function with Translation Property | Let $f$ be a real function.
Let $f$ have the translation property.
Let $c$ be a real number.
Let $\map {f'} c$ exist.
Then:
:$f'$ exists
:$f'$ is a constant function | Let $x$ be a real number.
We have:
{{begin-eqn}}
{{eqn | l = \map {f'} x
| r = \lim_{y \mathop \to x} \frac {\map f y - \map f x} {y - x}
| c = {{Defof|Differentiable Mapping|subdef = Real Function/Point|index = 1}}
}}
{{eqn | r = \lim_{y \mathop \to x} \frac {\map f {y - x + x - c + c} - \map f {x - c + c}... | Let $f$ be a [[Definition:Real Function|real function]].
Let $f$ have the [[Definition:Translation Property|translation property]].
Let $c$ be a [[Definition:Real Number|real number]].
Let $\map {f'} c$ exist.
Then:
:$f'$ exists
:$f'$ is a [[Definition:Constant Mapping|constant function]] | Let $x$ be a [[Definition:Real Number|real number]].
We have:
{{begin-eqn}}
{{eqn | l = \map {f'} x
| r = \lim_{y \mathop \to x} \frac {\map f y - \map f x} {y - x}
| c = {{Defof|Differentiable Mapping|subdef = Real Function/Point|index = 1}}
}}
{{eqn | r = \lim_{y \mathop \to x} \frac {\map f {y - x + x ... | Differentiability of Function with Translation Property | https://proofwiki.org/wiki/Differentiability_of_Function_with_Translation_Property | https://proofwiki.org/wiki/Differentiability_of_Function_with_Translation_Property | [
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Translation Property",
"Definition:Real Number",
"Definition:Constant Mapping"
] | [
"Definition:Real Number",
"Definition:Constant Mapping",
"Definition:Real Number",
"Category:Differential Calculus"
] |
proofwiki-18537 | Linearity of Function with Translation Property | Let $f$ be a real function.
Let $c$ be a real number.
Then:
:$f$ has the translation property
{{improve|Obviously, what needs to happen here is that rather than use the suboptimal page translation property, we need to make it so that it directly references the page which defines what a translation is. That is to say "$... | === Sufficient Condition ===
Let:
:$f$ have the translation property
:$\map {f'} c$ exist
We need to show that:
:$f$ is linear
The fact that $f$ has the translation property means:
:$\forall x_1, x_2, t \in \R: \map f {x_1 + t} - \map f {x_2 + t} = \map f {x_1} - \map f {x_2}$
$f$ being linear means:
:$\forall x \in \R... | Let $f$ be a [[Definition:Real Function|real function]].
Let $c$ be a [[Definition:Real Number|real number]].
Then:
:$f$ has the [[Definition:Translation Property|translation property]]
{{improve|Obviously, what needs to happen here is that rather than use the suboptimal page [[Definition:Translation Property|trans... | === Sufficient Condition ===
Let:
:$f$ have the [[Definition:Translation Property|translation property]]
:$\map {f'} c$ exist
We need to show that:
:$f$ is [[Definition:Linear Real Function|linear]]
The fact that $f$ has the [[Definition:Translation Property|translation property]] means:
:$\forall x_1, x_2, t \in ... | Linearity of Function with Translation Property | https://proofwiki.org/wiki/Linearity_of_Function_with_Translation_Property | https://proofwiki.org/wiki/Linearity_of_Function_with_Translation_Property | [
"Linear Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Number",
"Definition:Translation Property",
"Definition:Translation Property",
"Definition:Linear Real Function"
] | [
"Definition:Translation Property",
"Definition:Linear Real Function",
"Definition:Translation Property",
"Definition:Linear Real Function",
"Definition:Real Number",
"Differentiability of Function with Translation Property",
"Definition:Constant Mapping",
"Definition:Real Number",
"Definition:Real N... |
proofwiki-18538 | Linearity of Function defined using Function with Translation Property | Let $f$ be a real function.
Let $f$ have the translation property.
Let $x$ and $l$ be real numbers.
Define:
:$\map {f_x} l = \map f {x + l} - \map f x$
Then:
:$\forall q \in \Q: \map {f_x} {q l} = q \map {f_x} l$ | === Lemma ===
{{:Linearity of Function defined using Function with Translation Property/Lemma}} {{qed|lemma}}
Let $q$ be a rational number.
Choose integers $n$, $m$ such that:
:$\dfrac n m = q$
We need to prove that:
:$\map {f_x} {q l} = q \map {f_x} l$
We have:
{{begin-eqn}}
{{eqn | l = \map {f_x} {q l}
| r = \m... | Let $f$ be a [[Definition:Real Function|real function]].
Let $f$ have the [[Definition:Translation Property|translation property]].
Let $x$ and $l$ be [[Definition:Real Number|real numbers]].
Define:
:$\map {f_x} l = \map f {x + l} - \map f x$
Then:
:$\forall q \in \Q: \map {f_x} {q l} = q \map {f_x} l$ | === [[Linearity of Function defined using Function with Translation Property/Lemma|Lemma]] ===
{{:Linearity of Function defined using Function with Translation Property/Lemma}} {{qed|lemma}}
Let $q$ be a [[Definition:Rational Number|rational number]].
Choose [[Definition:Integer|integers]] $n$, $m$ such that:
:$\dfr... | Linearity of Function defined using Function with Translation Property | https://proofwiki.org/wiki/Linearity_of_Function_defined_using_Function_with_Translation_Property | https://proofwiki.org/wiki/Linearity_of_Function_defined_using_Function_with_Translation_Property | [
"Linear Real Functions"
] | [
"Definition:Real Function",
"Definition:Translation Property",
"Definition:Real Number"
] | [
"Linearity of Function defined using Function with Translation Property/Lemma",
"Definition:Rational Number",
"Definition:Integer",
"Linearity of Function defined using Function with Translation Property/Lemma",
"Linearity of Function defined using Function with Translation Property/Lemma",
"Category:Line... |
proofwiki-18539 | Baer's Criterion | Let $R$ be a ring with unity.
Let $M$ be a left $R$-module.
Then $M$ is injective {{iff}} the following condition holds:
:For all left ideals $I$ of $R$ with inclusion map $\iota : I \to R$, and for all $R$-module homomorphisms $f : I \to M$, there exists an $R$-module homomorphism $\tilde f : R \to M$ such that:
::$\t... | {{ProofWanted}}
{{Namedfor|Reinhold Baer|cat = Baer}}
Category:Homological Algebra
Category:Module Theory
Category:Ring Theory
9q1r8gxmef4xghp9rp3oif3hzsg0762 | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M$ be a [[Definition:Left Module over Ring|left $R$-module]].
Then $M$ is [[Definition:Injective Module|injective]] {{iff}} the following condition holds:
:For all [[Definition:Left Ideal of Ring|left ideals]] $I$ of $R$ with [[Definition:Inclusion Ma... | {{ProofWanted}}
{{Namedfor|Reinhold Baer|cat = Baer}}
[[Category:Homological Algebra]]
[[Category:Module Theory]]
[[Category:Ring Theory]]
9q1r8gxmef4xghp9rp3oif3hzsg0762 | Baer's Criterion | https://proofwiki.org/wiki/Baer's_Criterion | https://proofwiki.org/wiki/Baer's_Criterion | [
"Homological Algebra",
"Module Theory",
"Ring Theory"
] | [
"Definition:Ring with Unity",
"Definition:Left Module over Ring",
"Definition:Injective Module",
"Definition:Ideal of Ring/Left Ideal",
"Definition:Inclusion Mapping",
"Definition:Linear Transformation",
"Definition:Linear Transformation"
] | [
"Category:Homological Algebra",
"Category:Module Theory",
"Category:Ring Theory"
] |
proofwiki-18540 | Injective Module over Dedekind Domain | Let $D$ be a Dedekind domain.
Let $M$ be a $D$-module.
Then $M$ is injective {{iff}} it is divisible. | {{ProofWanted}}
Category:Commutative Algebra
Category:Homological Algebra
Category:Module Theory
7ejvfxep5vvl24spbd569d1yiwvwttq | Let $D$ be a [[Definition:Dedekind Domain|Dedekind domain]].
Let $M$ be a [[Definition:Module over Ring|$D$-module]].
Then $M$ is [[Definition:Injective Module|injective]] {{iff}} it is [[Definition:Divisible Module|divisible]]. | {{ProofWanted}}
[[Category:Commutative Algebra]]
[[Category:Homological Algebra]]
[[Category:Module Theory]]
7ejvfxep5vvl24spbd569d1yiwvwttq | Injective Module over Dedekind Domain | https://proofwiki.org/wiki/Injective_Module_over_Dedekind_Domain | https://proofwiki.org/wiki/Injective_Module_over_Dedekind_Domain | [
"Commutative Algebra",
"Homological Algebra",
"Module Theory"
] | [
"Definition:Dedekind Domain",
"Definition:Module over Ring",
"Definition:Injective Module",
"Definition:Divisible Module"
] | [
"Category:Commutative Algebra",
"Category:Homological Algebra",
"Category:Module Theory"
] |
proofwiki-18541 | Injective Module over Principal Ideal Domain | Let $D$ be a principal ideal domain.
Let $M$ be a $D$-module.
Then $M$ is injective {{iff}} it is divisible. | By Principal Ideal Domain is Dedekind Domain $D$ is a Dedekind domain.
By Injective Module over Dedekind Domain $M$ is injective {{iff}} it is divisible.
{{qed}}
Category:Principal Ideal Domains
Category:Commutative Algebra
Category:Homological Algebra
Category:Module Theory
s91awp7825oi1x931w87o1txnfzyqls | Let $D$ be a [[Definition:Principal Ideal Domain|principal ideal domain]].
Let $M$ be a [[Definition:Module over Ring|$D$-module]].
Then $M$ is [[Definition:Injective Module|injective]] {{iff}} it is [[Definition:Divisible Module|divisible]]. | By [[Principal Ideal Domain is Dedekind Domain]] $D$ is a [[Definition:Dedekind Domain|Dedekind domain]].
By [[Injective Module over Dedekind Domain]] $M$ is [[Definition:Injective Module|injective]] {{iff}} it is [[Definition:Divisible Module|divisible]].
{{qed}}
[[Category:Principal Ideal Domains]]
[[Category:Comm... | Injective Module over Principal Ideal Domain | https://proofwiki.org/wiki/Injective_Module_over_Principal_Ideal_Domain | https://proofwiki.org/wiki/Injective_Module_over_Principal_Ideal_Domain | [
"Principal Ideal Domains",
"Commutative Algebra",
"Homological Algebra",
"Module Theory"
] | [
"Definition:Principal Ideal Domain",
"Definition:Module over Ring",
"Definition:Injective Module",
"Definition:Divisible Module"
] | [
"Principal Ideal Domain is Dedekind Domain",
"Definition:Dedekind Domain",
"Injective Module over Dedekind Domain",
"Definition:Injective Module",
"Definition:Divisible Module",
"Category:Principal Ideal Domains",
"Category:Commutative Algebra",
"Category:Homological Algebra",
"Category:Module Theor... |
proofwiki-18542 | Principal Ideal Domain is Dedekind Domain | Let $D$ be a principal ideal domain which is specifically not a field.
Then $D$ is a Dedekind domain. | By definition of principal ideal domain $D$ is an integral domain.
By Principal Ideal Domain is Noetherian $D$ is noetherian.
By Principal Ideal Domain is Integrally Closed $D$ is integrally closed.
By Prime Ideal of Principal Ideal Domain is Maximal $D$ has Krull dimension $\le 1$.
By Integral Domain has Dimension Zer... | Let $D$ be a [[Definition:Principal Ideal Domain|principal ideal domain]] which is specifically not a [[Definition:Field (Abstract Algebra)|field]].
Then $D$ is a [[Definition:Dedekind Domain|Dedekind domain]]. | By definition of [[Definition:Principal Ideal Domain|principal ideal domain]] $D$ is an [[Definition:Integral Domain|integral domain]].
By [[Principal Ideal Domain is Noetherian]] $D$ is [[Definition:Noetherian Ring|noetherian]].
By [[Principal Ideal Domain is Integrally Closed]] $D$ is [[Definition:Integrally Closed... | Principal Ideal Domain is Dedekind Domain | https://proofwiki.org/wiki/Principal_Ideal_Domain_is_Dedekind_Domain | https://proofwiki.org/wiki/Principal_Ideal_Domain_is_Dedekind_Domain | [
"Principal Ideal Domains",
"Dedekind Domains"
] | [
"Definition:Principal Ideal Domain",
"Definition:Field (Abstract Algebra)",
"Definition:Dedekind Domain"
] | [
"Definition:Principal Ideal Domain",
"Definition:Integral Domain",
"Principal Ideal Domain is Noetherian",
"Definition:Noetherian Ring",
"Principal Ideal Domain is Integrally Closed",
"Definition:Integrally Closed Integral Domain",
"Prime Ideal of Principal Ideal Domain is Maximal",
"Definition:Krull ... |
proofwiki-18543 | Principal Ideal Domain is Integrally Closed | Let $A$ be a principal ideal domain.
Then $A$ is integrally closed. | By Principal Ideal Domain is Unique Factorization Domain $A$ is a unique factorization domain.
By Unique Factorization Domain is Integrally Closed $A$ is integrally closed.
{{qed}}
Category:Principal Ideal Domains
onnfqze63bvk4dhb9ga9bj3vngezdtc | Let $A$ be a [[Definition:Principal Ideal Domain|principal ideal domain]].
Then $A$ is [[Definition:Integrally Closed Integral Domain|integrally closed]]. | By [[Principal Ideal Domain is Unique Factorization Domain]] $A$ is a [[Definition:Unique Factorization Domain|unique factorization domain]].
By [[Unique Factorization Domain is Integrally Closed]] $A$ is [[Definition:Integrally Closed Integral Domain|integrally closed]].
{{qed}}
[[Category:Principal Ideal Domains]]
... | Principal Ideal Domain is Integrally Closed | https://proofwiki.org/wiki/Principal_Ideal_Domain_is_Integrally_Closed | https://proofwiki.org/wiki/Principal_Ideal_Domain_is_Integrally_Closed | [
"Principal Ideal Domains"
] | [
"Definition:Principal Ideal Domain",
"Definition:Integrally Closed Integral Domain"
] | [
"Principal Ideal Domain is Unique Factorization Domain",
"Definition:Unique Factorization Domain",
"Unique Factorization Domain is Integrally Closed",
"Definition:Integrally Closed Integral Domain",
"Category:Principal Ideal Domains"
] |
proofwiki-18544 | Field of Quotients is Divisible Module | Let $D$ be an integral domain.
Let $\map {\operatorname {Quot} } D$ be the field of quotients of $D$.
Then $\map {\operatorname {Quot} } D$ is a divisible $D$-module. | Let $a \in D$ be a non zero divisor.
Let $x, y \in D$ such that $y \ne 0$.
Then $\dfrac x y \in \map {\operatorname {Quot} } D$.
By definition of integral domain:
:$a \ne 0$
Thus $\dfrac x {a y}$ is defined in $\map {\operatorname {Quot} } D$.
It follows that:
:$a \cdot \dfrac x {a y} = \dfrac x y$
Thus $\map {\operato... | Let $D$ be an [[Definition:Integral Domain|integral domain]].
Let $\map {\operatorname {Quot} } D$ be the [[Definition:Field of Quotients|field of quotients]] of $D$.
Then $\map {\operatorname {Quot} } D$ is a [[Definition:Divisible Module|divisible]] [[Definition:Module over Ring|$D$-module]]. | Let $a \in D$ be a non [[Definition:Zero Divisor of Ring|zero divisor]].
Let $x, y \in D$ such that $y \ne 0$.
Then $\dfrac x y \in \map {\operatorname {Quot} } D$.
By definition of [[Definition:Integral Domain|integral domain]]:
:$a \ne 0$
Thus $\dfrac x {a y}$ is defined in $\map {\operatorname {Quot} } D$.
It f... | Field of Quotients is Divisible Module | https://proofwiki.org/wiki/Field_of_Quotients_is_Divisible_Module | https://proofwiki.org/wiki/Field_of_Quotients_is_Divisible_Module | [
"Integral Domains",
"Module Theory",
"Homological Algebra"
] | [
"Definition:Integral Domain",
"Definition:Field of Quotients",
"Definition:Divisible Module",
"Definition:Module over Ring"
] | [
"Definition:Zero Divisor/Ring",
"Definition:Integral Domain",
"Definition:Divisible Module",
"Definition:Module over Ring",
"Category:Integral Domains",
"Category:Module Theory",
"Category:Homological Algebra"
] |
proofwiki-18545 | Quotient of Divisible Module is Divisible | Let $R$ be a ring with unity.
Let $M$ be a divisible left $R$-module.
Let $N \subseteq M$ be an $R$-submodule.
Then the quotient module $M / N$ is divisible. | Let $r \in R$ be a regular element of $R$.
Hence by definition $r$ is not a zero divisor of $R$.
Let $\eqclass m {} \in M / N$ be an arbitrary element represented by $m \in M$.
Since $M$ is divisible, there exists some $m' \in M$ such that $m = r m'$.
By definition of scalar multiplication on the quotient module $M / N... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M$ be a [[Definition:Divisible Module|divisible]] [[Definition:Left Module over Ring|left $R$-module]].
Let $N \subseteq M$ be an $R$-[[Definition:Submodule|submodule]].
Then the [[Definition:Quotient Module|quotient module]] $M / N$ is [[Definition:... | Let $r \in R$ be a [[Definition:Regular Element of Ring|regular element]] of $R$.
Hence by definition $r$ is not a [[Definition:Zero Divisor of Ring|zero divisor]] of $R$.
Let $\eqclass m {} \in M / N$ be an arbitrary [[Definition:Element|element]] represented by $m \in M$.
Since $M$ is [[Definition:Divisible Module... | Quotient of Divisible Module is Divisible | https://proofwiki.org/wiki/Quotient_of_Divisible_Module_is_Divisible | https://proofwiki.org/wiki/Quotient_of_Divisible_Module_is_Divisible | [
"Homological Algebra",
"Module Theory"
] | [
"Definition:Ring with Unity",
"Definition:Divisible Module",
"Definition:Left Module over Ring",
"Definition:Submodule",
"Definition:Quotient Module",
"Definition:Divisible Module"
] | [
"Definition:Regular Element of Ring",
"Definition:Zero Divisor/Ring",
"Definition:Element",
"Definition:Divisible Module",
"Definition:Existential Quantifier",
"Definition:Scalar Multiplication/Module",
"Definition:Quotient Module",
"Definition:Divisible Module",
"Category:Homological Algebra",
"C... |
proofwiki-18546 | Quotient of Rationals by Integers is Injective | Let $\struct {\Q, +}$ be the abelian group of rational numbers.
Let $\struct {\Z, +}$ be the abelian group of integers, considered as a subgroup of $\struct {\Q, +}$.
Then the quotient group $\Q / \Z$ is an injective object in the category of abelian groups. | By definition, $\struct {\Q, +, \times}$ is the field of quotients of the ring of integers $\struct {\Z, +, \times}$.
By Field of Quotients is Divisible Module $\Q$ is a divisible $\Z$-module.
By Quotient of Divisible Module is Divisible $\Q / \Z$ is a divisible $\Z$-module.
By Ring of Integers is Principal Ideal Domai... | Let $\struct {\Q, +}$ be the [[Definition:Abelian Group|abelian group]] of [[Definition:Field of Rational Numbers|rational numbers]].
Let $\struct {\Z, +}$ be the [[Definition:Abelian Group|abelian group]] of [[Definition:Integer|integers]], considered as a [[Definition:Subgroup|subgroup]] of $\struct {\Q, +}$.
Then... | By definition, $\struct {\Q, +, \times}$ is the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Ring of Integers|ring of integers]] $\struct {\Z, +, \times}$.
By [[Field of Quotients is Divisible Module]] $\Q$ is a [[Definition:Divisible Module|divisible]] $\Z$-[[Definition:Module over Ring|mo... | Quotient of Rationals by Integers is Injective | https://proofwiki.org/wiki/Quotient_of_Rationals_by_Integers_is_Injective | https://proofwiki.org/wiki/Quotient_of_Rationals_by_Integers_is_Injective | [
"Homological Algebra",
"Abelian Groups"
] | [
"Definition:Abelian Group",
"Definition:Field of Rational Numbers",
"Definition:Abelian Group",
"Definition:Integer",
"Definition:Subgroup",
"Definition:Quotient Group",
"Definition:Injective Object",
"Definition:Category of Abelian Groups"
] | [
"Definition:Field of Quotients",
"Definition:Ring of Integers",
"Field of Quotients is Divisible Module",
"Definition:Divisible Module",
"Definition:Module over Ring",
"Quotient of Divisible Module is Divisible",
"Definition:Divisible Module",
"Definition:Module over Ring",
"Ring of Integers is Prin... |
proofwiki-18547 | Ring by Idempotent | Let $\struct {A, +, \circ}$ be a commutative ring.
Let $e$ be an idempotent element of $A$.
Then the ideal $I := \ideal e$ generated by $e$ is a commutative ring with unity $\struct {I, +, \circ}$ with unity $e$. | Because $\struct {I, +}$ is an ideal of $\struct {A, +, \circ}$, it follows that $\struct {I, +, \circ}$ is a ring (not necessarily unital).
{{explain|Why is $\struct {I, +}$ an ideal?
By definition it is the ideal generated by $e$. --Wandynsky (talk) 13:04, 30 July 2021 (UTC)
I don't see this anywhere in the Definitio... | Let $\struct {A, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $e$ be an [[Definition:Idempotent Element|idempotent element]] of $A$.
Then the [[Definition:Ideal of Ring|ideal]] $I := \ideal e$ [[Definition:Generator of Ideal|generated]] by $e$ is a [[Definition:Commutative Ring with Unity|co... | Because $\struct {I, +}$ is an [[Definition:Ideal of Ring|ideal]] of $\struct {A, +, \circ}$, it follows that $\struct {I, +, \circ}$ is a [[Definition:Commutative Ring|ring]] (not necessarily [[Definition:Ring with Unity|unital]]).
{{explain|Why is $\struct {I, +}$ an [[Definition:Ideal of Ring|ideal]]?
By definitio... | Ring by Idempotent | https://proofwiki.org/wiki/Ring_by_Idempotent | https://proofwiki.org/wiki/Ring_by_Idempotent | [
"Ring Theory"
] | [
"Definition:Commutative Ring",
"Definition:Idempotence/Element",
"Definition:Ideal of Ring",
"Definition:Generator of Ideal of Ring",
"Definition:Commutative and Unitary Ring",
"Definition:Unity (Abstract Algebra)/Ring"
] | [
"Definition:Ideal of Ring",
"Definition:Commutative Ring",
"Definition:Ring with Unity",
"Definition:Ideal of Ring",
"User:Wandynsky",
"User talk:Wandynsky",
"Definition:Generator of Ideal of Ring",
"Definition:Generated Ideal of Ring",
"Definition:Generated Ideal of Ring",
"Definition:Intersectio... |
proofwiki-18548 | Ring Homomorphism by Idempotent | Let $A$ be a commutative ring.
Let $e \in A$ be an idempotent element.
Let $\ideal e$ be the ideal of $A$ generated by $e$.
Then the mapping:
:$f: A \to \ideal e: a \mapsto e a$
is a surjective ring homomorphism with kernel the ideal $\ideal {1 - e}$ generated by $1 - e$. | Let $e a \in \ideal e$ for some arbitrary $a \in A$.
{{explain|Technically, one needs to prove that a general element of $\ideal e$ has this form. --Wandynsky (talk) 15:38, 30 July 2021 (UTC)
Go to it.}}
Then
{{begin-eqn}}
{{eqn | l = \map f {e a}
| r = e e a
| c = Definition of $f$
}}
{{eqn | r = ea
... | Let $A$ be a [[Definition:Commutative Ring|commutative ring]].
Let $e \in A$ be an [[Definition:Idempotent Element|idempotent element]].
Let $\ideal e$ be the [[Definition:Ideal of Ring|ideal]] of $A$ [[Definition:Generator of Ideal|generated]] by $e$.
Then the [[Definition:Mapping|mapping]]:
:$f: A \to \ideal e: a... | Let $e a \in \ideal e$ for some arbitrary $a \in A$.
{{explain|Technically, one needs to prove that a general element of $\ideal e$ has this form. --[[User:Wandynsky|Wandynsky]] ([[User talk:Wandynsky|talk]]) 15:38, 30 July 2021 (UTC)
Go to it.}}
Then
{{begin-eqn}}
{{eqn | l = \map f {e a}
| r = e e a
| ... | Ring Homomorphism by Idempotent | https://proofwiki.org/wiki/Ring_Homomorphism_by_Idempotent | https://proofwiki.org/wiki/Ring_Homomorphism_by_Idempotent | [
"Ring Theory",
"Commutative Rings",
"Ring Homomorphisms"
] | [
"Definition:Commutative Ring",
"Definition:Idempotence/Element",
"Definition:Ideal of Ring",
"Definition:Generator of Ideal of Ring",
"Definition:Mapping",
"Definition:Surjection",
"Definition:Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Definition:Ideal of Ring",
"Definition:Gen... | [
"User:Wandynsky",
"User talk:Wandynsky",
"Definition:Idempotence/Element",
"Definition:Surjection",
"Definition:Idempotence/Element",
"Definition:Commutative Ring",
"Definition:Ring Homomorphism",
"Definition:Idempotence/Element",
"User:Wandynsky",
"User talk:Wandynsky"
] |
proofwiki-18549 | Inner Product is Sesquilinear | Let $\mathbb F$ be a subfield of $\C$.
Let $V$ be a inner product space over $V$ with inner product $\innerprod \cdot \cdot$.
Define the $f : V \times V \to \mathbb F$ by:
:$\map f {x, y} = \innerprod x y$
for each $x, y \in V$.
Then $f$ is sesquilinear. | Let $\alpha \in \mathbb F$.
Let $x_1, x_2, y \in V$.
By the definition of the inner product, $f$ is linear in its first argument.
So, we have:
:$\innerprod {\alpha x_1 + x_2} y = \alpha \innerprod {x_1} y + \innerprod {x_2} y$
From the definition of the inner product, we also have that $f$ is conjugate symmetric, so... | Let $\mathbb F$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $V$ be a [[Definition:Inner Product Space|inner product space]] over $V$ with [[Definition:Inner Product|inner product]] $\innerprod \cdot \cdot$.
Define the $f : V \times V \to \mathbb F$ by:
:$\map f {x, y} = \innerprod x y$
for each $x, y \in ... | Let $\alpha \in \mathbb F$.
Let $x_1, x_2, y \in V$.
By the definition of the [[Definition:Inner Product|inner product]], $f$ is [[Definition:Linear Mapping|linear]] in its first argument.
So, we have:
:$\innerprod {\alpha x_1 + x_2} y = \alpha \innerprod {x_1} y + \innerprod {x_2} y$
From the definition of the... | Inner Product is Sesquilinear | https://proofwiki.org/wiki/Inner_Product_is_Sesquilinear | https://proofwiki.org/wiki/Inner_Product_is_Sesquilinear | [
"Inner Product Spaces"
] | [
"Definition:Subfield",
"Definition:Inner Product Space",
"Definition:Inner Product",
"Definition:Sesquilinear Form"
] | [
"Definition:Inner Product",
"Definition:Linear Transformation",
"Definition:Inner Product",
"Definition:Conjugate Symmetric Mapping",
"Sum of Complex Conjugates",
"Product of Complex Conjugates",
"Definition:Conjugate Symmetric Mapping",
"Definition:Inner Product",
"Definition:Sesquilinear Form",
... |
proofwiki-18550 | Unital Ring Homomorphism by Idempotent | Let $A$ be a commutative ring with unity.
Let $e \in A$ be an idempotent element.
Let $\ideal e$ be the ideal of $A$ generated by $e$.
Then the mapping:
:$f: A \to \ideal e: a \mapsto e a$
is a surjective unital ring homomorphism from $\struct {A, +, \circ}$ to $\struct {\ideal e, +, \circ}$ with kernel the ideal $\id... | By Ring Homomorphism by Idempotent $f$ is a surjective ring homomorphism with kernel $\ideal {1 - e}$.
By Ring by Idempotent $A$ is a commutative ring with unity whose unity is $e$.
Then
{{begin-eqn}}
{{eqn | l = \map f 1
| r = e \cdot 1
| c = Definition of $f$
}}
{{eqn | r = e
| c = $1$ is the unity ... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $e \in A$ be an [[Definition:Idempotent Element|idempotent element]].
Let $\ideal e$ be the [[Definition:Ideal of Ring|ideal]] of $A$ [[Definition:Generator of Ideal|generated]] by $e$.
Then the [[Definition:Mapping|mapping]]:
... | By [[Ring Homomorphism by Idempotent]] $f$ is a [[Definition:Surjection|surjective]] [[Definition:Ring Homomorphism|ring homomorphism]] with [[Definition:Kernel of Ring Homomorphism|kernel]] $\ideal {1 - e}$.
By [[Ring by Idempotent]] $A$ is a [[Definition:Commutative Ring with Unity|commutative ring with unity]] whos... | Unital Ring Homomorphism by Idempotent | https://proofwiki.org/wiki/Unital_Ring_Homomorphism_by_Idempotent | https://proofwiki.org/wiki/Unital_Ring_Homomorphism_by_Idempotent | [
"Ring Theory",
"Commutative Rings",
"Ring Homomorphisms"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Idempotence/Element",
"Definition:Ideal of Ring",
"Definition:Generator of Ideal of Ring",
"Definition:Mapping",
"Definition:Surjection",
"Definition:Unital Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Definition:Ideal of Ring"... | [
"Ring Homomorphism by Idempotent",
"Definition:Surjection",
"Definition:Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Ring by Idempotent",
"Definition:Commutative and Unitary Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:U... |
proofwiki-18551 | Equivalence of Definitions of Generated Normal Subgroup | {{TFAE|def = Generated Normal Subgroup}}
Let $G$ be a group.
Let $S \subseteq G$ be a subset. | Let $H$ be the smallest normal subgroup containing $S$.
Let $\mathbb S$ be the set of normal subgroups containing $S$.
To show the equivalence of the two definitions, we need to show that $\ds H = \bigcap \mathbb S$.
Since $H$ is a normal subgroup containing $S$:
:$H \in \mathbb S$
By Intersection is Subset:
:$\ds \big... | {{TFAE|def = Generated Normal Subgroup}}
Let $G$ be a [[Definition:Group|group]].
Let $S \subseteq G$ be a [[Definition:Subset|subset]]. | Let $H$ be the [[Definition:Smallest Set by Set Inclusion|smallest]] [[Definition:Normal Subgroup|normal subgroup]] containing $S$.
Let $\mathbb S$ be the set of [[Definition:Normal Subgroup|normal subgroups]] containing $S$.
To show the equivalence of the two definitions, we need to show that $\ds H = \bigcap \mathb... | Equivalence of Definitions of Generated Normal Subgroup | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generated_Normal_Subgroup | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generated_Normal_Subgroup | [
"Generated Normal Subgroups"
] | [
"Definition:Group",
"Definition:Subset"
] | [
"Definition:Smallest Set by Set Inclusion",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Intersection is Subset/General Result",
"Intersection of Normal Subgroups is Normal",
"Definition:Normal Subgroup",
"Definition:Smallest Set by Set Inclusion",
"Defi... |
proofwiki-18552 | Grothendieck Universe is Closed under Mappings | Let $\mathbb U$ be a Grothendieck universe.
Let $u, v \in \mathbb U$.
Let $f: u \to v$ be a mapping realized as a relation consisting of ordered pairs in Kuratowski formalization.
Then $f \in \mathbb U$. | Let $u \times v$ be the finite cartesian product of $u$ and $v$ realized as a set of ordered pairs in Kuratowski formalization.
By definition of mapping, we have $f \subseteq u \times v$.
Then:
{{begin-eqn}}
{{eqn | l = u, v
| o = \in
| r = \mathbb U
}}
{{eqn | ll= \leadsto
| l = u \times v
| o ... | Let $\mathbb U$ be a [[Definition:Grothendieck Universe|Grothendieck universe]].
Let $u, v \in \mathbb U$.
Let $f: u \to v$ be a [[Definition:Mapping|mapping]] realized as a [[Definition:Relation as Subset of Cartesian Product|relation]] consisting of [[Definition:Kuratowski Formalization of Ordered Pair|ordered pair... | Let $u \times v$ be the [[Definition:Finite Cartesian Product|finite cartesian product]] of $u$ and $v$ realized as a [[Definition:Set|set]] of [[Definition:Kuratowski Formalization of Ordered Pair |ordered pairs in Kuratowski formalization]].
By definition of [[Definition:Mapping|mapping]], we have $f \subseteq u \ti... | Grothendieck Universe is Closed under Mappings | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Mappings | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Mappings | [
"Grothendieck Universes"
] | [
"Definition:Grothendieck Universe",
"Definition:Mapping",
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definition:Ordered Pair/Kuratowski Formalization"
] | [
"Definition:Cartesian Product/Finite",
"Definition:Set",
"Definition:Kuratowski Formalization of Ordered Pair ",
"Definition:Mapping",
"Grothendieck Universe is Closed under Binary Cartesian Product",
"Grothendieck Universe is Closed under Subset",
"Category:Grothendieck Universes"
] |
proofwiki-18553 | Grothendieck Universe is Closed under Subset | Let $\mathbb U$ be a Grothendieck universe.
Let $u \in \mathbb U$.
Let $v \subseteq u$ be a subset of $u$.
Then $v \in \mathbb U$. | {{begin-eqn}}
{{eqn | l = v
| o = \in
| r = \powerset u
| c = {{Defof|Power Set}}
}}
{{eqn | l = u
| o = \in
| r = \mathbb U
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = \powerset u
| o = \in
| r = \mathbb U
| c = Grothendieck Universe: Axiom $(3)$
}}
{{... | Let $\mathbb U$ be a [[Definition:Grothendieck Universe|Grothendieck universe]].
Let $u \in \mathbb U$.
Let $v \subseteq u$ be a [[Definition:Subset|subset]] of $u$.
Then $v \in \mathbb U$. | {{begin-eqn}}
{{eqn | l = v
| o = \in
| r = \powerset u
| c = {{Defof|Power Set}}
}}
{{eqn | l = u
| o = \in
| r = \mathbb U
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = \powerset u
| o = \in
| r = \mathbb U
| c = [[Definition:Grothendieck Universe|Groth... | Grothendieck Universe is Closed under Subset | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Subset | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Subset | [
"Grothendieck Universes"
] | [
"Definition:Grothendieck Universe",
"Definition:Subset"
] | [
"Definition:Grothendieck Universe",
"Definition:Grothendieck Universe",
"Category:Grothendieck Universes"
] |
proofwiki-18554 | Grothendieck Universe is Closed under Binary Cartesian Product | Let $\mathbb U$ be a Grothendieck universe.
Let $u, v \in \mathbb U$.
Let $u \times v$ be the binary cartesian product of $u$ and $v$ realized as a set of ordered pairs in Kuratowski formalization.
Then $u \times v \in \mathbb U$. | From Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union:
:$u \times v \subseteq \powerset {\powerset {u \cup v} }$
Then:
{{begin-eqn}}
{{eqn | l = u \cup v
| o = \in
| r = \mathbb U
| c = Grothendieck Universe is Closed under Binary Union
}}
{{eqn | ll=... | Let $\mathbb U$ be a [[Definition:Grothendieck Universe|Grothendieck universe]].
Let $u, v \in \mathbb U$.
Let $u \times v$ be the [[Definition:Finite Cartesian Product|binary cartesian product]] of $u$ and $v$ realized as a [[Definition:Set|set]] of [[Definition:Kuratowski Formalization of Ordered Pair|ordered pairs... | From [[Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union]]:
:$u \times v \subseteq \powerset {\powerset {u \cup v} }$
Then:
{{begin-eqn}}
{{eqn | l = u \cup v
| o = \in
| r = \mathbb U
| c = [[Grothendieck Universe is Closed under Binary Union]]
}}
... | Grothendieck Universe is Closed under Binary Cartesian Product | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Binary_Cartesian_Product | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Binary_Cartesian_Product | [
"Grothendieck Universes"
] | [
"Definition:Grothendieck Universe",
"Definition:Cartesian Product/Finite",
"Definition:Set",
"Definition:Ordered Pair/Kuratowski Formalization"
] | [
"Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union",
"Grothendieck Universe is Closed under Binary Union",
"Definition:Grothendieck Universe",
"Definition:Grothendieck Universe",
"Grothendieck Universe is Closed under Subset",
"Category:Grothendieck Universe... |
proofwiki-18555 | Grothendieck Universe is Closed under Binary Union | Let $\mathbb U$ be a Grothendieck universe.
Let $u, v \in \mathbb U$.
Then $u \cup v \in \mathbb U$. | If $\mathbb U = \O$, the claim is true.
Assume $\mathbb U \ne \O$.
By Nonempty Grothendieck Universe contains Von Neumann Natural Numbers, every von Neumann natural number is an element of $\mathbb U$.
In particular:
:$2 = \set {\O, \set \O} \in \mathbb U$
Using $2$ as an indexing set, we remember that $0 = \O$ and $1 ... | Let $\mathbb U$ be a [[Definition:Grothendieck Universe|Grothendieck universe]].
Let $u, v \in \mathbb U$.
Then $u \cup v \in \mathbb U$. | If $\mathbb U = \O$, the claim is true.
Assume $\mathbb U \ne \O$.
By [[Nonempty Grothendieck Universe contains Von Neumann Natural Numbers]], every [[Definition:Von Neumann Construction of Natural Numbers|von Neumann natural number]] is an element of $\mathbb U$.
In particular:
:$2 = \set {\O, \set \O} \in \mathbb ... | Grothendieck Universe is Closed under Binary Union | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Binary_Union | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Binary_Union | [
"Grothendieck Universes"
] | [
"Definition:Grothendieck Universe"
] | [
"Nonempty Grothendieck Universe contains Von Neumann Natural Numbers",
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Indexing Set",
"Definition:Grothendieck Universe",
"Definition:Grothendieck Universe",
"Definition:Grothendieck Universe",
"Category:Grothendieck Universes"
] |
proofwiki-18556 | Faithful Functor Reflects Monomorphisms | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F: \mathbf C \to \mathbf D$ be a faithful functor.
Let $x$ and $y$ be objects in $\mathbf C$.
Let $f: x \to y$ be a morphism in $\mathbf C$.
Let $\map F f : \map F x \to \map F y$ be a monomorphism in $\mathbf D$.
Then $f$ is a monomorphism in $\mathbf C$. | Let $z$ be an object in $\mathbf C$.
Let $g: z \to x$ and $h: z \to x$ be morphisms in $\mathbf C$ such that $f \circ g = f \circ h$.
{{begin-eqn}}
{{eqn | l = f \circ g
| r = f \circ h
}}
{{eqn | ll= \leadsto
| l = \map F f \circ \map F g
| r = \map F f \circ \map F h
| c = {{Defof|Functor}}
}}... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F: \mathbf C \to \mathbf D$ be a [[Definition:Faithful Functor|faithful functor]].
Let $x$ and $y$ be [[Definition:Object (Category Theory)|objects]] in $\mathbf C$.
Let $f: x \to y$ be a [[Definition:Morphism|morphism]] in $\mathbf C$.
Let... | Let $z$ be an [[Definition:Object (Category Theory)|object]] in $\mathbf C$.
Let $g: z \to x$ and $h: z \to x$ be [[Definition:Morphism|morphisms]] in $\mathbf C$ such that $f \circ g = f \circ h$.
{{begin-eqn}}
{{eqn | l = f \circ g
| r = f \circ h
}}
{{eqn | ll= \leadsto
| l = \map F f \circ \map F g
... | Faithful Functor Reflects Monomorphisms | https://proofwiki.org/wiki/Faithful_Functor_Reflects_Monomorphisms | https://proofwiki.org/wiki/Faithful_Functor_Reflects_Monomorphisms | [
"Functors",
"Monomorphisms (Category Theory)"
] | [
"Definition:Category",
"Definition:Faithful Functor",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Monomorphism (Category Theory)",
"Definition:Monomorphism (Category Theory)"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Monomorphism (Category Theory)",
"Definition:Monomorphism (Category Theory)",
"Category:Functors",
"Category:Monomorphisms (Category Theory)"
] |
proofwiki-18557 | Subring Module is Module/Special Case/Unitary Module | Let $\struct {R, +, \circ}$ be a ring with unity such that $1_R$ is that unity.
Let $1_R \in S$.
Then $\struct {R, +, \circ_S}_S$ is a unitary $S$-module. | From Subring Module is Module: Special Case, we have that $\struct {R, +, \circ_S}_S$ is an $S$-module.
Then {{hypothesis}} $1_R$ is the unity of $\struct {S, +, \circ_S}$.
Thus $\struct {S, +, \circ_S}$ is also a ring with unity.
It follows from Ring with Unity is Module over Itself that $\struct {R, +, \circ_S}_S$ is... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]] such that $1_R$ is that [[Definition:Unity of Ring|unity]].
Let $1_R \in S$.
Then $\struct {R, +, \circ_S}_S$ is a [[Definition:Unitary Module over Ring|unitary $S$-module]]. | From [[Subring Module is Module/Special Case|Subring Module is Module: Special Case]], we have that $\struct {R, +, \circ_S}_S$ is an [[Definition:Module over Ring|$S$-module]].
Then {{hypothesis}} $1_R$ is the [[Definition:Unity of Ring|unity]] of $\struct {S, +, \circ_S}$.
Thus $\struct {S, +, \circ_S}$ is also a ... | Subring Module is Module/Special Case/Unitary Module | https://proofwiki.org/wiki/Subring_Module_is_Module/Special_Case/Unitary_Module | https://proofwiki.org/wiki/Subring_Module_is_Module/Special_Case/Unitary_Module | [
"Subring Module is Module"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Unitary Module over Ring"
] | [
"Subring Module is Module/Special Case",
"Definition:Module over Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Ring with Unity",
"Ring with Unity is Module over Itself",
"Definition:Unitary Module over Ring"
] |
proofwiki-18558 | Integral Ideal iff Set of Integer Multiples | Let $J$ be a non-empty subset of the set of integers $\Z$.
Then:
:$J$ is an integral ideal
{{iff}}:
:$\exists m \in \Z: J = m \Z$. | === Sufficient Condition ===
{{:Integral Ideal is Set of Integer Multiples}}
{{qed|lemma}} | Let $J$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Integer|set of integers]] $\Z$.
Then:
:$J$ is an [[Definition:Integral Ideal|integral ideal]]
{{iff}}:
:$\exists m \in \Z: J = m \Z$. | === [[Integral Ideal is Set of Integer Multiples|Sufficient Condition]] ===
{{:Integral Ideal is Set of Integer Multiples}}
{{qed|lemma}} | Integral Ideal iff Set of Integer Multiples | https://proofwiki.org/wiki/Integral_Ideal_iff_Set_of_Integer_Multiples | https://proofwiki.org/wiki/Integral_Ideal_iff_Set_of_Integer_Multiples | [
"Integral Ideals",
"Sets of Integer Multiples"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Integer",
"Definition:Integral Ideal"
] | [
"Integral Ideal is Set of Integer Multiples"
] |
proofwiki-18559 | Principal Ideal Domain is Bézout Domain | Let $D$ be a principal ideal domain.
Then $D$ is a Bézout domain. | Let $J_1, J_2$ be ideals of $D$.
From Sum of Ideals is Ideal, $J_1 + J_2$ is an ideal of $D$.
By definition of principal ideal domain, $J_1 + J_2$ is principal.
Hence $D$ is a Bézout domain.
{{qed}}
Category:Bézout Domains
Category:Principal Ideal Domains
t17929ufe1vmkt57g3c8426k5448zdn | Let $D$ be a [[Definition:Principal Ideal Domain|principal ideal domain]].
Then $D$ is a [[Definition:Bézout Domain|Bézout domain]]. | Let $J_1, J_2$ be [[Definition:Ideal of Ring|ideals]] of $D$.
From [[Sum of Ideals is Ideal]], $J_1 + J_2$ is an [[Definition:Ideal of Ring|ideal]] of $D$.
By definition of [[Definition:Principal Ideal Domain|principal ideal domain]], $J_1 + J_2$ is [[Definition:Principal Ideal of Ring|principal]].
Hence $D$ is a [[... | Principal Ideal Domain is Bézout Domain | https://proofwiki.org/wiki/Principal_Ideal_Domain_is_Bézout_Domain | https://proofwiki.org/wiki/Principal_Ideal_Domain_is_Bézout_Domain | [
"Bézout Domains",
"Principal Ideal Domains"
] | [
"Definition:Principal Ideal Domain",
"Definition:Bézout Domain"
] | [
"Definition:Ideal of Ring",
"Sum of Ideals is Ideal",
"Definition:Ideal of Ring",
"Definition:Principal Ideal Domain",
"Definition:Principal Ideal of Ring",
"Definition:Bézout Domain",
"Category:Bézout Domains",
"Category:Principal Ideal Domains"
] |
proofwiki-18560 | Ring with Unity is Module over Itself | Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.
Then $\struct {R, +, \circ}_R$ is a unitary $R$-module. | From Ring is Module over Itself we have that $\struct {R, +, \circ}_R$ is an $R$-module.
We have {{hypothesis}} that $\struct {R, +, \circ}$ has a unity $1_R$.
For $\struct {R, +, \circ}_R$ to be unitary, it must satisfy the additional axiom:
{{begin-axiom}}
{{axiom | n = 4
| q = \forall x \in R
| m =... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
Then $\struct {R, +, \circ}_R$ is a [[Definition:Unitary Module over Ring|unitary $R$-module]]. | From [[Ring is Module over Itself]] we have that $\struct {R, +, \circ}_R$ is an [[Definition:Module over Ring|$R$-module]].
We have {{hypothesis}} that $\struct {R, +, \circ}$ has a [[Definition:Unity of Ring|unity]] $1_R$.
For $\struct {R, +, \circ}_R$ to be [[Definition:Unitary Module over Ring|unitary]], it mus... | Ring with Unity is Module over Itself | https://proofwiki.org/wiki/Ring_with_Unity_is_Module_over_Itself | https://proofwiki.org/wiki/Ring_with_Unity_is_Module_over_Itself | [
"Module Theory",
"Ring is Module over Itself"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Unitary Module over Ring"
] | [
"Ring is Module over Itself",
"Definition:Module over Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Unitary Module over Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Category:Module Theory",
"Category:Ring is Module over Itself"
] |
proofwiki-18561 | Subring Module is Module/Unitary | Let $\struct {R, +, \times}$ be a ring with unity.
Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module.
Let $1_R \in S$.
Then $\struct{G, +_G, \circ_S}_S$ is also unitary. | From Subring Module is Module, we have that $\struct {G, +_G, \circ_S}_S$ is an $S$-module.
It remains to be demonstrated that $\struct{G, +_G, \circ_S}_S$ is unitary.
To show this, we must prove that:
:$\forall x \in G: 1_R \circ_S x = x$
Since $1_R \in S$ by assumption, the product $1_R \circ_S x$ is defined.
We now... | Let $\struct {R, +, \times}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $\struct {G, +_G, \circ}_R$ be a [[Definition:Unitary Module over Ring|unitary $R$-module]].
Let $1_R \in S$.
Then $\struct{G, +_G, \circ_S}_S$ is also [[Definition:Unitary Module over Ring|unitary]]. | From [[Subring Module is Module]], we have that $\struct {G, +_G, \circ_S}_S$ is an [[Definition:Module over Ring|$S$-module]].
It remains to be demonstrated that $\struct{G, +_G, \circ_S}_S$ is [[Definition:Unitary Module over Ring|unitary]].
To show this, we must prove that:
:$\forall x \in G: 1_R \circ_S x = x$
... | Subring Module is Module/Unitary | https://proofwiki.org/wiki/Subring_Module_is_Module/Unitary | https://proofwiki.org/wiki/Subring_Module_is_Module/Unitary | [
"Subring Module is Module"
] | [
"Definition:Ring with Unity",
"Definition:Unitary Module over Ring",
"Definition:Unitary Module over Ring"
] | [
"Subring Module is Module",
"Definition:Module over Ring",
"Definition:Unitary Module over Ring",
"Definition:Scalar Multiplication/Module"
] |
proofwiki-18562 | Nonempty Grothendieck Universe contains Von Neumann Natural Numbers | Let $\mathbb U$ be a non-empty Grothendieck universe.
Let $\N$ denote the set of von Neumann natural numbers.
Then $\N$ is a subset of $\mathbb U$. | We prove the claim by induction. | Let $\mathbb U$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Grothendieck Universe|Grothendieck universe]].
Let $\N$ denote the [[Definition:Set|set]] of [[Definition:Von Neumann Construction of Natural Numbers|von Neumann natural numbers]].
Then $\N$ is a [[Definition:Subset|subset]] of $\mathbb U$. | We prove the claim by [[Definition:Principle of Mathematical Induction|induction]]. | Nonempty Grothendieck Universe contains Von Neumann Natural Numbers | https://proofwiki.org/wiki/Nonempty_Grothendieck_Universe_contains_Von_Neumann_Natural_Numbers | https://proofwiki.org/wiki/Nonempty_Grothendieck_Universe_contains_Von_Neumann_Natural_Numbers | [
"Grothendieck Universes"
] | [
"Definition:Non-Empty Set",
"Definition:Grothendieck Universe",
"Definition:Set",
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Subset"
] | [
"Principle of Mathematical Induction"
] |
proofwiki-18563 | Empty Set is Element of Nonempty Grothendieck Universe | Let $\mathbb U$ be a non-empty Grothendieck universe.
Then $\O \in \mathbb U$. | Let $A \in \mathbb U$ be an arbitrary element.
{{begin-eqn}}
{{eqn | l = \O
| o = \subseteq
| r = A
| c = Empty Set is Subset of All Sets
}}
{{eqn | ll = \leadsto
| l = \O
| o = \in
| r = \mathbb U
| c = Grothendieck Universe is Closed under Subset
}}
{{end-eqn}}
{{qed}}
Catego... | Let $\mathbb U$ be a non-[[Definition:Empty Set|empty]] [[Definition:Grothendieck Universe|Grothendieck universe]].
Then $\O \in \mathbb U$. | Let $A \in \mathbb U$ be an arbitrary [[Definition:Element|element]].
{{begin-eqn}}
{{eqn | l = \O
| o = \subseteq
| r = A
| c = [[Empty Set is Subset of All Sets]]
}}
{{eqn | ll = \leadsto
| l = \O
| o = \in
| r = \mathbb U
| c = [[Grothendieck Universe is Closed under Subset... | Empty Set is Element of Nonempty Grothendieck Universe | https://proofwiki.org/wiki/Empty_Set_is_Element_of_Nonempty_Grothendieck_Universe | https://proofwiki.org/wiki/Empty_Set_is_Element_of_Nonempty_Grothendieck_Universe | [
"Grothendieck Universes"
] | [
"Definition:Empty Set",
"Definition:Grothendieck Universe"
] | [
"Definition:Element",
"Empty Set is Subset of All Sets",
"Grothendieck Universe is Closed under Subset",
"Category:Grothendieck Universes"
] |
proofwiki-18564 | Adjoint of Identity Transformation | Let $\tuple {\HH, \innerprod \cdot \cdot_\HH}$ be a Hilbert space.
Let $I_\HH$ be the identity transformation on $\HH$.
Then:
:${I_\HH}^* = I_\HH$
where ${I_\HH}^*$ denotes the adjoint of $I_\HH$. | From Identity Mapping on Normed Vector Space is Bounded Linear Operator:
:$I_\HH$ is a bounded linear transformation.
So, from the existence part of Existence and Uniqueness of Adjoint:
:$I_\HH$ has an adjoint ${I_\HH}^*$.
That is:
:$\innerprod {I_\HH h} g_\HH = \innerprod h { {I_\HH}^* g}_\HH$
for all $h, g \in \HH$.
... | Let $\tuple {\HH, \innerprod \cdot \cdot_\HH}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $I_\HH$ be the [[Definition:Identity Mapping|identity transformation]] on $\HH$.
Then:
:${I_\HH}^* = I_\HH$
where ${I_\HH}^*$ denotes the [[Definition:Adjoint Linear Transformation|adjoint]] of $I_\HH$. | From [[Identity Mapping on Normed Vector Space is Bounded Linear Operator]]:
:$I_\HH$ is a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
So, from the existence part of [[Existence and Uniqueness of Adjoint]]:
:$I_\HH$ has an [[Definition:Adjoint Linear Transformation|adjoint]] ${I_\HH}^... | Adjoint of Identity Transformation | https://proofwiki.org/wiki/Adjoint_of_Identity_Transformation | https://proofwiki.org/wiki/Adjoint_of_Identity_Transformation | [
"Adjoints",
"Identity Mappings"
] | [
"Definition:Hilbert Space",
"Definition:Identity Mapping",
"Definition:Adjoint Linear Transformation"
] | [
"Identity Mapping on Normed Vector Space is Bounded Linear Operator",
"Definition:Bounded Linear Transformation",
"Existence and Uniqueness of Adjoint",
"Definition:Adjoint Linear Transformation",
"Definition:Identity Mapping",
"Existence and Uniqueness of Adjoint",
"Category:Adjoints",
"Category:Iden... |
proofwiki-18565 | Characterization of Invertible Bounded Linear Transformations | <onlyinclude>
Let $\struct {U, \norm \cdot_U}$ and $\struct {V, \norm \cdot_V}$ be normed vector spaces.
Let $A : V \to U$ be a linear transformation with inverse $A^{-1} : U \to V$.
Then $A^{-1}$ is a bounded linear transformation {{iff}}:
:there exists a real number $c > 0$ such that $\norm {A x}_U \ge c \norm x_V$ f... | From Inverse of Linear Transformation is Linear Transformation, we have:
:$A^{-1}$ is a linear transformation.
So we are interested in determining when $A^{-1}$ is bounded. | <onlyinclude>
Let $\struct {U, \norm \cdot_U}$ and $\struct {V, \norm \cdot_V}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $A : V \to U$ be a [[Definition:Linear Transformation|linear transformation]] with [[Definition:Inverse Mapping|inverse]] $A^{-1} : U \to V$.
Then $A^{-1}$ is a [[Definition... | From [[Inverse of Linear Transformation is Linear Transformation]], we have:
:$A^{-1}$ is a [[Definition:Linear Transformation|linear transformation]].
So we are interested in determining when $A^{-1}$ is [[Definition:Bounded Linear Transformation|bounded]]. | Characterization of Invertible Bounded Linear Transformations | https://proofwiki.org/wiki/Characterization_of_Invertible_Bounded_Linear_Transformations | https://proofwiki.org/wiki/Characterization_of_Invertible_Bounded_Linear_Transformations | [
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Transformation",
"Definition:Inverse Mapping",
"Definition:Bounded Linear Transformation",
"Definition:Real Number",
"Definition:Invertible Bounded Linear Transformation",
"Definition:Real Number"
] | [
"Inverse of Linear Transformation is Linear Transformation",
"Definition:Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation"
] |
proofwiki-18566 | Sum of Bounded Linear Transformations is Bounded Linear Transformation | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ and $\struct {\KK, \innerprod \cdot \cdot_\KK}$ be Hilbert spaces over $\mathbb F$.
Let $A, B : \HH \to \KK$ be bounded linear transformations.
Let $\norm \cdot$ be the norm on the space of bounded linear transformations.
Then:
:$A + ... | From Addition of Linear Transformations, we have that:
:$A + B$ is a linear transformation.
It remains to show that $A + B$ is bounded.
Let $\norm \cdot_\HH$ be the inner product norm on $\HH$.
Let $\norm \cdot_\KK$ be the inner product norm on $\KK$.
Since $A$ is a bounded linear transformation, from Fundamental Pro... | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ and $\struct {\KK, \innerprod \cdot \cdot_\KK}$ be [[Definition:Hilbert Space|Hilbert spaces]] over $\mathbb F$.
Let $A, B : \HH \to \KK$ be [[Definition:Bounded Linear Transformation|bounded linear transformations]].
Let $\norm \cd... | From [[Addition of Linear Transformations]], we have that:
:$A + B$ is a [[Definition:Linear Transformation|linear transformation]].
It remains to show that $A + B$ is [[Definition:Bounded Linear Transformation|bounded]].
Let $\norm \cdot_\HH$ be the [[Definition:Inner Product Norm|inner product norm]] on $\HH$.
... | Sum of Bounded Linear Transformations is Bounded Linear Transformation | https://proofwiki.org/wiki/Sum_of_Bounded_Linear_Transformations_is_Bounded_Linear_Transformation | https://proofwiki.org/wiki/Sum_of_Bounded_Linear_Transformations_is_Bounded_Linear_Transformation | [
"Linear Transformations on Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Transformation",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Bounded Linear Transformation"
] | [
"Addition of Linear Transformations",
"Definition:Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Inner Product Norm",
"Definition:Inner Product Norm",
"Definition:Bounded Linear Transformation",
"Fundamental Property of Norm on Bounded Linear Transformation",
"Definiti... |
proofwiki-18567 | Suprema and Infima of Combined Bounded Functions/Bounded Below | Let both $f$ and $g$ be bounded below on $S \subseteq \R$.
Then:
:$\ds \map {\inf_{x \mathop \in S} } {\map f x + c} = c + \map {\inf_{x \mathop \in S} } {\map f x}$
:$\ds \map {\inf_{x \mathop \in S} } {\map f x + \map g x} \ge \map {\inf_{x \mathop \in S} } {\map f x} + \map {\inf_{x \mathop \in S} } {\map g x}$
wher... | First we show that:
:$\ds \map {\inf_{x \mathop \in S} } {\map f x + c} = c + \map {\inf_{x \mathop \in S} } {\map f x}$
Let $T = \set {\map f x: x \in S}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\inf_{x \mathop \in S} } {\map f x + c}
| r = \map {\inf_{y \mathop \in T} } {y + c}
| c =
}}
{{eqn | r = c + \m... | Let both $f$ and $g$ be [[Definition:Bounded Below Real-Valued Function|bounded below]] on $S \subseteq \R$.
Then:
:$\ds \map {\inf_{x \mathop \in S} } {\map f x + c} = c + \map {\inf_{x \mathop \in S} } {\map f x}$
:$\ds \map {\inf_{x \mathop \in S} } {\map f x + \map g x} \ge \map {\inf_{x \mathop \in S} } {\map f ... | First we show that:
:$\ds \map {\inf_{x \mathop \in S} } {\map f x + c} = c + \map {\inf_{x \mathop \in S} } {\map f x}$
Let $T = \set {\map f x: x \in S}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\inf_{x \mathop \in S} } {\map f x + c}
| r = \map {\inf_{y \mathop \in T} } {y + c}
| c =
}}
{{eqn | r = c +... | Suprema and Infima of Combined Bounded Functions/Bounded Below | https://proofwiki.org/wiki/Suprema_and_Infima_of_Combined_Bounded_Functions/Bounded_Below | https://proofwiki.org/wiki/Suprema_and_Infima_of_Combined_Bounded_Functions/Bounded_Below | [
"Suprema and Infima of Combined Bounded Functions"
] | [
"Definition:Bounded Below Mapping/Real-Valued",
"Definition:Infimum of Mapping/Real-Valued Function"
] | [
"Infimum Plus Constant",
"Definition:Lower Bound of Mapping/Real-Valued"
] |
proofwiki-18568 | Suprema and Infima of Combined Bounded Functions/Bounded Above | Let both $f$ and $g$ be bounded above on $S \subseteq \R$.
Then:
:$\ds \map {\sup_{x \mathop \in S} } {\map f x + c} = c + \map {\sup_{x \mathop \in S} } {\map f x}$
:$\ds \map {\sup_{x \mathop \in S} } {\map f x + \map g x} \le \map {\sup_{x \mathop \in S} } {\map f x} + \map {\sup_{x \mathop \in S} } {\map g x}$
wher... | First we show that:
:$\ds \map {\sup_{x \mathop \in S} } {\map f x + c} = c + \map {\sup_{x \mathop \in S} } {\map f x}$
Let $T = \set {\map f x: x \in S}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\sup_{x \mathop \in S} } {\map f x + c}
| r = \map {\sup_{y \mathop \in T} } {y + c}
| c =
}}
{{eqn | r = c + \m... | Let both $f$ and $g$ be [[Definition:Bounded Above Real-Valued Function|bounded above]] on $S \subseteq \R$.
Then:
:$\ds \map {\sup_{x \mathop \in S} } {\map f x + c} = c + \map {\sup_{x \mathop \in S} } {\map f x}$
:$\ds \map {\sup_{x \mathop \in S} } {\map f x + \map g x} \le \map {\sup_{x \mathop \in S} } {\map f ... | First we show that:
:$\ds \map {\sup_{x \mathop \in S} } {\map f x + c} = c + \map {\sup_{x \mathop \in S} } {\map f x}$
Let $T = \set {\map f x: x \in S}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\sup_{x \mathop \in S} } {\map f x + c}
| r = \map {\sup_{y \mathop \in T} } {y + c}
| c =
}}
{{eqn | r = c +... | Suprema and Infima of Combined Bounded Functions/Bounded Above | https://proofwiki.org/wiki/Suprema_and_Infima_of_Combined_Bounded_Functions/Bounded_Above | https://proofwiki.org/wiki/Suprema_and_Infima_of_Combined_Bounded_Functions/Bounded_Above | [
"Suprema and Infima of Combined Bounded Functions"
] | [
"Definition:Bounded Above Mapping/Real-Valued",
"Definition:Supremum of Mapping/Real-Valued Function"
] | [
"Supremum Plus Constant",
"Definition:Upper Bound of Mapping/Real-Valued"
] |
proofwiki-18569 | Kernel of Linear Transformation contained in Kernel of different Linear Transformation implies Transformations are Proportional | Let $V$ be a complex vector space.
Let $\map \LL {V, \C}$ be the space of all linear transformations from $V$ to complex numbers $\C$.
Let $\ell, L \in \map \LL {V, \C}$ be such that:
:$\ker \ell \subseteq \ker L$
where $\ker$ denotes the kernel.
Then:
:$\exists c \in \C : L = c \ell$ | Suppose $\ell = \mathbf 0$.
Then:
:$\ker \ell = V$
That is, the kernel of $\ell$ is the entire vector space $V$.
Moreover:
:$\ker \ell \subseteq \ker L \implies \ker L = V$
Therefore:
:$L = \mathbf 0$
and we can set $c = 0$ to have:
:$L = \mathbf 0 = 0 \cdot \ell$
Suppose $\ell \ne \mathbf 0$.
By Linear Transformation ... | Let $V$ be a [[Definition:Complex Vector Space|complex vector space]].
Let $\map \LL {V, \C}$ be the [[Definition:Set of All Linear Transformations/Vector Space|space of all linear transformations]] from $V$ to [[Definition:Complex Number|complex numbers]] $\C$.
Let $\ell, L \in \map \LL {V, \C}$ be such that:
:$\ke... | Suppose $\ell = \mathbf 0$.
Then:
:$\ker \ell = V$
That is, the [[Definition:Kernel of Linear Transformation|kernel]] of $\ell$ is the entire [[Definition:Complex Vector Space|vector space]] $V$.
Moreover:
:$\ker \ell \subseteq \ker L \implies \ker L = V$
Therefore:
:$L = \mathbf 0$
and we can set $c = 0$ to hav... | Kernel of Linear Transformation contained in Kernel of different Linear Transformation implies Transformations are Proportional | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_contained_in_Kernel_of_different_Linear_Transformation_implies_Transformations_are_Proportional | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_contained_in_Kernel_of_different_Linear_Transformation_implies_Transformations_are_Proportional | [
"Linear Transformations"
] | [
"Definition:Complex Vector Space",
"Definition:Set of All Linear Transformations/Vector Space",
"Definition:Complex Number",
"Definition:Kernel"
] | [
"Definition:Kernel of Linear Transformation",
"Definition:Complex Vector Space",
"Linear Transformation Maps Zero Vector to Zero Vector"
] |
proofwiki-18570 | Norm on Bounded Linear Transformation is Submultiplicative | Let $\struct {X, \norm \cdot_X}$, $\struct {Y, \norm \cdot_Y}$ and $\struct {Z, \norm \cdot_Z}$ be normed vector spaces.
Let $A : X \to Y$ and $B : Y \to Z$ be bounded linear transformations.
Let $\norm \cdot _{\map B {X,Y} }$ be the norm for bounded linear transformations $X \to Y$.
Let $\norm \cdot _{\map B {Y,Z} }$... | From Composition of Linear Transformations is Linear Transformation, we have:
:$B \circ A$ is a linear transformation
Let $x \in X$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \norm {\paren {B \circ A} x}_Z
| o = \le
| r = \norm B \norm {A x}_Y
| c = Fundamental Property of Norm on Bounded Linear Transformation
}}
{... | Let $\struct {X, \norm \cdot_X}$, $\struct {Y, \norm \cdot_Y}$ and $\struct {Z, \norm \cdot_Z}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $A : X \to Y$ and $B : Y \to Z$ be [[Definition:Bounded Linear Transformation|bounded linear transformations]].
Let $\norm \cdot _{\map B {X,Y} }$ be the [[D... | From [[Composition of Linear Transformations is Linear Transformation]], we have:
:$B \circ A$ is a [[Definition:Linear Transformation|linear transformation]]
Let $x \in X$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \norm {\paren {B \circ A} x}_Z
| o = \le
| r = \norm B \norm {A x}_Y
| c = [[Fundamental Propert... | Norm on Bounded Linear Transformation is Submultiplicative | https://proofwiki.org/wiki/Norm_on_Bounded_Linear_Transformation_is_Submultiplicative | https://proofwiki.org/wiki/Norm_on_Bounded_Linear_Transformation_is_Submultiplicative | [
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Transformation",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Norm/Bounded Linear Transfor... | [
"Composition of Linear Transformations is Linear Transformation",
"Definition:Linear Transformation",
"Fundamental Property of Norm on Bounded Linear Transformation",
"Fundamental Property of Norm on Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Supremum of Set/Re... |
proofwiki-18571 | Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.
Let $\sequence{x_n}$ be a rational sequence.
Then:
:$\sequence{x_n}$ converges to $a$ {{iff}} $\sequence{x_n}$ is a representative of $a$
=== Corollary ===
{{:Ration... | Let $\norm {\,\cdot\,}^\Q_p$ be the p-adic norm on the rationals $\Q$.
By definition of the $p$-adic numbers:
:$\Q_p$ is the quotient ring $\CC \, \big / \NN$
where:
:$\CC$ is the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.
and
:$\NN$ is the set of null sequences in $\struct {\Q, \... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a$ be a [[Definition:P-adic Number|$p$-adic number]], that is [[Definition:Left Coset|left coset]], in $\Q_p$.
Let $\sequence{x_n}$ be a [[Defin... | Let $\norm {\,\cdot\,}^\Q_p$ be the [[Definition:P-adic Norm|p-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]].
By definition of the [[Definition:Field of P-adic Numbers|$p$-adic numbers]]:
:$\Q_p$ is the [[Definition:Quotient Ring|quotient ring]] $\CC \, \big / \NN$
where:
:$\CC$ is the [[Definition... | Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit | https://proofwiki.org/wiki/Rational_Sequence_Converges_in_P-adic_Numbers_iff_Sequence_Represents_Limit | https://proofwiki.org/wiki/Rational_Sequence_Converges_in_P-adic_Numbers_iff_Sequence_Represents_Limit | [
"P-adic Number Theory",
"Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Number",
"Definition:Coset/Left Coset",
"Definition:Rational Sequence",
"Definition:Convergent Sequence/P-adic Numbers",
"Definition:P-adic Number/Representative",
"Rational Sequence Converges in P-adic Numbers ... | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Field of P-adic Numbers",
"Definition:Quotient Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Null Sequence",
"Definition:Valued Field of P-adic Numbers",
"Quotient Ring of Cauchy Sequences is Normed Div... |
proofwiki-18572 | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit | Let $\struct {R, \norm{\,\cdot\,}_R}$ be a normed division ring.
Let $\CC$ be the ring of Cauchy sequences over $R$
Let $\NN$ be the set of null sequences.
Let $Q = \CC / \NN$ where $\CC / \NN$ denotes a quotient ring.
Let $\norm {\, \cdot \,}_Q: Q \to \R_{\ge 0}$ be the norm on the quotient ring $Q$ defined by:
:$\d... | === Lemma 1 ===
{{:Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 1}}{{qed|lemma}} | Let $\struct {R, \norm{\,\cdot\,}_R}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\CC$ be the [[Definition:Ring of Cauchy Sequences|ring of Cauchy sequences over $R$]]
Let $\NN$ be the [[Definition:Set|set]] of [[Definition:Null Sequence in Normed Division Ring|null sequences]].
Let $Q = \CC... | === [[Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 1|Lemma 1]] ===
{{:Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 1}}{{qed|lemma}} | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit | [
"Normed Division Rings",
"Complete Metric Spaces",
"Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit"
] | [
"Definition:Normed Division Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Null Sequence/Normed Division Ring",
"Definition:Quotient Ring",
"Definition:Norm/Division Ring",
"Definition:Quotient Ring",
"Definition:Mapping",
"Definition:Quotient Ring",
"Definition:Coset/... | [
"Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 1"
] |
proofwiki-18573 | Vanishing Distributional Derivative of Distribution implies Distribution is Constant | Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Let $\mathbf 0$ be the zero distribution.
Suppose the distributional derivative of $T$ vanishes:
:$\ds \dfrac \d {\d x} T = \mathbf 0$
Then $T$ is a constant distribution. | Let $\phi \in \map \DD \R$ be a test function.
Then:
{{begin-eqn}}
{{eqn | l = 0
| r = \map {\mathbf 0} \phi
}}
{{eqn | r = \map {T'} \phi
| c = Assumption of the Theorem
}}
{{eqn | r = - \map T {\phi'}
| c = {{Defof|Distributional Derivative}}
}}
{{end-eqn}}
Hence:
:$\set {\phi' : \phi \in \map \DD ... | Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Let $\mathbf 0$ be the [[Definition:Zero Distribution|zero distribution]].
Suppose the [[Definition:Distributional Derivative|distributional derivative]] of $T$ vanishes:
:$\ds \dfrac \d {\d x} T = \mathbf 0$
Then $T$ is a ... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
Then:
{{begin-eqn}}
{{eqn | l = 0
| r = \map {\mathbf 0} \phi
}}
{{eqn | r = \map {T'} \phi
| c = Assumption of the Theorem
}}
{{eqn | r = - \map T {\phi'}
| c = {{Defof|Distributional Derivative}}
}}
{{end-eqn}}
Hence:
:... | Vanishing Distributional Derivative of Distribution implies Distribution is Constant | https://proofwiki.org/wiki/Vanishing_Distributional_Derivative_of_Distribution_implies_Distribution_is_Constant | https://proofwiki.org/wiki/Vanishing_Distributional_Derivative_of_Distribution_implies_Distribution_is_Constant | [
"Examples of Distributional Derivatives"
] | [
"Definition:Schwartz Distribution",
"Definition:Zero Mapping/Schwartz Distribution",
"Definition:Distributional Derivative",
"Definition:Constant Distribution"
] | [
"Definition:Test Function",
"Definition:Kernel",
"Definition:Constant Mapping",
"Definition:Schwartz Distribution",
"Characterization of Derivative of Test Function",
"Test Function Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Kernel of Linear Transformation co... |
proofwiki-18574 | Product Rule for Distributional Derivatives of Distributions multiplied by Smooth Functions | Let $\alpha \in \map {C^\infty} \R$ be a smooth real function.
Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Then in the distributional sense it holds that:
:$\paren {\alpha T}' = \alpha' T + \alpha T'$ | Let $\phi \in \map \DD \R$ be a test function.
By the Product Rule for Derivatives:
:$\paren {\alpha \phi}' = \alpha' \phi + \alpha \phi'$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\paren {\alpha T}'} \phi
| r = -\map {\paren {\alpha T} } {\phi'}
| c = {{Defof|Distributional Derivative}}
}}
{{eqn | r = -\map T... | Let $\alpha \in \map {C^\infty} \R$ be a [[Definition:Smooth Real Function|smooth real function]].
Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Then in the [[Definition:Distributional Derivative|distributional sense]] it holds that:
:$\paren {\alpha T}' = \alpha' T + ... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
By the [[Product Rule for Derivatives]]:
:$\paren {\alpha \phi}' = \alpha' \phi + \alpha \phi'$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\paren {\alpha T}'} \phi
| r = -\map {\paren {\alpha T} } {\phi'}
| c = {{Defof|Distributiona... | Product Rule for Distributional Derivatives of Distributions multiplied by Smooth Functions | https://proofwiki.org/wiki/Product_Rule_for_Distributional_Derivatives_of_Distributions_multiplied_by_Smooth_Functions | https://proofwiki.org/wiki/Product_Rule_for_Distributional_Derivatives_of_Distributions_multiplied_by_Smooth_Functions | [
"Distributional Derivatives"
] | [
"Definition:Smooth Real Function",
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative"
] | [
"Definition:Test Function",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Addition of Distributions"
] |
proofwiki-18575 | Distributional Solution to y' - k y = 0 | Let $f \in \map {C^1} \R$ be a continuously differentiable function.
Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Let $T_f$ be a Schwartz distribution associated with $f$.
Let $\mathbf 0 \in \map {\DD'} \R$ be the zero distribution.
Let $T$ be a distributional solution to the following distributional differen... | In the distributional sense we have:
{{begin-eqn}}
{{eqn | l = \paren {\map \exp {- k x} T}'
| r = -\map \exp {-k x} T + \map \exp {-k x} T'
| c = Product Rule for Distributional Derivatives of Distributions multiplied by Smooth Functions
}}
{{eqn | r = \map \exp {-k x} \paren {-k T + T'}
}}
{{eqn | r = \ma... | Let $f \in \map {C^1} \R$ be a [[Definition:Continuously Differentiable|continuously differentiable function]].
Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Let $T_f$ be a [[Definition:Schwartz Distribution|Schwartz distribution]] associated with $f$.
Let $\mathbf 0 \in... | In the [[Definition:Schwartz Distribution|distributional]] sense we have:
{{begin-eqn}}
{{eqn | l = \paren {\map \exp {- k x} T}'
| r = -\map \exp {-k x} T + \map \exp {-k x} T'
| c = [[Product Rule for Distributional Derivatives of Distributions multiplied by Smooth Functions]]
}}
{{eqn | r = \map \exp {-... | Distributional Solution to y' - k y = 0 | https://proofwiki.org/wiki/Distributional_Solution_to_y'_-_k_y_=_0 | https://proofwiki.org/wiki/Distributional_Solution_to_y'_-_k_y_=_0 | [
"Examples of Distributional Solutions",
"Distributional Derivatives"
] | [
"Definition:Continuously Differentiable",
"Definition:Schwartz Distribution",
"Definition:Schwartz Distribution",
"Definition:Zero Mapping/Schwartz Distribution",
"Weak Solution/Examples/Distributional Solution",
"Definition:Differential Equation/Distributional",
"Definition:Differential Equation/Soluti... | [
"Definition:Schwartz Distribution",
"Product Rule for Distributional Derivatives of Distributions multiplied by Smooth Functions",
"Vanishing Distributional Derivative of Distribution implies Distribution is Constant"
] |
proofwiki-18576 | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 1 | :$\sequence{x_n}$ is a Cauchy sequence {{iff}} $\sequence{\map \phi {x_n}}$ is a Cauchy sequence | From Embedding Division Ring into Quotient Ring of Cauchy Sequences:
:the mapping $\phi: R \to Q$ is a distance-preserving monomorphism
We have
{{begin-eqn}}
{{eqn | r = \sequence{x_n} \text{is a Cauchy sequence}
| o =
}}
{{eqn | ll = \leadstoandfrom
| o =
| r = \forall \epsilon > 0 : \exists N \in ... | :$\sequence{x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] {{iff}} $\sequence{\map \phi {x_n}}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] | From [[Embedding Division Ring into Quotient Ring of Cauchy Sequences]]:
:the [[Definition:Mapping|mapping]] $\phi: R \to Q$ is a [[Definition:Distance-Preserving Mapping|distance-preserving]] [[Definition:Ring Monomorphism|monomorphism]]
We have
{{begin-eqn}}
{{eqn | r = \sequence{x_n} \text{is a Cauchy sequence}
... | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 1 | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit/Lemma_1 | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit/Lemma_1 | [
"Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit"
] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Embedding Division Ring into Quotient Ring of Cauchy Sequences",
"Definition:Mapping",
"Definition:Distance-Preserving Mapping",
"Definition:Ring Monomorphism",
"Category:Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit"
] |
proofwiki-18577 | Dx - k is Hypoelliptic | Let $f \in \map {C^\infty} \R$ be a smooth real function.
Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Let $T_f$ be the Schwartz distribution associated with $f$.
Let $k \in \R$.
Suppose in the distributional sense it holds that:
:$\paren {\dfrac \d {\d x} - k} T = T_f \quad (1)$
Then there is a $c \in \R$ su... | By assumption $f \in \map {C^\infty} \R$.
By Solution to Linear First Order ODE with Constant Coefficients:
:$\exists F \in \map {C^\infty} \R : \paren {\dfrac \d {\d x} - k} F = f$
By Differentiable Function as Distribution and multiplication of Schwartz distribution by a smooth function:
:$\exists F \in \map {C^\inft... | Let $f \in \map {C^\infty} \R$ be a [[Definition:Smooth Real Function|smooth real function]].
Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Let $T_f$ be the [[Definition:Schwartz Distribution|Schwartz distribution]] associated with $f$.
Let $k \in \R$.
Suppose in the [[... | By assumption $f \in \map {C^\infty} \R$.
By [[Solution to Linear First Order ODE with Constant Coefficients]]:
:$\exists F \in \map {C^\infty} \R : \paren {\dfrac \d {\d x} - k} F = f$
By [[Differentiable Function as Distribution]] and [[Definition:Multiplication of Schwartz Distribution by Smooth Function|multipli... | Dx - k is Hypoelliptic | https://proofwiki.org/wiki/Dx_-_k_is_Hypoelliptic | https://proofwiki.org/wiki/Dx_-_k_is_Hypoelliptic | [
"Examples of Hypoelliptic Operators",
"Examples of Distributional Solutions"
] | [
"Definition:Smooth Real Function",
"Definition:Schwartz Distribution",
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Hypoelliptic Operator"
] | [
"Solution to Linear First Order ODE with Constant Coefficients",
"Differentiable Function as Distribution",
"Definition:Multiplication of Schwartz Distribution by Smooth Function",
"Definition:Zero Mapping/Schwartz Distribution",
"Distributional Solution to y' - k y = 0"
] |
proofwiki-18578 | Riemann Uniformization Theorem | Every connected $2$-dimensional manifold has a complete Riemannian metric with the constant Gaussian curvature.
{{Stub|More precise statement}} | {{ProofWanted}}
{{Namedfor|Georg Friedrich Bernhard Riemann|cat = Riemann}} | Every [[Definition:Connected Manifold|connected]] [[Definition:Dimension (Topology)|$2$-dimensional]] [[Definition:Topological Manifold|manifold]] has a [[Definition:Complete Riemannian Metric|complete]] [[Definition:Riemannian Metric|Riemannian metric]] with the [[Definition:Constant|constant]] [[Definition:Gaussian C... | {{ProofWanted}}
{{Namedfor|Georg Friedrich Bernhard Riemann|cat = Riemann}} | Riemann Uniformization Theorem | https://proofwiki.org/wiki/Riemann_Uniformization_Theorem | https://proofwiki.org/wiki/Riemann_Uniformization_Theorem | [
"Riemannian Manifolds",
"Curvature"
] | [
"Definition:Connected Manifold",
"Definition:Dimension (Topology)",
"Definition:Topological Manifold",
"Definition:Complete Riemannian Metric",
"Definition:Riemannian Metric",
"Definition:Constant",
"Definition:Gaussian Curvature"
] | [] |
proofwiki-18579 | Cartan-Hadamard Theorem | Let $M$ be a complete connected $n$-dimensional Riemannian manifold.
Suppose all sectional curvatures of $M$ are less than or equal to zero.
Then the universal covering space of $M$ is diffeomorphic to $\R^n$.
{{explain|What is the universal covering space of $M$? Does such a space exist? The book uses this theorem as ... | {{ProofWanted}}
{{Namedfor|Élie Joseph Cartan|name2 = Jacques Salomon Hadamard|cat = Cartan|cat2 = Hadamard}} | Let $M$ be a [[Definition:Metrically Complete Connected Riemannian Manifold|complete]] [[Definition:Connected Manifold|connected]] [[Definition:Dimension of Riemannian Manifold|$n$-dimensional]] [[Definition:Riemannian Manifold|Riemannian manifold]].
Suppose all [[Definition:Sectional Curvature|sectional curvatures]] ... | {{ProofWanted}}
{{Namedfor|Élie Joseph Cartan|name2 = Jacques Salomon Hadamard|cat = Cartan|cat2 = Hadamard}} | Cartan-Hadamard Theorem | https://proofwiki.org/wiki/Cartan-Hadamard_Theorem | https://proofwiki.org/wiki/Cartan-Hadamard_Theorem | [
"Riemannian Manifolds"
] | [
"Definition:Metrically Complete Connected Riemannian Manifold",
"Definition:Connected Manifold",
"Definition:Riemannian Manifold/Dimension",
"Definition:Riemannian Manifold",
"Definition:Sectional Curvature",
"Definition:Universal Cover",
"Definition:Diffeomorphism",
"Definition:Universal Cover"
] | [] |
proofwiki-18580 | Bonnet-Myers Theorem | Let $M$ be a complete connected Riemannian manifold.
Suppose all the sectional curvatures of $M$ are bounded below by a positive constant.
Then $M$ is compact and has a finite fundamental group. | {{ProofWanted}}
{{Namedfor|Pierre Ossian Bonnet|name2 = Sumner Byron Myers|cat = Bonnet|cat2 = Myers}} | Let $M$ be a [[Definition:Metrically Complete Connected Riemannian Manifold|complete]] [[Definition:Connnected Manifold|connected]] [[Definition:Riemannian Manifold|Riemannian manifold]].
Suppose all the [[Definition:Sectional Curvature|sectional curvatures]] of $M$ are [[Definition:Lower Bound of Subset of Real Numbe... | {{ProofWanted}}
{{Namedfor|Pierre Ossian Bonnet|name2 = Sumner Byron Myers|cat = Bonnet|cat2 = Myers}} | Bonnet-Myers Theorem | https://proofwiki.org/wiki/Bonnet-Myers_Theorem | https://proofwiki.org/wiki/Bonnet-Myers_Theorem | [
"Riemannian Manifolds"
] | [
"Definition:Metrically Complete Connected Riemannian Manifold",
"Definition:Connnected Manifold",
"Definition:Riemannian Manifold",
"Definition:Sectional Curvature",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Positive/Number",
"Definition:Compact Manifold",
"Definition:Finite",
"Defin... | [] |
proofwiki-18581 | Norm of Compact Hermitian Operator is Equal to Greatest Modulus of Eigenvalue | Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ be a Hilbert space over $\C$.
Let $T : \HH \to \HH$ be a compact Hermitian operator.
Let $\map {\sigma_p} T$ be the point spectrum of $T$.
Then:
:$\norm T_{\map \BB \HH} = \max \set {\cmod \lambda : \lambda \in \map {\sigma_p} T}$
where $\norm \cdot_{\map \BB \HH}$ is t... | Suppose that $T = 0$.
Then $\norm T_{\map \BB \HH} = 0$ from {{NormAxiomVector|1}}.
Then $-\lambda I$ is injective {{iff}} $\lambda \ne 0$.
So $\map {\sigma_p} T = \set 0$ in this case and so:
:$\max \set {\cmod \lambda : \lambda \in \map {\sigma_p} T} = 0 = \norm T_{\map \BB \HH}$.
Now let $T \ne 0$ so that $\norm ... | Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $T : \HH \to \HH$ be a [[Definition:Compact Linear Operator|compact]] [[Definition:Hermitian Operator|Hermitian]] [[Definition:Linear Operator|operator]].
Let $\map {\sigma_p} T$ be the [[Definition:Point Sp... | Suppose that $T = 0$.
Then $\norm T_{\map \BB \HH} = 0$ from {{NormAxiomVector|1}}.
Then $-\lambda I$ is [[Definition:Injective|injective]] {{iff}} $\lambda \ne 0$.
So $\map {\sigma_p} T = \set 0$ in this case and so:
:$\max \set {\cmod \lambda : \lambda \in \map {\sigma_p} T} = 0 = \norm T_{\map \BB \HH}$.
No... | Norm of Compact Hermitian Operator is Equal to Greatest Modulus of Eigenvalue | https://proofwiki.org/wiki/Norm_of_Compact_Hermitian_Operator_is_Equal_to_Greatest_Modulus_of_Eigenvalue | https://proofwiki.org/wiki/Norm_of_Compact_Hermitian_Operator_is_Equal_to_Greatest_Modulus_of_Eigenvalue | [
"Compact Linear Transformations",
"Hermitian Operators"
] | [
"Definition:Hilbert Space",
"Definition:Compact Linear Operator",
"Definition:Hermitian Operator",
"Definition:Linear Operator",
"Definition:Point Spectrum of Linear Operator",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Definition:Injective",
"Norm of Hermitian Operator",
"Definition:Supremum of Set/Real Numbers",
"Definition:Sequence",
"Squeeze Theorem",
"Operator is Hermitian iff Numerical Range is Real",
"Definition:Convergent Sequence",
"Definition:Subsequence",
"Definition:Convergent Sequence",
"Modulus of ... |
proofwiki-18582 | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 2 | For each $n \in \N$:
:$\norm{\map \phi {x_n} - y}_Q = \ds \lim_{m \mathop \to \infty} \norm{x_n - x_m}_R$ | Let $n \in \N$.
We have:
:$\tuple {x_n, x_n, x_n, \ldots} \in \map \phi {x_n}$
:$\sequence{x_m} \in y$
From Element of Group is in Unique Coset of Subgroup:
:$\tuple {x_n, x_n, x_n, \ldots} + \NN = \map \phi {x_n}$
:$\sequence{x_m} + \NN = y$
Then:
{{begin-eqn}}
{{eqn | l = \sequence{x_n - x_m}_{m \in \N} + \NN
... | For each $n \in \N$:
:$\norm{\map \phi {x_n} - y}_Q = \ds \lim_{m \mathop \to \infty} \norm{x_n - x_m}_R$ | Let $n \in \N$.
We have:
:$\tuple {x_n, x_n, x_n, \ldots} \in \map \phi {x_n}$
:$\sequence{x_m} \in y$
From [[Element of Group is in Unique Coset of Subgroup]]:
:$\tuple {x_n, x_n, x_n, \ldots} + \NN = \map \phi {x_n}$
:$\sequence{x_m} + \NN = y$
Then:
{{begin-eqn}}
{{eqn | l = \sequence{x_n - x_m}_{m \in \N} + \N... | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 2 | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit/Lemma_2 | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit/Lemma_2 | [
"Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit"
] | [] | [
"Element of Group is in Unique Coset of Subgroup",
"Combination Theorem for Cauchy Sequences/Difference Rule",
"Element of Group is in its own Coset",
"Definition:Induced Norm on Quotient of Cauchy Sequences",
"Category:Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit"
] |
proofwiki-18583 | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Necessary Condition | {{:Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit}}
If $\sequence{\map \phi {x_n}}$ converges to $y$ then $\sequence{x_n} \in y$ | Let $\sequence{\map \phi {x_n}}$ converge to $y$.
From Convergent Sequence in Normed Division Ring is Cauchy Sequence:
:$\sequence{\map \phi {x_n}}$ is a Cauchy Sequence
From Lemma 1:
:$\sequence{x_n}$ is a Cauchy Sequence
Let $y'$ be the left coset that contains $\sequence{x_n}$.
From sufficient condition:
:$\ds \lim_... | {{:Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit}}
If $\sequence{\map \phi {x_n}}$ [[Definition:Convergent Sequence in Normed Division Ring|converges]] to $y$ then $\sequence{x_n} \in y$ | Let $\sequence{\map \phi {x_n}}$ [[Definition:Convergent Sequence in Normed Division Ring|converge]] to $y$.
From [[Convergent Sequence in Normed Division Ring is Cauchy Sequence]]:
:$\sequence{\map \phi {x_n}}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy Sequence]]
From [[Normed Division Ring Se... | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Necessary Condition | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit/Necessary_Condition | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit/Necessary_Condition | [
"Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit"
] | [
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Convergent Sequence is Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 1",
"Definition:Cauchy Sequence/Normed Division ... |
proofwiki-18584 | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Sufficient Condition | {{:Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit}}
If $\sequence{x_n} \in y$ then $\sequence{\map \phi {x_n}}$ converges to $y$ | Let $\sequence{x_n} \in y$.
Then $\sequence{x_n}$ is a Cauchy Sequence by definition of $y$.
Let $\epsilon > 0$ be arbitrary.
By definition of a Cauchy sequence:
:$\exists N \in \N: \forall n, m \ge N : \norm{x_n - x_m}_R < \dfrac \epsilon 2$
Let $n \ge N$ be arbitrary.
From Difference Rule for Cauchy Sequences in Norm... | {{:Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit}}
If $\sequence{x_n} \in y$ then $\sequence{\map \phi {x_n}}$ [[Definition:Convergent Sequence in Normed Division Ring|converges]] to $y$ | Let $\sequence{x_n} \in y$.
Then $\sequence{x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy Sequence]] by definition of $y$.
Let $\epsilon > 0$ be arbitrary.
By definition of a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]:
:$\exists N \in \N: \forall n, m \ge N : \no... | Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Sufficient Condition | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit/Sufficient_Condition | https://proofwiki.org/wiki/Normed_Division_Ring_Sequence_Converges_in_Completion_iff_Sequence_Represents_Limit/Sufficient_Condition | [
"Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit"
] | [
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Combination Theorem for Cauchy Sequences/Difference Rule",
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Norm Sequence of Cauchy Sequence has Limit",
"Inequality Rule for R... |
proofwiki-18585 | Gram-Schmidt Orthogonalization/Corollary 2 | Let $\struct {V, \innerprod \cdot \cdot}$ be an $n$-dimensional inner product space over $\R$ or $\C$.
Let $\tuple {v_1, \ldots, v_n}$ be any ordered basis for $V$.
Then there is an orthonormal ordered basis $\tuple {b_1, \ldots, b_n}$ satisfying the following conditions:
:$\forall k \in \set {1, \ldots, n} : \span \se... | By the definition of basis, it follows that $\set{ v_1, \ldots, v_n }$ is a linearly independent subset of $V$.
From {{Corollary|Gram-Schmidt Orthogonalization|1}}, it follows that there exists an orthonormal subset $\set {b_1, \ldots, b_n}$ of $V$ such that:
:$\forall k \in \set {1, \ldots, n}: \span \set {v_1, \ldots... | Let $\struct {V, \innerprod \cdot \cdot}$ be an [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Inner Product Space|inner product space]] over $\R$ or $\C$.
Let $\tuple {v_1, \ldots, v_n}$ be any [[Definition:Ordered Basis|ordered basis]] for $V$.
Then there is an [[Definition:Orthonormal Basis... | By the definition of [[Definition:Basis of Vector Space|basis]], it follows that $\set{ v_1, \ldots, v_n }$ is a [[Definition:Linearly Independent Set|linearly independent]] [[Definition:Subset|subset]] of $V$.
From {{Corollary|Gram-Schmidt Orthogonalization|1}}, it follows that there exists an [[Definition:Orthonorma... | Gram-Schmidt Orthogonalization/Corollary 2 | https://proofwiki.org/wiki/Gram-Schmidt_Orthogonalization/Corollary_2 | https://proofwiki.org/wiki/Gram-Schmidt_Orthogonalization/Corollary_2 | [
"Vector Algebra",
"Linear Algebra",
"Gram-Schmidt Orthogonalization"
] | [
"Definition:Dimension of Vector Space",
"Definition:Inner Product Space",
"Definition:Ordered Basis",
"Definition:Orthonormal Basis of Vector Space",
"Definition:Ordered Basis"
] | [
"Definition:Basis of Vector Space",
"Definition:Linearly Independent/Set",
"Definition:Subset",
"Definition:Orthonormal Subset",
"Definition:Generated Submodule/Linear Span",
"Orthogonal Set is Linearly Independent Set",
"Definition:Linearly Independent/Set",
"Definition:Subset",
"Sufficient Conditi... |
proofwiki-18586 | Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit/P-adic Expansion | Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion.
Then:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$ converges to $a$ {{iff}} $\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a representative of $a$ | By definition of a $p$-adic expansion:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a rational sequence.
The theorem follows from Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit.
{{qed}}
Category:Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit
rqerqmrf1d8slcacex4... | Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be a [[Definition:P-adic Expansion|$p$-adic expansion]].
Then:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$ [[Definition:Convergent P-adic Sequence|converges]] to $a$ {{iff}} $\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a [[Definition:Representative of P-adic Number|represe... | By definition of a [[Definition:P-adic Expansion|$p$-adic expansion]]:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a [[Definition:Rational Sequence|rational sequence]].
The [[Definition:Theorem|theorem]] follows from [[Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit]].
{{qed}}
[[Category... | Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit/P-adic Expansion | https://proofwiki.org/wiki/Rational_Sequence_Converges_in_P-adic_Numbers_iff_Sequence_Represents_Limit/P-adic_Expansion | https://proofwiki.org/wiki/Rational_Sequence_Converges_in_P-adic_Numbers_iff_Sequence_Represents_Limit/P-adic_Expansion | [
"Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit"
] | [
"Definition:P-adic Expansion",
"Definition:Convergent Sequence/P-adic Numbers",
"Definition:P-adic Number/Representative"
] | [
"Definition:P-adic Expansion",
"Definition:Rational Sequence",
"Definition:Theorem",
"Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit",
"Category:Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit"
] |
proofwiki-18587 | Solution to Distributional Ordinary Differential Equation with Constant Coefficients | Let $D$ be an ordinary differential operator with constant complex coefficients:
:$\ds D = \sum_{k \mathop = 0}^n a_k \paren {\dfrac \d {\d x}}^k$
Let $f \in \map {C^\infty} \R$ be a smooth real function.
Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Let $T_f$ be a Schwartz distribution associated with $f$.
Su... | Let $\map P \xi$ be a polynomial over complex numbers such that:
:$\ds \map P \xi = \sum_{k \mathop = 0}^n a_k \xi^k = a_n \prod_{k \mathop = 0}^n \paren {\xi - \lambda_k}$
where $a_n \ne 0$.
Then there exists a polynomial $\map Q \xi$ such that:
:$\map P \xi = \paren {\xi - \lambda_n} \map Q \lambda$
Let:
:$\ds D = \s... | Let $D$ be an [[Definition:Ordinary Derivative|ordinary]] [[Definition:Differential Operator|differential operator]] with [[Definition:Constant|constant]] [[Definition:Complex Number|complex]] [[Definition:Coefficient|coefficients]]:
:$\ds D = \sum_{k \mathop = 0}^n a_k \paren {\dfrac \d {\d x}}^k$
Let $f \in \map {C... | Let $\map P \xi$ be a [[Definition:Polynomial over Complex Numbers|polynomial over complex numbers]] such that:
:$\ds \map P \xi = \sum_{k \mathop = 0}^n a_k \xi^k = a_n \prod_{k \mathop = 0}^n \paren {\xi - \lambda_k}$
where $a_n \ne 0$.
Then there exists a [[Definition:Polynomial over Complex Numbers|polynomial]] ... | Solution to Distributional Ordinary Differential Equation with Constant Coefficients | https://proofwiki.org/wiki/Solution_to_Distributional_Ordinary_Differential_Equation_with_Constant_Coefficients | https://proofwiki.org/wiki/Solution_to_Distributional_Ordinary_Differential_Equation_with_Constant_Coefficients | [
"Examples of Hypoelliptic Operators",
"Examples of Distributional Solutions",
"Distributional Derivatives"
] | [
"Definition:Derivative",
"Definition:Differential Operator",
"Definition:Constant",
"Definition:Complex Number",
"Definition:Coefficient",
"Definition:Smooth Real Function",
"Definition:Schwartz Distribution",
"Definition:Schwartz Distribution",
"Weak Solution/Examples/Distributional Solution",
"D... | [
"Definition:Polynomial/Complex Numbers",
"Definition:Polynomial/Complex Numbers",
"Principle of Mathematical Induction"
] |
proofwiki-18588 | Vector Space over Division Subring is Vector Space/Special Case | Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$.
Let $S$ be a division subring of $R$, such that $1_R \in S$.
The vector space $\struct {R, +, \circ_S}_S$ over $\circ_S$ is a $S$-vector space. | A vector space over a division ring $D$ is by definition a unitary module over $D$.
$S$ is a division ring by assumption.
$\struct {R, +, \circ_S}_S$ is a unitary module by Subring Module is Module/Special Case.
{{qed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]] whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $S$ be a [[Definition:Division Subring|division subring]] of $R$, such that $1_R \in S$.
The [[Definition:Vector Space over Division Subring|vector space $\struct {R, +, \circ_S}_S$ ... | A [[Definition:Vector Space over Division Ring|vector space]] over a [[Definition:Division Ring|division ring]] $D$ is by definition a [[Definition:Unitary Module|unitary module]] over $D$.
$S$ is a [[Definition:Division Ring|division ring]] by assumption.
$\struct {R, +, \circ_S}_S$ is a [[Definition:Unitary Modul... | Vector Space over Division Subring is Vector Space/Special Case | https://proofwiki.org/wiki/Vector_Space_over_Division_Subring_is_Vector_Space/Special_Case | https://proofwiki.org/wiki/Vector_Space_over_Division_Subring_is_Vector_Space/Special_Case | [
"Examples of Vector Spaces"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Division Subring",
"Definition:Vector Space over Division Subring",
"Definition:Vector Space/Division Ring"
] | [
"Definition:Vector Space/Division Ring",
"Definition:Division Ring",
"Definition:Unitary Module over Ring",
"Definition:Division Ring",
"Definition:Unitary Module over Ring",
"Subring Module is Module/Special Case"
] |
proofwiki-18589 | Prime-Counting Function is Theta of x over Logarithm of x | We have:
:$\map \pi x = \map \Theta {\dfrac x {\ln x} }$
where:
:$\Theta$ is $\Theta$ notation
:$\pi$ is the prime counting function. | From Second Chebyshev Function is $\map \Theta x$, there exists real numbers $A, B, x_0 > 0$ such that:
:$A x \le \map \psi x \le B x$
for $x \ge x_0$, where $\psi$ is the second Chebyshev function.
From Bounds for Prime-Counting Function in terms of Second Chebyshev Function, there exists a real function $R : \hoint... | We have:
:$\map \pi x = \map \Theta {\dfrac x {\ln x} }$
where:
:$\Theta$ is [[Definition:Theta Notation|$\Theta$ notation]]
:$\pi$ is the [[Definition:Prime-Counting Function|prime counting function]]. | From [[Second Chebyshev Function is Theta of x|Second Chebyshev Function is $\map \Theta x$]], there exists [[Definition:Real Number|real numbers]] $A, B, x_0 > 0$ such that:
:$A x \le \map \psi x \le B x$
for $x \ge x_0$, where $\psi$ is the [[Definition:Second Chebyshev Function|second Chebyshev function]].
From... | Prime-Counting Function is Theta of x over Logarithm of x | https://proofwiki.org/wiki/Prime-Counting_Function_is_Theta_of_x_over_Logarithm_of_x | https://proofwiki.org/wiki/Prime-Counting_Function_is_Theta_of_x_over_Logarithm_of_x | [
"Prime-Counting Function",
"Theta Notation"
] | [
"Definition:Theta Notation",
"Definition:Prime-Counting Function"
] | [
"Second Chebyshev Function is Theta of x",
"Definition:Real Number",
"Definition:Second Chebyshev Function",
"Bounds for Prime-Counting Function in terms of Second Chebyshev Function",
"Definition:Real Function",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Real Number",
"Order of... |
proofwiki-18590 | Intersection of Submodules is Submodule | Let $R$ be a ring.
Let $\struct {G, +_G}$ be an abelian group.
Let $M = \struct {G, +, \circ}_R$ be an $R$-module.
Let $H$ and $K$ be submodules of $M$.
Then $H \cap K$ is also a submodule of $M$. | This is a special case of the General Result with $S = \set {H, K}$.
The proof follows immediately from the proof of the General Result.
{{qed}} | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {G, +_G}$ be an [[Definition:Abelian Group|abelian group]].
Let $M = \struct {G, +, \circ}_R$ be an [[Definition:Module over Ring|$R$-module]].
Let $H$ and $K$ be [[Definition:Submodule|submodules]] of $M$.
Then $H \cap K$ is also a [[Definitio... | This is a special case of the [[Intersection of Submodules is Submodule/General Result|General Result]] with $S = \set {H, K}$.
The proof follows immediately from the proof of the [[Intersection of Submodules is Submodule/General Result|General Result]].
{{qed}} | Intersection of Submodules is Submodule | https://proofwiki.org/wiki/Intersection_of_Submodules_is_Submodule | https://proofwiki.org/wiki/Intersection_of_Submodules_is_Submodule | [
"Submodules"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Abelian Group",
"Definition:Module over Ring",
"Definition:Submodule",
"Definition:Submodule"
] | [
"Intersection of Submodules is Submodule/General Result",
"Intersection of Submodules is Submodule/General Result"
] |
proofwiki-18591 | Intersection of Set of Submodules containing Subset is Smallest Submodule | Let $R$ be a ring.
Let $\struct {G, +_G}$ be an abelian group.
Let $M = \struct {G, +, \circ}_R$ be an $R$-module.
Let $S \subset M$ be a subset of $M$.
Let $T$ be the set of all submodules of $M$ which contain $S$ as a subset.
Then the intersection $\bigcap T$ is the smallest submodule of $M$ containing $S$. | By hypothesis, we have:
:$\ds T = \bigcap \set { M' \subseteq M : S \subseteq M', \textrm {$M'$ is a submodule of $M$} }$
From Intersection of Submodules is Submodule:General Result, it follows that $\bigcap T$ is a submodule of $M$.
As $S \subseteq M'$ for all $M' \in T$, it follows that $S \subseteq \bigcap T$.
Let $... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {G, +_G}$ be an [[Definition:Abelian Group|abelian group]].
Let $M = \struct {G, +, \circ}_R$ be an [[Definition:Module over Ring|$R$-module]].
Let $S \subset M$ be a [[Definition:Subset|subset]] of $M$.
Let $T$ be the [[Definition:Set|set]] of ... | [[Definition:By Hypothesis|By hypothesis]], we have:
:$\ds T = \bigcap \set { M' \subseteq M : S \subseteq M', \textrm {$M'$ is a submodule of $M$} }$
From [[Intersection of Submodules is Submodule/General Result|Intersection of Submodules is Submodule:General Result]], it follows that $\bigcap T$ is a [[Definition:Su... | Intersection of Set of Submodules containing Subset is Smallest Submodule | https://proofwiki.org/wiki/Intersection_of_Set_of_Submodules_containing_Subset_is_Smallest_Submodule | https://proofwiki.org/wiki/Intersection_of_Set_of_Submodules_containing_Subset_is_Smallest_Submodule | [
"Submodules"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Abelian Group",
"Definition:Module over Ring",
"Definition:Subset",
"Definition:Set",
"Definition:Submodule",
"Definition:Subset",
"Definition:Set Intersection/Set of Sets",
"Definition:Smallest Set by Set Inclusion",
"Definition:Submodule"
] | [
"Definition:By Hypothesis",
"Intersection of Submodules is Submodule/General Result",
"Definition:Submodule",
"Definition:Submodule",
"Intersection is Subset/General Result"
] |
proofwiki-18592 | Fundamental Solutions to Distributional Homogeneous ODE with Constant Coefficients differ by Classical Solution | Let $E_*, E, T \in \map {\DD'} \R$ be Schwartz distributions.
Let $D$ be an ordinary differential operator with constant coefficients.
Let $f$ be a function differentiable by $D$.
Let $T_f \in \map {\DD'} \R$ be a Schwartz distribution associated with $f$.
Let $\delta$ be the Dirac delta distribution.
Let $E_*$ be the ... | === Necessary Condition ===
Suppose both $E_*$ and $E$ are fundamental solutions:
:$DE = \delta$
:$DE_* = \delta$
Taking the difference yields:
:$D \paren {E - E_*} = \mathbf 0$
where $\mathbf 0 \in \map {\DD'} \R$ is the zero distribution.
By Solution to Distributional Ordinary Differential Equation with Constant Coef... | Let $E_*, E, T \in \map {\DD'} \R$ be [[Definition:Schwartz Distribution|Schwartz distributions]].
Let $D$ be an [[Definition:Ordinary Derivative|ordinary]] [[Definition:Differential Operator|differential operator]] with [[Definition:Constant Mapping|constant]] [[Definition:Coefficient|coefficients]].
Let $f$ be a [[... | === Necessary Condition ===
Suppose both $E_*$ and $E$ are [[Definition:Fundamental Solution|fundamental solutions]]:
:$DE = \delta$
:$DE_* = \delta$
Taking the [[Definition:Subtraction|difference]] yields:
:$D \paren {E - E_*} = \mathbf 0$
where $\mathbf 0 \in \map {\DD'} \R$ is the [[Definition:Zero Distribution... | Fundamental Solutions to Distributional Homogeneous ODE with Constant Coefficients differ by Classical Solution | https://proofwiki.org/wiki/Fundamental_Solutions_to_Distributional_Homogeneous_ODE_with_Constant_Coefficients_differ_by_Classical_Solution | https://proofwiki.org/wiki/Fundamental_Solutions_to_Distributional_Homogeneous_ODE_with_Constant_Coefficients_differ_by_Classical_Solution | [
"Examples of Distributional Solutions",
"Distributional Derivatives",
"Fundamental Solutions"
] | [
"Definition:Schwartz Distribution",
"Definition:Derivative",
"Definition:Differential Operator",
"Definition:Constant Mapping",
"Definition:Coefficient",
"Definition:Differentiable Mapping",
"Definition:Schwartz Distribution",
"Definition:Dirac Delta Distribution",
"Definition:Fundamental Solution",... | [
"Definition:Fundamental Solution",
"Definition:Subtraction",
"Definition:Zero Mapping/Schwartz Distribution",
"Solution to Distributional Ordinary Differential Equation with Constant Coefficients",
"Definition:Fundamental Solution"
] |
proofwiki-18593 | Elements of Submodule form Subgroup | Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {G, +_G}$ be an abelian group.
Let $\struct {G, +_G, \circ_G}_R$ be an $R$-module.
Let $\struct {H, +_H, \circ_H}_R$ be a submodule of $\struct {G, +_G, \circ_G}_R$.
Then $\struct {H, +_H}$ is a subgroup of $\struct {G, +_G}$. | By definition of submodule, $\struct {H, +_H}$ is an abelian group.
The result follows by definition of subgroup.
{{qed}}
Category:Submodules
i9qlsqmy3a2vb08mi2qavzv6cj87tat | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {G, +_G}$ be an [[Definition:Abelian Group|abelian group]].
Let $\struct {G, +_G, \circ_G}_R$ be an [[Definition:Module over Ring|$R$-module]].
Let $\struct {H, +_H, \circ_H}_R$ be a [[Definition:Submodule|submodule]] of $\str... | By definition of [[Definition:Submodule|submodule]], $\struct {H, +_H}$ is an [[Definition:Abelian Group|abelian group]].
The result follows by definition of [[Definition:Subgroup|subgroup]].
{{qed}}
[[Category:Submodules]]
i9qlsqmy3a2vb08mi2qavzv6cj87tat | Elements of Submodule form Subgroup | https://proofwiki.org/wiki/Elements_of_Submodule_form_Subgroup | https://proofwiki.org/wiki/Elements_of_Submodule_form_Subgroup | [
"Submodules"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Abelian Group",
"Definition:Module over Ring",
"Definition:Submodule",
"Definition:Subgroup"
] | [
"Definition:Submodule",
"Definition:Abelian Group",
"Definition:Subgroup",
"Category:Submodules"
] |
proofwiki-18594 | Image of Submodule under Linear Transformation is Submodule | Let $M$ be a submodule of $G$.
Then $\phi \sqbrk M$ is a submodule of $H$. | Let $N = \phi \sqbrk M$ be the image set of $M$ under $\phi$.
By definition, a linear transformation $\phi: G \to H$ is, in particular, a (group) homomorphism from the group $\struct {G, +_G}$ to the group $\struct {H, +_H}$.
We have {{hypothesis}} that $M$ is a submodule of $G$.
So from Elements of Submodule form Subg... | Let $M$ be a [[Definition:Submodule|submodule]] of $G$.
Then $\phi \sqbrk M$ is a [[Definition:Submodule|submodule]] of $H$. | Let $N = \phi \sqbrk M$ be the [[Definition:Image of Subset under Mapping|image set]] of $M$ under $\phi$.
By definition, a [[Definition:Linear Transformation|linear transformation]] $\phi: G \to H$ is, in particular, a [[Definition:Group Homomorphism|(group) homomorphism]] from the [[Definition:Group|group]] $\struc... | Image of Submodule under Linear Transformation is Submodule | https://proofwiki.org/wiki/Image_of_Submodule_under_Linear_Transformation_is_Submodule | https://proofwiki.org/wiki/Image_of_Submodule_under_Linear_Transformation_is_Submodule | [
"Linear Transformations"
] | [
"Definition:Submodule",
"Definition:Submodule"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Linear Transformation",
"Definition:Group Homomorphism",
"Definition:Group",
"Definition:Group",
"Definition:Submodule",
"Elements of Submodule form Subgroup",
"Definition:Subgroup",
"Group Homomorphism Preserves Subgroups",
"Definition:S... |
proofwiki-18595 | Preimage of Submodule under Linear Transformation is Submodule | Let $N$ be a submodule of $H$.
Then $\phi^{-1} \sqbrk N$ is a submodule of $G$. | Let $M = \phi^{-1} \sqbrk N$ be the preimage of $N$ under $\phi$.
{{AimForCont}} $M$ is not a submodule of $G$.
This means that $M$ does not fulfil all the module axioms.
First suppose that:
:$(1): \quad \struct {M, +_G}$ is not a subgroup of $G$.
Then $M$ is not a group.
Then by the One-Step Subgroup Test:
:$\exists x... | Let $N$ be a [[Definition:Submodule|submodule]] of $H$.
Then $\phi^{-1} \sqbrk N$ is a [[Definition:Submodule|submodule]] of $G$. | Let $M = \phi^{-1} \sqbrk N$ be the [[Definition:Preimage of Subset under Mapping|preimage]] of $N$ under $\phi$.
{{AimForCont}} $M$ is not a [[Definition:Submodule|submodule]] of $G$.
This means that $M$ does not fulfil all the [[Axiom:Module Axioms|module axioms]].
First suppose that:
:$(1): \quad \struct {M, +_G... | Preimage of Submodule under Linear Transformation is Submodule | https://proofwiki.org/wiki/Preimage_of_Submodule_under_Linear_Transformation_is_Submodule | https://proofwiki.org/wiki/Preimage_of_Submodule_under_Linear_Transformation_is_Submodule | [
"Linear Transformations"
] | [
"Definition:Submodule",
"Definition:Submodule"
] | [
"Definition:Preimage/Mapping/Subset",
"Definition:Submodule",
"Axiom:Left Module Axioms",
"Definition:Subgroup",
"Definition:Group",
"One-Step Subgroup Test",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Group",
"Definition:Subgroup",
"Definition:Contradiction",
"Defi... |
proofwiki-18596 | Test Function Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space | Let $\map \DD {\R^d}$ be the test function space.
Let $\struct {\C, +_\C, \times_\C}$ be the field of complex numbers.
Let $\paren +$ be the pointwise addition of test functions.
Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplication of test functions over $\C$.
Then $\struct {\map \DD {\R^d}, +, \, \cdot \,... | Let $f, g, h \in \map \DD {\R^d}$ be test functions with the compact support $K$.
Let $\lambda, \mu \in \C$.
Let $\map 0 x$ be a real-valued function such that:
:$\map 0 x : \R^d \to 0$.
Let us use real number addition and multiplication.
$\forall x \in \R^d$ define pointwise addition as:
:$\map {\paren {f + g}} x := \... | Let $\map \DD {\R^d}$ be the [[Definition:Test Function Space|test function space]].
Let $\struct {\C, +_\C, \times_\C}$ be the [[Definition:Field of Complex Numbers|field of complex numbers]].
Let $\paren +$ be the [[Definition:Pointwise Addition of Mappings|pointwise addition]] of [[Definition:Test Function|test fu... | Let $f, g, h \in \map \DD {\R^d}$ be [[Definition:Test Function|test functions]] with the [[Definition:Compact Subset of Real Euclidean Space|compact]] [[Definition:Support of Continuous Mapping|support]] $K$.
Let $\lambda, \mu \in \C$.
Let $\map 0 x$ be a [[Definition:Complex-Valued Function|real-valued function]] s... | Test Function Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space | https://proofwiki.org/wiki/Test_Function_Space_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_forms_Vector_Space | https://proofwiki.org/wiki/Test_Function_Space_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_forms_Vector_Space | [
"Examples of Vector Spaces",
"Functional Analysis"
] | [
"Definition:Test Function Space",
"Definition:Field of Complex Numbers",
"Definition:Pointwise Addition of Mappings",
"Definition:Test Function",
"Definition:Pointwise Scalar Multiplication of Mappings",
"Definition:Test Function",
"Definition:Vector Space"
] | [
"Definition:Test Function",
"Definition:Compact Space/Euclidean Space",
"Definition:Support of Continuous Mapping",
"Definition:Complex-Valued Function",
"Definition:Complex Number",
"Definition:Addition/Complex Numbers",
"Definition:Multiplication/Complex Numbers",
"Definition:Pointwise Addition of C... |
proofwiki-18597 | Image of Linear Transformation is Submodule | Let $\Img \phi$ denote the image set of $\phi$.
Then $\Img \phi$ is a submodule of $H$. | By Module is Submodule of Itself, $\struct {G, +_G, \circ_G}_R$ is a submodule of $\struct {G, +_G, \circ_G}_R$.
The result follows from Image of Submodule under Linear Transformation is Submodule.
{{Qed}} | Let $\Img \phi$ denote the [[Definition:Image of Mapping|image set]] of $\phi$.
Then $\Img \phi$ is a [[Definition:Submodule|submodule]] of $H$. | By [[Module is Submodule of Itself]], $\struct {G, +_G, \circ_G}_R$ is a [[Definition:Submodule|submodule]] of $\struct {G, +_G, \circ_G}_R$.
The result follows from [[Image of Submodule under Linear Transformation is Submodule]].
{{Qed}} | Image of Linear Transformation is Submodule | https://proofwiki.org/wiki/Image_of_Linear_Transformation_is_Submodule | https://proofwiki.org/wiki/Image_of_Linear_Transformation_is_Submodule | [
"Linear Transformations"
] | [
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Submodule"
] | [
"Module is Submodule of Itself",
"Definition:Submodule",
"Image of Submodule under Linear Transformation is Submodule"
] |
proofwiki-18598 | Kernel of Linear Transformation is Submodule | Let $\map \ker \phi$ denote the kernel of $\phi$.
Then $\map \ker \phi$ is a submodule of $G$. | By definition, the kernel of $\phi$ is defined as:
:$\map \ker \phi := \phi^{-1} \sqbrk {\set {e_H} }$
where $e_H$ is the identity of $\struct {H, +_H}$.
where $\phi^{-1} \sqbrk S$ denotes the preimage of $S$ under $\phi$.
From Null Module is Module:
:$\struct {\set {e_H}, +_H, \circ_H}_R$ is a module
where $\struct {\... | Let $\map \ker \phi$ denote the [[Definition:Kernel of Linear Transformation|kernel]] of $\phi$.
Then $\map \ker \phi$ is a [[Definition:Submodule|submodule]] of $G$. | By definition, the [[Definition:Kernel of Linear Transformation|kernel]] of $\phi$ is defined as:
:$\map \ker \phi := \phi^{-1} \sqbrk {\set {e_H} }$
where $e_H$ is the [[Definition:Identity Element|identity]] of $\struct {H, +_H}$.
where $\phi^{-1} \sqbrk S$ denotes the [[Definition:Preimage of Subset under Mapping|p... | Kernel of Linear Transformation is Submodule | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_is_Submodule | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_is_Submodule | [
"Linear Transformations"
] | [
"Definition:Kernel of Linear Transformation",
"Definition:Submodule"
] | [
"Definition:Kernel of Linear Transformation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Preimage/Mapping/Subset",
"Null Module is Module",
"Definition:Module over Ring",
"Definition:Null Module",
"Definition:Module over Ring",
"Definition:Subset",
"Definition:Submodule... |
proofwiki-18599 | Locally Integrable Function defines Distribution | Let $f \in \map {L^1_{loc}} {\R^d}$ be a locally integrable function.
Let $\map \DD {\R^d}$ be the test function space.
Let $T_f : \map \DD {\R^d} \to \C$ be a mapping.
Then $T_f$ is a Schwartz distribution. | === Existence ===
Let $\phi \in \map \DD {\R^d}$ be a test function.
Let $T_f$ be defined as
:$\ds T_f = \int_{\R^d} \map f {\mathbf x} \map \phi {\mathbf x} \rd \mathbf x$
By definition, $\phi$ has the compact support.
Together with the properties of $f$ we have that $T_f$ is bounded with respect to any compact range ... | Let $f \in \map {L^1_{loc}} {\R^d}$ be a [[Definition:Locally Integrable Function|locally integrable function]].
Let $\map \DD {\R^d}$ be the [[Definition:Test Function Space|test function space]].
Let $T_f : \map \DD {\R^d} \to \C$ be a [[Definition:Mapping|mapping]].
Then $T_f$ is a [[Definition:Schwartz Distribu... | === Existence ===
Let $\phi \in \map \DD {\R^d}$ be a [[Definition:Test Function|test function]].
Let $T_f$ be defined as
:$\ds T_f = \int_{\R^d} \map f {\mathbf x} \map \phi {\mathbf x} \rd \mathbf x$
By [[Definition:Test Function|definition]], $\phi$ has the [[Definition:Compact Subset of Real Euclidean Space|com... | Locally Integrable Function defines Distribution | https://proofwiki.org/wiki/Locally_Integrable_Function_defines_Distribution | https://proofwiki.org/wiki/Locally_Integrable_Function_defines_Distribution | [
"Examples of Schwartz Distributions"
] | [
"Definition:Integrable Function/Locally Integrable Function",
"Definition:Test Function Space",
"Definition:Mapping",
"Definition:Schwartz Distribution"
] | [
"Definition:Test Function",
"Definition:Test Function",
"Definition:Compact Space/Euclidean Space",
"Definition:Support of Continuous Mapping/Real-Valued",
"Definition:Integrable Function/Locally Integrable Function",
"Definition:Bounded Mapping",
"Definition:Compact Space/Euclidean Space",
"Definitio... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.