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-21900 | Kakutani's Fixed Point Theorem | Let $S \subset \R^n$ be nonempty, compact, and convex.
Let $\Phi : S \to 2^S$ be a correspondence.
Let the following conditions be satisfied:
:$(1): \quad \map \Phi x$ is nonempty and convex for all $x$
:$(2): \quad \map \Phi \cdot$ is upper hemi-continuous
Then $\Phi$ has a fixed point. | {{ProofWanted}}
{{Namedfor|Shizuo Kakutani|Kakutani}} | Let $S \subset \R^n$ be nonempty, compact, and convex.
Let $\Phi : S \to 2^S$ be a correspondence.
Let the following conditions be satisfied:
:$(1): \quad \map \Phi x$ is nonempty and convex for all $x$
:$(2): \quad \map \Phi \cdot$ is upper hemi-continuous
Then $\Phi$ has a fixed point. | {{ProofWanted}}
{{Namedfor|Shizuo Kakutani|Kakutani}} | Kakutani's Fixed Point Theorem | https://proofwiki.org/wiki/Kakutani's_Fixed_Point_Theorem | https://proofwiki.org/wiki/Kakutani's_Fixed_Point_Theorem | [] | [] | [] |
proofwiki-21901 | Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom | Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $\mathscr B$ be the set of bases of the matroid $M$.
Then $\mathscr B$ satisfies formulation $7$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 7}} | Let $\mathscr C$ denote the set of circuits of $M$.
From Circuits of Matroid iff Matroid Circuit Axioms, $\mathscr C$ satisifies:
:{{:Axiom:Circuit Axioms (Matroid)/Formulation 1}} | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\mathscr B$ be the [[Definition:Set|set]] of [[Definition:Base of Matroid|bases]] of the [[Definition:Matroid|matroid]] $M$.
Then $\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 3|formulation $7$ of base axiom]]:
{{:Axiom:Base... | Let $\mathscr C$ denote the [[Definition:Set|set]] of [[Definition:Circuit (Matroid)|circuits]] of $M$.
From [[Circuits of Matroid iff Matroid Circuit Axioms]], $\mathscr C$ satisifies:
:{{:Axiom:Circuit Axioms (Matroid)/Formulation 1}} | Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom | https://proofwiki.org/wiki/Matroid_Bases_Satisfy_Formulation_7_of_Matroid_Base_Axiom | https://proofwiki.org/wiki/Matroid_Bases_Satisfy_Formulation_7_of_Matroid_Base_Axiom | [
"Matroid Bases",
"Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom"
] | [
"Definition:Matroid",
"Definition:Set",
"Definition:Base of Matroid",
"Definition:Matroid",
"Axiom:Base Axiom (Matroid)/Formulation 3"
] | [
"Definition:Set",
"Definition:Circuit (Matroid)",
"Circuits of Matroid iff Matroid Circuit Axioms"
] |
proofwiki-21902 | Bonnet's Recursion Formula | Let $\map {P_n} x$ denote the Legendre polynomial of order $n$.
'''Bonnet's Recursion Formula''' states:
:$\paren {n + 1} \map {P_{n + 1} } x = \paren {2 n + 1} x \map {P_n} x - n \map {P_{n - 1} } x$ | From Generating Function for Legendre Polynomials, the generating function for $P_n$ is:
:$(1): \quad \ds \frac 1 {\sqrt {1 - 2 x t + t^2} } = \sum_{n \mathop = 0}^\infty \map {P_n} x t^n$
Differentiating both sides of $(1)$ {{WRT|Differentiation}} $t$:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d t} } {\paren {1 - 2 ... | Let $\map {P_n} x$ denote the [[Definition:Legendre Polynomial|Legendre polynomial of order $n$]].
'''[[Bonnet's Recursion Formula]]''' states:
:$\paren {n + 1} \map {P_{n + 1} } x = \paren {2 n + 1} x \map {P_n} x - n \map {P_{n - 1} } x$ | From [[Generating Function for Legendre Polynomials]], the [[Definition:Generating Function|generating function]] for $P_n$ is:
:$(1): \quad \ds \frac 1 {\sqrt {1 - 2 x t + t^2} } = \sum_{n \mathop = 0}^\infty \map {P_n} x t^n$
[[Definition:Differentiation|Differentiating]] both sides of $(1)$ {{WRT|Differentiation}} ... | Bonnet's Recursion Formula | https://proofwiki.org/wiki/Bonnet's_Recursion_Formula | https://proofwiki.org/wiki/Bonnet's_Recursion_Formula | [
"Bonnet's Recursion Formula",
"Legendre Polynomials"
] | [
"Definition:Legendre Polynomial",
"Bonnet's Recursion Formula"
] | [
"Generating Function for Legendre Polynomials",
"Definition:Generating Function",
"Definition:Differentiation",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Translation of Index Variable of Summation",
"Definition:Coefficient of Polynomial"
] |
proofwiki-21903 | Length of Legendre Polynomial | Let $\map {P_n} x$ denote the '''Legendre polynomial of order $n$'''.
Let $\norm {\map {P_n} x}$ denote the '''length''' of $\map {P_n} x$.
Then:
:$\norm {\map {P_n} x} := \sqrt {\dfrac 2 {2 n + 1} }$ | Applying Bonnet's Recursion Formula for $n - 1$:
:$n \map {P_n} x = \paren {2 n - 1} x \map {P_{n - 1} } x - \paren {n - 1} \map {P_{n - 2} } x$
so:
:$\map {P_n} x = \dfrac {2 n - 1} n x \map {P_{n - 1} } x - \dfrac {n - 1} n \map {P_{n - 2} } x$
Substituting for $\map {P_n} x$:
{{begin-eqn}}
{{eqn | l = \norm {\map {P... | Let $\map {P_n} x$ denote the '''[[Definition:Legendre Polynomial|Legendre polynomial of order $n$]]'''.
Let $\norm {\map {P_n} x}$ denote the '''[[Definition:Length of Legendre Polynomial|length]]''' of $\map {P_n} x$.
Then:
:$\norm {\map {P_n} x} := \sqrt {\dfrac 2 {2 n + 1} }$ | Applying [[Bonnet's Recursion Formula]] for $n - 1$:
:$n \map {P_n} x = \paren {2 n - 1} x \map {P_{n - 1} } x - \paren {n - 1} \map {P_{n - 2} } x$
so:
:$\map {P_n} x = \dfrac {2 n - 1} n x \map {P_{n - 1} } x - \dfrac {n - 1} n \map {P_{n - 2} } x$
Substituting for $\map {P_n} x$:
{{begin-eqn}}
{{eqn | l = \norm... | Length of Legendre Polynomial | https://proofwiki.org/wiki/Length_of_Legendre_Polynomial | https://proofwiki.org/wiki/Length_of_Legendre_Polynomial | [
"Legendre Polynomials"
] | [
"Definition:Legendre Polynomial",
"Definition:Legendre Polynomial/Length"
] | [
"Bonnet's Recursion Formula",
"Linear Combination of Integrals/Definite",
"Orthogonality of Legendre Polynomials",
"Bonnet's Recursion Formula",
"Definition:Legendre Polynomial/Length",
"Legendre Polynomial/Examples",
"Primitive of Constant",
"Definition:Square Root",
"Category:Legendre Polynomials"... |
proofwiki-21904 | Lemniscate of Bernoulli from Tangents to Rectangular Hyperbola | Let $\KK$ be a rectangular hyperbola.
Let $\LL$ be the locus of the foot of the perpendicular from the origin to the tangents to $\KK$.
Then $\LL$ is the '''lemniscate of Bernoulli'''. | :600px
{{ProofWanted}} | Let $\KK$ be a [[Definition:Rectangular Hyperbola|rectangular hyperbola]].
Let $\LL$ be the [[Definition:Locus|locus]] of the [[Definition:Foot of Perpendicular|foot]] of the [[Definition:Perpendicular|perpendicular]] from the [[Definition:Origin|origin]] to the [[Definition:Tangent Line|tangents]] to $\KK$.
Then $\... | :[[File:Lemniscate-by-Tangents-to-Hyperbola.png|600px]]
{{ProofWanted}} | Lemniscate of Bernoulli from Tangents to Rectangular Hyperbola | https://proofwiki.org/wiki/Lemniscate_of_Bernoulli_from_Tangents_to_Rectangular_Hyperbola | https://proofwiki.org/wiki/Lemniscate_of_Bernoulli_from_Tangents_to_Rectangular_Hyperbola | [
"Lemniscate of Bernoulli",
"Rectangular Hyperbolas"
] | [
"Definition:Rectangular Hyperbola",
"Definition:Locus",
"Definition:Right Angle/Perpendicular/Foot",
"Definition:Right Angle/Perpendicular",
"Definition:Coordinate System/Origin",
"Definition:Tangent Line",
"Definition:Lemniscate of Bernoulli"
] | [
"File:Lemniscate-by-Tangents-to-Hyperbola.png"
] |
proofwiki-21905 | Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom/Lemma 1 | Let $n \in \N_{>0}$.
Let $C_1, C_2, \ldots, C_n \in \mathscr C$ satisfy:
:$\forall 0 \le k \le n : C_k \nsubseteq \ds \bigcup_{i \ne k} C_i$
Let:
:$A \subseteq S : \size A < n$
Let $r = \size A$.
Then:
:$\ds \exists D_1, \ldots, D_{n - r} \in \mathscr C : \bigcup_{i = 1}^{n - r} D_i \subseteq \paren{\bigcup_{i = 1}^{n}... | === Case 1 : $n = 1$ ===
Let $n = 1$.
Hence $\size A = 0$.
From Cardinality of Empty Set:
:$A = \O$
It follows that $C_1$ suffices for $D_1$.
{{qed|lemma}} | Let $n \in \N_{>0}$.
Let $C_1, C_2, \ldots, C_n \in \mathscr C$ satisfy:
:$\forall 0 \le k \le n : C_k \nsubseteq \ds \bigcup_{i \ne k} C_i$
Let:
:$A \subseteq S : \size A < n$
Let $r = \size A$.
Then:
:$\ds \exists D_1, \ldots, D_{n - r} \in \mathscr C : \bigcup_{i = 1}^{n - r} D_i \subseteq \paren{\bigcup_{i = 1... | === Case 1 : $n = 1$ ===
Let $n = 1$.
Hence $\size A = 0$.
From [[Cardinality of Empty Set]]:
:$A = \O$
It follows that $C_1$ suffices for $D_1$.
{{qed|lemma}} | Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom/Lemma 1 | https://proofwiki.org/wiki/Matroid_Bases_Satisfy_Formulation_7_of_Matroid_Base_Axiom/Lemma_1 | https://proofwiki.org/wiki/Matroid_Bases_Satisfy_Formulation_7_of_Matroid_Base_Axiom/Lemma_1 | [
"Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom"
] | [] | [
"Cardinality of Empty Set"
] |
proofwiki-21906 | Mechanical Advantage of Type 3 Lever is Less than 1 | Let $M$ be a type $3$ lever.
Then the mechanical advantage of $M$ is less than $1$.
Hence, instead of amplifying the force applied, a type $3$ lever amplifies the velocity ratio. | From Principle of Lever:
:$\text {MA} = \dfrac b a$
where:
:$b$ is the distance of the effort from the fulcrum
:$a$ is the distance of the load from the fulcrum.
But in a type $3$ lever, the effort acts between the load and the fulcrum.
That is:
:$a > b$
and hence the result.
{{qed}} | Let $M$ be a [[Definition:Type 3 Lever|type $3$ lever]].
Then the [[Definition:Mechanical Advantage|mechanical advantage]] of $M$ is less than $1$.
Hence, instead of amplifying the [[Definition:Force|force]] applied, a [[Definition:Type 3 Lever|type $3$ lever]] amplifies the [[Definition:Velocity Ratio|velocity rati... | From [[Principle of Lever]]:
:$\text {MA} = \dfrac b a$
where:
:$b$ is the [[Definition:Distance between Points|distance]] of the [[Definition:Effort|effort]] from the [[Definition:Fulcrum|fulcrum]]
:$a$ is the [[Definition:Distance between Points|distance]] of the [[Definition:Load|load]] from the [[Definition:Fulcrum... | Mechanical Advantage of Type 3 Lever is Less than 1 | https://proofwiki.org/wiki/Mechanical_Advantage_of_Type_3_Lever_is_Less_than_1 | https://proofwiki.org/wiki/Mechanical_Advantage_of_Type_3_Lever_is_Less_than_1 | [
"Type 3 Levers"
] | [
"Definition:Lever/Type 3",
"Definition:Mechanical Advantage",
"Definition:Force",
"Definition:Lever/Type 3",
"Definition:Velocity Ratio"
] | [
"Principle of Lever",
"Definition:Distance between Points",
"Definition:Effort",
"Definition:Lever/Fulcrum",
"Definition:Distance between Points",
"Definition:Load",
"Definition:Lever/Fulcrum",
"Definition:Lever/Type 3",
"Definition:Effort",
"Definition:Load",
"Definition:Lever/Fulcrum"
] |
proofwiki-21907 | Independent Subset Contains No Dependent Subset | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $X \subseteq S$ be any independent subset of $M$.
Then:
:No dependent subset $D$ of $M$ is a subset of $X$. | By definition of independent subset:
:$X \in \mathscr I$
By definition of matroid, specifically matroid axiom $( \text I 2)$:
:$\forall Y \subseteq X : Y \in \mathscr I$
By definition of dependent subset:
:$\forall Y \subseteq X : Y$ is not a dependent subset
{{qed}}
Category:Matroid Independent Subsets
Category:Matroi... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $X \subseteq S$ be any [[Definition:Independent Subset (Matroid)|independent subset]] of $M$.
Then:
:No [[Definition:Dependent Subset (Matroid)|dependent subset]] $D$ of $M$ is a [[Definition:Subset|subset]] of $X$. | By definition of [[Definition:Independent Subset (Matroid)|independent subset]]:
:$X \in \mathscr I$
By definition of [[Definition:Matroid|matroid]], specifically [[Axiom:Matroid Axioms|matroid axiom $( \text I 2)$]]:
:$\forall Y \subseteq X : Y \in \mathscr I$
By definition of [[Definition:Dependent Subset (Matroid)... | Independent Subset Contains No Dependent Subset | https://proofwiki.org/wiki/Independent_Subset_Contains_No_Dependent_Subset | https://proofwiki.org/wiki/Independent_Subset_Contains_No_Dependent_Subset | [
"Matroid Independent Subsets",
"Matroid Dependent Subsets",
"Independent Subset Contains No Dependent Subset"
] | [
"Definition:Matroid",
"Definition:Matroid/Independent Set",
"Definition:Matroid/Dependent Set",
"Definition:Subset"
] | [
"Definition:Matroid/Independent Set",
"Definition:Matroid",
"Axiom:Matroid Axioms",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Category:Matroid Independent Subsets",
"Category:Matroid Dependent Subsets",
"Category:Independent Subset Contains No Dependent Subset"
] |
proofwiki-21908 | Independent Subset Contains No Dependent Subset/Corollary 1 | Let $B \subseteq S$ be any base of $M$.
Then:
:No dependent subset $D$ of $M$ is a subset of $B$. | By definition of matroid base:
:$B$ is an independent subset of $M$
From Independent Subset Contains No Dependent Subset:
:No dependent subset $D$ of $M$ is a subset of $B$.
{{qed}}
Category:Independent Subset Contains No Dependent Subset
8f6pz3re1rnjnbt5gg4yznwcxg61tvh | Let $B \subseteq S$ be any [[Definition:Base of Matroid|base]] of $M$.
Then:
:No [[Definition:Dependent Subset (Matroid)|dependent subset]] $D$ of $M$ is a [[Definition:Subset|subset]] of $B$. | By definition of [[Definition:Base of Matroid|matroid base]]:
:$B$ is an [[Definition:Independent Subset (Matroid)|independent subset]] of $M$
From [[Independent Subset Contains No Dependent Subset]]:
:No [[Definition:Dependent Subset (Matroid)|dependent subset]] $D$ of $M$ is a [[Definition:Subset|subset]] of $B$.
{{... | Independent Subset Contains No Dependent Subset/Corollary 1 | https://proofwiki.org/wiki/Independent_Subset_Contains_No_Dependent_Subset/Corollary_1 | https://proofwiki.org/wiki/Independent_Subset_Contains_No_Dependent_Subset/Corollary_1 | [
"Independent Subset Contains No Dependent Subset"
] | [
"Definition:Base of Matroid",
"Definition:Matroid/Dependent Set",
"Definition:Subset"
] | [
"Definition:Base of Matroid",
"Definition:Matroid/Independent Set",
"Independent Subset Contains No Dependent Subset",
"Definition:Matroid/Dependent Set",
"Definition:Subset",
"Category:Independent Subset Contains No Dependent Subset"
] |
proofwiki-21909 | Independent Subset Contains No Dependent Subset/Corollary 2 | Let $X \subseteq S$ be any independent subset of $M$.
Then:
:No circuit $C$ of $M$ is a subset of $X$. | Let $C$ be a circuit of $M$.
By definition of matroid circuit:
:$C$ is a dependent subset of $M$
From Independent Subset Contains No Dependent Subset:
:$C$ is not a subset of $X$.
The result follows.
{{qed}}
Category:Independent Subset Contains No Dependent Subset
blh296a5mwoh80eyt1356eq39y7gy0v | Let $X \subseteq S$ be any [[Definition:Independent Subset (Matroid)|independent subset]] of $M$.
Then:
:No [[Definition:Circuit (Matroid)|circuit]] $C$ of $M$ is a [[Definition:Subset|subset]] of $X$. | Let $C$ be a [[Definition:Circuit (Matroid)|circuit]] of $M$.
By definition of [[Definition:Circuit (Matroid)|matroid circuit]]:
:$C$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$
From [[Independent Subset Contains No Dependent Subset]]:
:$C$ is not a [[Definition:Subset|subset]] of $X$.
The... | Independent Subset Contains No Dependent Subset/Corollary 2 | https://proofwiki.org/wiki/Independent_Subset_Contains_No_Dependent_Subset/Corollary_2 | https://proofwiki.org/wiki/Independent_Subset_Contains_No_Dependent_Subset/Corollary_2 | [
"Independent Subset Contains No Dependent Subset"
] | [
"Definition:Matroid/Independent Set",
"Definition:Circuit (Matroid)",
"Definition:Subset"
] | [
"Definition:Circuit (Matroid)",
"Definition:Circuit (Matroid)",
"Definition:Matroid/Dependent Set",
"Independent Subset Contains No Dependent Subset",
"Definition:Subset",
"Category:Independent Subset Contains No Dependent Subset"
] |
proofwiki-21910 | Independent Subset Contains No Dependent Subset/Corollary 3 | Let $B \subseteq S$ be any base of $M$.
Then:
:No circuit $C$ of $M$ is a subset of $B$. | By definition of matroid base:
:$B$ is an independent subset of $M$
From Independent Subset Contains No Circuit:
:No circuit $C$ of $M$ is a subset of $B$.
{{qed}}
Category:Independent Subset Contains No Dependent Subset
rcfy54uuv91ecynli40mjz9j12l5m43 | Let $B \subseteq S$ be any [[Definition:Base of Matroid|base]] of $M$.
Then:
:No [[Definition:Circuit (Matroid)|circuit]] $C$ of $M$ is a [[Definition:Subset|subset]] of $B$. | By definition of [[Definition:Base of Matroid|matroid base]]:
:$B$ is an [[Definition:Independent Subset (Matroid)|independent subset]] of $M$
From [[Independent Subset Contains No Circuit]]:
:No [[Definition:Circuit (Matroid)|circuit]] $C$ of $M$ is a [[Definition:Subset|subset]] of $B$.
{{qed}}
[[Category:Independe... | Independent Subset Contains No Dependent Subset/Corollary 3 | https://proofwiki.org/wiki/Independent_Subset_Contains_No_Dependent_Subset/Corollary_3 | https://proofwiki.org/wiki/Independent_Subset_Contains_No_Dependent_Subset/Corollary_3 | [
"Independent Subset Contains No Dependent Subset"
] | [
"Definition:Base of Matroid",
"Definition:Circuit (Matroid)",
"Definition:Subset"
] | [
"Definition:Base of Matroid",
"Definition:Matroid/Independent Set",
"Independent Subset Contains No Dependent Subset/Corollary 1",
"Definition:Circuit (Matroid)",
"Definition:Subset",
"Category:Independent Subset Contains No Dependent Subset"
] |
proofwiki-21911 | Subset Intersection Set Difference is Empty Iff Subset of Second Set | Let $S$ and $T$ be sets.
Let $A \subseteq S$.
Then:
:$A \cap S \setminus T = \O$ {{iff}} $A \subseteq T$ | We have:
{{begin-eqn}}
{{eqn | l = A \cap \paren {S \setminus T}
| r = \paren {A \cap S} \setminus T
| c = Intersection with Set Difference is Set Difference with Intersection
}}
{{eqn | r = A \setminus T
| c = Intersection with Subset is Subset
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = A \cap \... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $A \subseteq S$.
Then:
:$A \cap S \setminus T = \O$ {{iff}} $A \subseteq T$ | We have:
{{begin-eqn}}
{{eqn | l = A \cap \paren {S \setminus T}
| r = \paren {A \cap S} \setminus T
| c = [[Intersection with Set Difference is Set Difference with Intersection]]
}}
{{eqn | r = A \setminus T
| c = [[Intersection with Subset is Subset]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l ... | Subset Intersection Set Difference is Empty Iff Subset of Second Set | https://proofwiki.org/wiki/Subset_Intersection_Set_Difference_is_Empty_Iff_Subset_of_Second_Set | https://proofwiki.org/wiki/Subset_Intersection_Set_Difference_is_Empty_Iff_Subset_of_Second_Set | [
"Set Difference",
"Set Intersection"
] | [
"Definition:Set"
] | [
"Intersection with Set Difference is Set Difference with Intersection",
"Intersection with Subset is Subset",
"Set Difference with Superset is Empty Set",
"Category:Set Difference",
"Category:Set Intersection"
] |
proofwiki-21912 | Negative Slope indicates Line slopes Downward from Left to Right | Let $\LL$ be a straight line with a slope which is negative.
Then $\LL$ slopes downward from left to right. | Let $\LL$ have a slope which is negative.
Expressed in slope-intercept form, $\LL$ can be written:
:$y = x \tan \psi + c$
where:
:$\psi$ is the angle between $\LL$ and the $x$-axis
:$c$ is the $y$-intercept.
:420px
By construction:
:$90 \degrees < \psi < 180 \degrees$
Hence by Shape of Tangent Function:
:$\tan \psi < ... | Let $\LL$ be a [[Definition:Straight Line|straight line]] with a [[Definition:Slope of Straight Line|slope]] which is [[Definition:Negative Real Number|negative]].
Then $\LL$ [[Definition:Slope of Straight Line|slopes]] [[Definition:Down|downward]] from [[Definition:Left|left]] to [[Definition:Right|right]]. | Let $\LL$ have a [[Definition:Slope of Straight Line|slope]] which is [[Definition:Negative Real Number|negative]].
Expressed in [[Equation of Straight Line in Plane/Slope-Intercept Form|slope-intercept form]], $\LL$ can be written:
:$y = x \tan \psi + c$
where:
:$\psi$ is the [[Definition:Angle|angle]] between $\LL$ ... | Negative Slope indicates Line slopes Downward from Left to Right | https://proofwiki.org/wiki/Negative_Slope_indicates_Line_slopes_Downward_from_Left_to_Right | https://proofwiki.org/wiki/Negative_Slope_indicates_Line_slopes_Downward_from_Left_to_Right | [
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Slope/Straight Line",
"Definition:Negative/Real Number",
"Definition:Slope/Straight Line",
"Definition:Down",
"Definition:Left",
"Definition:Right"
] | [
"Definition:Slope/Straight Line",
"Definition:Negative/Real Number",
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Intercept/Y-Intercept",
"File:Straight-line-downward-slope.png",
"Shape of Tangent Function"
] |
proofwiki-21913 | Graph of Linear Function of One Variable | Let $f: \R \to \R$ be a linear function of one variable expressed by the equation:
:$\map f x = a_0 + a_1 x$
The graph of $f$ in a cartesian plane consists of a straight line:
:whose $y$-intercept is $a_0$
:whose slope is $a_1$. | Expressing $f$ in the form $y = \map f x$, we have:
:$y = a_0 + a_1 x$
Thus it is in the same form as the slope-intercept form of an equation of a straight line in the plane.
Hence the result.
{{qed}} | Let $f: \R \to \R$ be a [[Definition:Linear Function of One Variable|linear function of one variable]] expressed by the [[Definition:Equation|equation]]:
:$\map f x = a_0 + a_1 x$
The [[Definition:Graph of Mapping|graph]] of $f$ in a [[Definition:Cartesian Plane|cartesian plane]] consists of a [[Definition:Straight Li... | Expressing $f$ in the form $y = \map f x$, we have:
:$y = a_0 + a_1 x$
Thus it is in the same form as the [[Equation of Straight Line in Plane/Slope-Intercept Form|slope-intercept form of an equation of a straight line in the plane]].
Hence the result.
{{qed}} | Graph of Linear Function of One Variable | https://proofwiki.org/wiki/Graph_of_Linear_Function_of_One_Variable | https://proofwiki.org/wiki/Graph_of_Linear_Function_of_One_Variable | [
"Linear Functions",
"Graphs of Mappings"
] | [
"Definition:Linear Function/One Variable",
"Definition:Equation",
"Definition:Graph of Mapping",
"Definition:Cartesian Plane",
"Definition:Line/Straight Line",
"Definition:Intercept/Y-Intercept",
"Definition:Slope/Straight Line"
] | [
"Equation of Straight Line in Plane/Slope-Intercept Form"
] |
proofwiki-21914 | Power of Ideal is Subset | Let $\struct {R, +, \cdot}$ be a ring.
Let $I \subseteq R$ be an ideal.
Let $n \in \Z$ be an integer such that $n \ge 1$.
Let $I^n$ denote the $n$th power of $I$.
Then $I^n$ is a subset of $I$:
:$I^n \subseteq I$ | Let $n = 1$.
Then by definition of Power of Ideal of Ring:
:$I^1 = I$
Hence by Set is Subset of Itself:
:$I^1 \subseteq I$.
{{qed|lemma}}
Let $n > 1$.
By definition of Power of Ideal of Ring, $I^n$ is an ideal generated by elements of $R$ the form:
:$a_1 \cdots a_n$
where each $a_i \in I$.
From Ideal is Subring, $I$ is... | Let $\struct {R, +, \cdot}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $I \subseteq R$ be an [[Definition:Ideal of Ring|ideal]].
Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ge 1$.
Let $I^n$ denote the [[Definition:Power of Ideal of Ring|$n$th power]] of $I$.
Then $I^n$ is a [[Defin... | Let $n = 1$.
Then by definition of [[Definition:Power of Ideal of Ring|Power of Ideal of Ring]]:
:$I^1 = I$
Hence by [[Set is Subset of Itself]]:
:$I^1 \subseteq I$.
{{qed|lemma}}
Let $n > 1$.
By definition of [[Definition:Power of Ideal of Ring|Power of Ideal of Ring]], $I^n$ is an [[Definition:Ideal of Ring|idea... | Power of Ideal is Subset | https://proofwiki.org/wiki/Power_of_Ideal_is_Subset | https://proofwiki.org/wiki/Power_of_Ideal_is_Subset | [
"Ideal Theory",
"Subsets",
"Power Set"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring",
"Definition:Integer",
"Definition:Power of Ideal of Ring",
"Definition:Subset"
] | [
"Definition:Power of Ideal of Ring",
"Set is Subset of Itself",
"Definition:Power of Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Generated Ideal of Ring",
"Definition:Element",
"Ideal is Subring",
"Definition:Subring",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Defin... |
proofwiki-21915 | Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue/Corollary | Let $\struct {\map D T, T}$ be a self-adjoint densely-defined linear operator. | Let $\map {\sigma_r} T$ be the residual spectrum of $\struct {\map D T, T}$.
From Self-Adjoint Densely-Defined Linear Operator has Empty Residual Spectrum, we have $\map {\sigma_r} T = \O$.
Hence we have $\lambda \in \map \sigma T \setminus \map {\sigma_r} T$.
From Element of Spectrum of Densely-Defined Linear Operato... | Let $\struct {\map D T, T}$ be a [[Definition:Self-Adjoint Densely-Defined Linear Operator|self-adjoint]] [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]]. | Let $\map {\sigma_r} T$ be the [[Definition:Residual Spectrum of Densely-Defined Linear Operator|residual spectrum]] of $\struct {\map D T, T}$.
From [[Self-Adjoint Densely-Defined Linear Operator has Empty Residual Spectrum]], we have $\map {\sigma_r} T = \O$.
Hence we have $\lambda \in \map \sigma T \setminus \map... | Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue/Corollary | https://proofwiki.org/wiki/Element_of_Spectrum_of_Densely-Defined_Linear_Operator_not_in_Residual_Spectrum_is_Approximate_Eigenvalue/Corollary | https://proofwiki.org/wiki/Element_of_Spectrum_of_Densely-Defined_Linear_Operator_not_in_Residual_Spectrum_is_Approximate_Eigenvalue/Corollary | [
"Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue"
] | [
"Definition:Self-Adjoint Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator"
] | [
"Definition:Residual Spectrum of Densely-Defined Linear Operator",
"Self-Adjoint Densely-Defined Linear Operator has Empty Residual Spectrum",
"Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue",
"Definition:Approximate Eigenvalue/Densely-Defined Linear ... |
proofwiki-21916 | Logarithmic Integral and Eulerian Logarithmic Integral Differ by Constant | Let $x \in \R$ be a real number such that $x > 2$.
Let $\map \li x$ denote the logarithmic integral of $x$.
Let $\map \Li x$ denote the Eulerian logarithmic integral of $x$.
Then $\map \li x - \map \Li x$ is a constant. | {{begin-eqn}}
{{eqn | l = \map \li x - \map \Li x
| r = \PV_0^x \frac {\d t} {\ln t} - \int_2^x \frac {\d t} {\ln t}
| c = {{Defof|Logarithmic Integral}}, {{Defof|Eulerian Logarithmic Integral}}
}}
{{eqn | r = \lim_{\varepsilon \mathop \to 0^+} \paren {\int_\varepsilon^{1 - \varepsilon} \frac {\rd t} {\ln t... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $x > 2$.
Let $\map \li x$ denote the [[Definition:Logarithmic Integral|logarithmic integral]] of $x$.
Let $\map \Li x$ denote the [[Definition:Eulerian Logarithmic Integral|Eulerian logarithmic integral]] of $x$.
Then $\map \li x - \map \Li x$ is a... | {{begin-eqn}}
{{eqn | l = \map \li x - \map \Li x
| r = \PV_0^x \frac {\d t} {\ln t} - \int_2^x \frac {\d t} {\ln t}
| c = {{Defof|Logarithmic Integral}}, {{Defof|Eulerian Logarithmic Integral}}
}}
{{eqn | r = \lim_{\varepsilon \mathop \to 0^+} \paren {\int_\varepsilon^{1 - \varepsilon} \frac {\rd t} {\ln t... | Logarithmic Integral and Eulerian Logarithmic Integral Differ by Constant | https://proofwiki.org/wiki/Logarithmic_Integral_and_Eulerian_Logarithmic_Integral_Differ_by_Constant | https://proofwiki.org/wiki/Logarithmic_Integral_and_Eulerian_Logarithmic_Integral_Differ_by_Constant | [
"Logarithmic Integral",
"Eulerian Logarithmic Integral"
] | [
"Definition:Real Number",
"Definition:Logarithmic Integral",
"Definition:Logarithmic Integral/Eulerian",
"Definition:Constant"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions"
] |
proofwiki-21917 | Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom/Lemma 2 | Let $m \in \Z: m \ge 2$.
Let $D_1, D_2, \ldots, D_m \in \mathscr C$ satisfy:
:$\ds \forall 0 \le i \le m : D_i \nsubseteq \bigcup_{j = 1, j \ne i}^m D_j$
Let $X, Y \subseteq S$:
:$\ds \bigcup_{i = 1}^m D_i \subseteq Y \setminus X$
Let $x \in S \setminus X$.
Then:
:$\exists E_1, E_2, \ldots, E_{m - 1} \in \mathscr C$:
:... | === Case 1: $x$ is not in any $D_i$ ===
Let:
:$\forall 1 \le i \le m : x \notin D_i$
We have:
{{begin-eqn}}
{{eqn | l = \bigcup_{i = 1}^m D_i
| r = \bigcup_{i = 1}^m \paren{D_i \setminus \set x}
| c = Set Difference with Disjoint Set
}}
{{eqn | r = \paren{\bigcup_{i = 1}^m D_i} \setminus \set x
| c = ... | Let $m \in \Z: m \ge 2$.
Let $D_1, D_2, \ldots, D_m \in \mathscr C$ satisfy:
:$\ds \forall 0 \le i \le m : D_i \nsubseteq \bigcup_{j = 1, j \ne i}^m D_j$
Let $X, Y \subseteq S$:
:$\ds \bigcup_{i = 1}^m D_i \subseteq Y \setminus X$
Let $x \in S \setminus X$.
Then:
:$\exists E_1, E_2, \ldots, E_{m - 1} \in \mathscr... | === Case 1: $x$ is not in any $D_i$ ===
Let:
:$\forall 1 \le i \le m : x \notin D_i$
We have:
{{begin-eqn}}
{{eqn | l = \bigcup_{i = 1}^m D_i
| r = \bigcup_{i = 1}^m \paren{D_i \setminus \set x}
| c = [[Set Difference with Disjoint Set]]
}}
{{eqn | r = \paren{\bigcup_{i = 1}^m D_i} \setminus \set x
... | Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom/Lemma 2 | https://proofwiki.org/wiki/Matroid_Bases_Satisfy_Formulation_7_of_Matroid_Base_Axiom/Lemma_2 | https://proofwiki.org/wiki/Matroid_Bases_Satisfy_Formulation_7_of_Matroid_Base_Axiom/Lemma_2 | [
"Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom"
] | [] | [
"Set Difference with Disjoint Set",
"Set Difference is Right Distributive over Union",
"Set Difference with Union",
"Set Difference is Right Distributive over Union"
] |
proofwiki-21918 | Form of Logit for Logistic Curve | Let $p$ denote the probability of the occurrence of an event.
Let $p$ satisfy a logistic relationship with an explanatory variable $x$ of the form:
:$p = \dfrac 1 {1 + \map \exp {-\paren {\alpha + \beta x} } }$
Let $Y$ be the logit of $p$.
Then:
:$Y = \alpha + \beta x$ | By definition, the logit of $p$ is given by:
:$Y = \map \ln {\dfrac p {1 - p} }$
In order to simplify the algebra, let $c = -\paren {\alpha + \beta x}$.
Then we have:
{{begin-eqn}}
{{eqn | l = Y
| r = \map \ln {\dfrac {\frac 1 {1 + \exp c} } {1 - \frac 1 {1 + \exp c} } }
| c =
}}
{{eqn | r = \map \ln {\dfr... | Let $p$ denote the [[Definition:Probability|probability]] of the [[Definition:Occurrence of Event|occurrence]] of an [[Definition:Event|event]].
Let $p$ satisfy a [[Definition:Logistic Curve|logistic relationship]] with an [[Definition:Explanatory Variable|explanatory variable]] $x$ of the form:
:$p = \dfrac 1 {1 + \m... | By definition, the [[Definition:Logit|logit]] of $p$ is given by:
:$Y = \map \ln {\dfrac p {1 - p} }$
In order to simplify the algebra, let $c = -\paren {\alpha + \beta x}$.
Then we have:
{{begin-eqn}}
{{eqn | l = Y
| r = \map \ln {\dfrac {\frac 1 {1 + \exp c} } {1 - \frac 1 {1 + \exp c} } }
| c =
}}
{{... | Form of Logit for Logistic Curve | https://proofwiki.org/wiki/Form_of_Logit_for_Logistic_Curve | https://proofwiki.org/wiki/Form_of_Logit_for_Logistic_Curve | [
"Logit",
"Logistic Curve"
] | [
"Definition:Probability",
"Definition:Event/Occurrence",
"Definition:Event",
"Definition:Logistic Curve",
"Definition:Regression/Cause Variable",
"Definition:Logit"
] | [
"Definition:Logit",
"Definition:Fraction",
"Definition:Common Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Logarithm of Reciprocal"
] |
proofwiki-21919 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 4 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Let $M = \struct{S, \mathscr I}$ where:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \card X}$
Then $M$... | We have:
{{begin-eqn}}
{{eqn | l = \map \rho \O
| r = 0
| c = Rank axiom $(\text R 1)$
}}
{{eqn | r = \card \O
| c = Cardinality of Empty Set
}}
{{end-eqn}}
So:
:$\O \in \mathscr I$
Hence:
:$M$ satisfies matroid axiom $(\text I 1)$. | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (M... | We have:
{{begin-eqn}}
{{eqn | l = \map \rho \O
| r = 0
| c = [[Axiom:Rank Axioms (Matroid)/Definition 1|Rank axiom $(\text R 1)$]]
}}
{{eqn | r = \card \O
| c = [[Cardinality of Empty Set]]
}}
{{end-eqn}}
So:
:$\O \in \mathscr I$
Hence:
:$M$ satisfies [[Axiom:Matroid Axioms|matroid axiom $(\text I ... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 4 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_4 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_4 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Axiom:Matroid Axioms"
] | [
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Cardinality of Empty Set",
"Axiom:Matroid Axioms"
] |
proofwiki-21920 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 5 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Let $M = \struct{S, \mathscr I}$ where:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \card X}$
Then $M$... | Let
:$X \in \mathscr I$
{{AimForCont}}
:$\exists Y \subseteq X : Y \notin \mathscr I$
Let:
:$Y_0 \subseteq X : \card {Y_0} = \max \set{\card Z : Z \subseteq X \land Z \notin \mathscr I}$
By definition of $\mathscr I$:
:$Y_0 \notin \mathscr I \leadsto \map \rho {Y_0} \ne \card {Y_0}$
From Lemma 2:
:$\map \rho {Y_0} < \c... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (M... | Let
:$X \in \mathscr I$
{{AimForCont}}
:$\exists Y \subseteq X : Y \notin \mathscr I$
Let:
:$Y_0 \subseteq X : \card {Y_0} = \max \set{\card Z : Z \subseteq X \land Z \notin \mathscr I}$
By definition of $\mathscr I$:
:$Y_0 \notin \mathscr I \leadsto \map \rho {Y_0} \ne \card {Y_0}$
From [[Formulation 1 Rank Axio... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 5/Proof 1 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_5 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_5/Proof_1 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Axiom:Matroid Axioms"
] | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2",
"Definition:Proper Subset",
"Set Difference with Proper Subset",
"Cardinality of Set Union/Corollary",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Definition:Contradiction",
"Axiom:Matroid Axioms"
] |
proofwiki-21921 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 5 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Let $M = \struct{S, \mathscr I}$ where:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \card X}$
Then $M$... | We prove the contrapositive statement:
:$\forall X, Y \subseteq S: Y \notin \mathscr I \land Y \subseteq X \implies X \notin \mathscr I$
Let $X, Y \subseteq S : Y \notin \mathscr I$ and $Y \subseteq X$.
====== Case 1: $Y = X$ ======
Let $Y = X$.
Then $X \notin \mathscr I$.
{{qed|lemma}}
====== Case 2: $Y \subset X$ ===... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (M... | We prove the [[Definition:Contrapositive Statement|contrapositive statement]]:
:$\forall X, Y \subseteq S: Y \notin \mathscr I \land Y \subseteq X \implies X \notin \mathscr I$
Let $X, Y \subseteq S : Y \notin \mathscr I$ and $Y \subseteq X$.
====== Case 1: $Y = X$ ======
Let $Y = X$.
Then $X \notin \mathscr I$.
{{... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 5/Proof 2 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_5 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_5/Proof_2 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Axiom:Matroid Axioms"
] | [
"Definition:Contrapositive Statement",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2",
"Cardinality of Set Union/Corollary",
"Definition:Contrapositive Statement",
"Axi... |
proofwiki-21922 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 6 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Let $M = \struct{S, \mathscr I}$ where:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \card X}$
Then $M$... | Let
:$U \in \mathscr I$
:$V \subseteq S$
:$\card U < \card V$
We prove the contrapositive statement:
:$\paren{\forall x \in V \setminus U : U \cup \set x \notin \mathscr I} \implies V \notin \mathscr I$
Let:
:$\forall x \in V \setminus U: U \cup \set x \notin \mathscr I$
That is, by definition of $\mathscr I$:
:$\foral... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (M... | Let
:$U \in \mathscr I$
:$V \subseteq S$
:$\card U < \card V$
We prove the [[Definition:Contrapositive Statement|contrapositive statement]]:
:$\paren{\forall x \in V \setminus U : U \cup \set x \notin \mathscr I} \implies V \notin \mathscr I$
Let:
:$\forall x \in V \setminus U: U \cup \set x \notin \mathscr I$
Tha... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 6 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_6 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_6 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Axiom:Matroid Axioms"
] | [
"Definition:Contrapositive Statement",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 3",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1",
"Axiom:Matroid Axioms"
] |
proofwiki-21923 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 7 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Let $M = \struct{S, \mathscr I}$ where:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \card X}$
Then $M$... | It is now proved that $\mathscr I$ satisifes the matroid Axiom $(\text I 4)$:
{{begin-axiom}}
{{axiom | n = \text I 4
| q = \forall U, V \in \mathscr I
| mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}}
Let $X, Y \in \mathscr I$ such that $\s... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (M... | It is now proved that $\mathscr I$ satisifes the [[Axiom:Matroid Axioms|matroid Axiom $(\text I 4)$]]:
{{begin-axiom}}
{{axiom | n = \text I 4
| q = \forall U, V \in \mathscr I
| mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}}
Let $X, Y \i... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 7 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_7 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_7 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Axiom:Matroid Axioms"
] | [
"Axiom:Matroid Axioms",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 3",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1",
"Definition:Contradiction",
"Axiom:Matroid Axioms"
] |
proofwiki-21924 | Operator with Zero Numerical Range is Zero Operator | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\norm {\, \cdot \,}$ be the inner product norm on $\struct {\HH, \innerprod \cdot \cdot}$.
Let $\struct {\map D T, T}$ be a densely-defined linear operator on $\HH$ such that:
:$\map W T = \set 0$
where $\map W T$ is the numerical range of ... | Let $x, y \in \map D T$.
Since $\map W T = \set 0$, we have:
:$\innerprod {\map T {x + y} } {x + y} = \innerprod {T x} x = \innerprod {T y} y = 0$
From Inner Product is Sesquilinear, we have:
:$\innerprod {T x} x + \innerprod {T x} y + \innerprod {T y} x + \innerprod {T y} y = 0$
That is:
:$\innerprod {T x} y = -\inne... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\norm {\, \cdot \,}$ be the [[Definition:Inner Product Norm|inner product norm]] on $\struct {\HH, \innerprod \cdot \cdot}$.
Let $\struct {\map D T, T}$ be a [[Definition:Densely-Defined Linear Operator|densel... | Let $x, y \in \map D T$.
Since $\map W T = \set 0$, we have:
:$\innerprod {\map T {x + y} } {x + y} = \innerprod {T x} x = \innerprod {T y} y = 0$
From [[Inner Product is Sesquilinear]], we have:
:$\innerprod {T x} x + \innerprod {T x} y + \innerprod {T y} x + \innerprod {T y} y = 0$
That is:
:$\innerprod {T x} y =... | Operator with Zero Numerical Range is Zero Operator | https://proofwiki.org/wiki/Operator_with_Zero_Numerical_Range_is_Zero_Operator | https://proofwiki.org/wiki/Operator_with_Zero_Numerical_Range_is_Zero_Operator | [
"Numerical Range",
"Operator with Zero Numerical Range is Zero Operator"
] | [
"Definition:Hilbert Space",
"Definition:Inner Product Norm",
"Definition:Densely-Defined Linear Operator",
"Definition:Numerical Range"
] | [
"Inner Product is Sesquilinear",
"Inner Product is Sesquilinear",
"Definition:Linear Transformation",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement",
"Linear Subspace Dense iff Zero Orthocomplement"
] |
proofwiki-21925 | Operator with Zero Numerical Range is Zero Operator/Corollary | Let $\DD$ be a dense linear subspace of $\HH$.
Let $\struct {\DD, T}$ and $\struct {\DD, S}$ be densely-defined linear operators on $\HH$ such that:
:$\innerprod {T x} x = \innerprod {S x} x$ for each $x \in \DD$.
Then $T = S$. | From Inner Product is Sesquilinear, we have:
:$\innerprod {\paren {T - S} x} x = 0$ for each $x \in \DD$.
Applying Operator with Zero Numerical Range is Zero Operator to the densely-defined linear operator $\struct {\DD, T - S}$, we have:
:$T - S = 0$
That is:
:$T = S$
{{qed}} | Let $\DD$ be a [[Definition:Dense Set|dense]] [[Definition:Linear Subspace|linear subspace]] of $\HH$.
Let $\struct {\DD, T}$ and $\struct {\DD, S}$ be [[Definition:Densely-Defined Linear Operator|densely-defined linear operators]] on $\HH$ such that:
:$\innerprod {T x} x = \innerprod {S x} x$ for each $x \in \DD$.
... | From [[Inner Product is Sesquilinear]], we have:
:$\innerprod {\paren {T - S} x} x = 0$ for each $x \in \DD$.
Applying [[Operator with Zero Numerical Range is Zero Operator]] to the [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]] $\struct {\DD, T - S}$, we have:
:$T - S = 0$
That is:
:$... | Operator with Zero Numerical Range is Zero Operator/Corollary | https://proofwiki.org/wiki/Operator_with_Zero_Numerical_Range_is_Zero_Operator/Corollary | https://proofwiki.org/wiki/Operator_with_Zero_Numerical_Range_is_Zero_Operator/Corollary | [
"Operator with Zero Numerical Range is Zero Operator"
] | [
"Definition:Everywhere Dense",
"Definition:Linear Subspace",
"Definition:Densely-Defined Linear Operator"
] | [
"Inner Product is Sesquilinear",
"Operator with Zero Numerical Range is Zero Operator",
"Definition:Densely-Defined Linear Operator"
] |
proofwiki-21926 | Square of Inner Product Norm of Sum | Let $\struct {X, \innerprod \cdot \cdot}$ be an inner product space.
Let $\norm {\, \cdot \,}$ be the inner product norm.
Let $x, y \in X$.
Then, we have:
:$\norm {x + y}^2 = \norm x^2 + 2 \map \Re {\innerprod x y} + \norm y^2$ | {{begin-eqn}}
{{eqn | l = \norm {x + y}^2
| r = \innerprod {x + y} {x + y}
| c = {{Defof|Inner Product Norm}}
}}
{{eqn | r = \innerprod x x + \innerprod y x + \innerprod x y + \innerprod y y
| c = Inner Product is Sesquilinear
}}
{{eqn | r = \norm x^2 + \overline {\innerprod x y} + \innerprod x y + \norm y^2
| ... | Let $\struct {X, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]].
Let $\norm {\, \cdot \,}$ be the [[Definition:Inner Product Norm|inner product norm]].
Let $x, y \in X$.
Then, we have:
:$\norm {x + y}^2 = \norm x^2 + 2 \map \Re {\innerprod x y} + \norm y^2$ | {{begin-eqn}}
{{eqn | l = \norm {x + y}^2
| r = \innerprod {x + y} {x + y}
| c = {{Defof|Inner Product Norm}}
}}
{{eqn | r = \innerprod x x + \innerprod y x + \innerprod x y + \innerprod y y
| c = [[Inner Product is Sesquilinear]]
}}
{{eqn | r = \norm x^2 + \overline {\innerprod x y} + \innerprod x y + \norm y^2
... | Square of Inner Product Norm of Sum | https://proofwiki.org/wiki/Square_of_Inner_Product_Norm_of_Sum | https://proofwiki.org/wiki/Square_of_Inner_Product_Norm_of_Sum | [
"Inner Product Spaces"
] | [
"Definition:Inner Product Space",
"Definition:Inner Product Norm"
] | [
"Inner Product is Sesquilinear",
"Definition:Conjugate Symmetric Mapping",
"Definition:Inner Product",
"Sum of Complex Number with Conjugate",
"Category:Inner Product Spaces"
] |
proofwiki-21927 | Fixed Points of Idempotent Operator | Let $X$ be a vector space.
Let $T : X \to X$ be an idempotent operator.
Then the set of fixed points of $T$ is precisely the range $\Rng T$. | If $x$ is a fixed point of $T$, then:
:$T x = x$
and we clearly have $x \in \Rng T$.
Conversely, if $y \in \Rng T$ then there exists $x \in X$ such that $y = T x$.
Then we have:
{{begin-eqn}}
{{eqn | l = T y
| r = T^2 x
}}
{{eqn | r = T x
| c = since $T^2 = T$
}}
{{eqn | r = y
}}
{{end-eqn}}
So $y$ is a fixed poi... | Let $X$ be a [[Definition:Vector Space|vector space]].
Let $T : X \to X$ be an [[Definition:Idempotent Operator|idempotent operator]].
Then the [[Definition:Set|set]] of [[Definition:Fixed Point|fixed points]] of $T$ is precisely the [[Definition:Range of Relation|range]] $\Rng T$. | If $x$ is a [[Definition:Fixed Point|fixed point]] of $T$, then:
:$T x = x$
and we clearly have $x \in \Rng T$.
Conversely, if $y \in \Rng T$ then there exists $x \in X$ such that $y = T x$.
Then we have:
{{begin-eqn}}
{{eqn | l = T y
| r = T^2 x
}}
{{eqn | r = T x
| c = since $T^2 = T$
}}
{{eqn | r = y
}}
{{en... | Fixed Points of Idempotent Operator | https://proofwiki.org/wiki/Fixed_Points_of_Idempotent_Operator | https://proofwiki.org/wiki/Fixed_Points_of_Idempotent_Operator | [
"Idempotent Operators"
] | [
"Definition:Vector Space",
"Definition:Idempotent Operator",
"Definition:Set",
"Definition:Fixed Point",
"Definition:Range of Relation"
] | [
"Definition:Fixed Point",
"Definition:Fixed Point",
"Category:Idempotent Operators"
] |
proofwiki-21928 | Identity Element in Unital *-Algebra is Hermitian | Let $\struct {A, \ast}$ be a unital $\ast$-algebra.
Let ${\mathbf 1}_A$ be the identity element of $A$.
Then we have:
:${\mathbf 1}_A^\ast = {\mathbf 1}_A$ | From Product of Element in *-Star Algebra with its Star is Hermitian, we have:
:$\paren { {\mathbf 1}_A {\mathbf 1}_A^\ast}^\ast = {\mathbf 1}_A {\mathbf 1}_A^\ast$
We have, since $\ast$ is an involution:
:$\paren { {\mathbf 1}_A {\mathbf 1}_A^\ast} = {\mathbf 1}_A^{\ast \ast} = {\mathbf 1}_A$
and:
:${\mathbf 1}_A {\m... | Let $\struct {A, \ast}$ be a [[Definition:Unital Algebra|unital]] [[Definition:*-Algebra|$\ast$-algebra]].
Let ${\mathbf 1}_A$ be the [[Definition:Identity Element|identity element]] of $A$.
Then we have:
:${\mathbf 1}_A^\ast = {\mathbf 1}_A$ | From [[Product of Element in *-Star Algebra with its Star is Hermitian]], we have:
:$\paren { {\mathbf 1}_A {\mathbf 1}_A^\ast}^\ast = {\mathbf 1}_A {\mathbf 1}_A^\ast$
We have, since $\ast$ is an [[Definition:Involution on Algebra|involution]]:
:$\paren { {\mathbf 1}_A {\mathbf 1}_A^\ast} = {\mathbf 1}_A^{\ast \ast}... | Identity Element in Unital *-Algebra is Hermitian | https://proofwiki.org/wiki/Identity_Element_in_Unital_*-Algebra_is_Hermitian | https://proofwiki.org/wiki/Identity_Element_in_Unital_*-Algebra_is_Hermitian | [
"Hermitian Elements of *-Algebras"
] | [
"Definition:Unital Algebra",
"Definition:*-Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Product of Element in *-Star Algebra with its Star is Hermitian",
"Definition:Involution on Algebra"
] |
proofwiki-21929 | Inverse of Star of Element in Unital *-Algebra | Let $\struct {A, \ast}$ be a unital $\ast$-algebra.
Let $a \in A$ be invertible.
Then $a^\ast$ is invertible and:
:$\paren {a^\ast}^{-1} = \paren {a^{-1} }^\ast$ | We have:
:$a a^{-1} = a^{-1} a = {\mathbf 1}_A$
From $(C^\ast 3)$ in the definition of an involution, we have:
:$\paren {a^{-1} }^\ast a^\ast = a^\ast \paren {a^{-1} }^\ast = {\mathbf 1}_A^\ast$
From Identity Element in Unital *-Algebra is Hermitian, we therefore have:
:$\paren {a^{-1} }^\ast a^\ast = a^\ast \paren {... | Let $\struct {A, \ast}$ be a [[Definition:Unital Algebra|unital]] [[Definition:*-Algebra|$\ast$-algebra]].
Let $a \in A$ be [[Definition:Invertible Element|invertible]].
Then $a^\ast$ is [[Definition:Invertible Element|invertible]] and:
:$\paren {a^\ast}^{-1} = \paren {a^{-1} }^\ast$ | We have:
:$a a^{-1} = a^{-1} a = {\mathbf 1}_A$
From $(C^\ast 3)$ in the definition of an [[Definition:Involution on Algebra|involution]], we have:
:$\paren {a^{-1} }^\ast a^\ast = a^\ast \paren {a^{-1} }^\ast = {\mathbf 1}_A^\ast$
From [[Identity Element in Unital *-Algebra is Hermitian]], we therefore have:
:$\pa... | Inverse of Star of Element in Unital *-Algebra | https://proofwiki.org/wiki/Inverse_of_Star_of_Element_in_Unital_*-Algebra | https://proofwiki.org/wiki/Inverse_of_Star_of_Element_in_Unital_*-Algebra | [
"*-Algebras",
"Inverse of Star of Element in Unital *-Algebra"
] | [
"Definition:Unital Algebra",
"Definition:*-Algebra",
"Definition:Invertible Element",
"Definition:Invertible Element"
] | [
"Definition:Involution on Algebra",
"Identity Element in Unital *-Algebra is Hermitian"
] |
proofwiki-21930 | Spectrum of Star of Element in *-Algebra | Let $\struct {A, \ast}$ be a unital $\ast$-algebra over $\C$.
Let $a \in A$.
Let $\sigma_A$ denote the spectrum.
Then:
:$\map {\sigma_A} {a^\ast} = \set {\overline \lambda : \lambda \in \map {\sigma_A} a}$ | From the definition of an involution, we have:
:$\paren {\lambda {\mathbf 1}_A - a}^\ast = \overline \lambda {\mathbf 1}_A^\ast - a^\ast$
From Identity Element in Unital *-Algebra is Hermitian, we therefore have:
:$\paren {\lambda {\mathbf 1}_A - a}^\ast = \overline \lambda {\mathbf 1}_A - a^\ast$
From Inverse of Sta... | Let $\struct {A, \ast}$ be a [[Definition:Unital Algebra|unital]] [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $a \in A$.
Let $\sigma_A$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]].
Then:
:$\map {\sigma_A} {a^\ast} = \set {\overline \lambda : \lambda \in \map {\sigma_A} a... | From the definition of an [[Definition:Involution on Algebra|involution]], we have:
:$\paren {\lambda {\mathbf 1}_A - a}^\ast = \overline \lambda {\mathbf 1}_A^\ast - a^\ast$
From [[Identity Element in Unital *-Algebra is Hermitian]], we therefore have:
:$\paren {\lambda {\mathbf 1}_A - a}^\ast = \overline \lambda {\... | Spectrum of Star of Element in *-Algebra | https://proofwiki.org/wiki/Spectrum_of_Star_of_Element_in_*-Algebra | https://proofwiki.org/wiki/Spectrum_of_Star_of_Element_in_*-Algebra | [
"Spectra (Spectral Theory)"
] | [
"Definition:Unital Algebra",
"Definition:*-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra"
] | [
"Definition:Involution on Algebra",
"Identity Element in Unital *-Algebra is Hermitian",
"Inverse of Star of Element in Unital *-Algebra/Corollary",
"Definition:Invertible Element",
"Definition:Invertible Element"
] |
proofwiki-21931 | Resolvent Set of Element in Subalgebra | Let $A$ be a Banach algebra.
Let $B$ be a closed unital subalgebra of $A$.
Let $x \in B$.
Let $\map {\rho_A} x$ and $\map {\rho_B} x$ be the resolvent sets of $x$ in $A$ and $B$ respectively.
Then:
:$\map {\rho_B} x \subseteq \map {\rho_A} x$ | Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the spectra of $x$ in $A$ and $B$ respectively.
From Spectrum of Element in Unital Subalgebra, we have:
:$\map {\sigma_A} x \subseteq \map {\sigma_B} x$
From Set Complement inverts Subsets, we have:
:$\C \setminus \map {\sigma_B} x \subseteq \C \setminus \map {\sigma_... | Let $A$ be a [[Definition:Banach Algebra|Banach algebra]].
Let $B$ be a [[Definition:Closed Set|closed]] [[Definition:Unital Subalgebra|unital subalgebra]] of $A$.
Let $x \in B$.
Let $\map {\rho_A} x$ and $\map {\rho_B} x$ be the [[Definition:Resolvent Set/Unital Algebra|resolvent sets]] of $x$ in $A$ and $B$ respec... | Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the [[Definition:Spectrum (Spectral Theory)|spectra]] of $x$ in $A$ and $B$ respectively.
From [[Spectrum of Element in Unital Subalgebra]], we have:
:$\map {\sigma_A} x \subseteq \map {\sigma_B} x$
From [[Set Complement inverts Subsets]], we have:
:$\C \setminus \m... | Resolvent Set of Element in Subalgebra | https://proofwiki.org/wiki/Resolvent_Set_of_Element_in_Subalgebra | https://proofwiki.org/wiki/Resolvent_Set_of_Element_in_Subalgebra | [
"Resolvent Sets"
] | [
"Definition:Banach Algebra",
"Definition:Closed Set",
"Definition:Unital Subalgebra",
"Definition:Resolvent Set/Unital Algebra"
] | [
"Definition:Spectrum (Spectral Theory)",
"Spectrum of Element in Unital Subalgebra",
"Set Complement inverts Subsets",
"Definition:Resolvent Set/Unital Algebra",
"Category:Resolvent Sets"
] |
proofwiki-21932 | Resolvent Set of Element in Closed Unital Subalgebra of Banach Algebra is Clopen in Resolvent Set | Let $A$ be a Banach algebra.
Let $B$ be a closed unital subalgebra of $A$.
Let $x \in B$.
Let $\map {\rho_A} x$ and $\map {\rho_B} x$ be the resolvent sets of $x$ in $A$ and $B$ respectively.
Then $\map {\rho_B} x$ is clopen in $\map {\rho_A} x$. | Let $\map G A$ and $\map G B$ be the group of units of $A$ and $B$ respectively.
First, from Resolvent Set of Element in Subalgebra we have:
:$\map {\rho_B} x \subseteq \map {\rho_A} x$
Define $R : \map {\rho_A} x \to A$ by:
:$\map R \lambda = \paren {\lambda {\mathbf 1}_A - x}^{-1}$
for each $\lambda \in \map {\rh... | Let $A$ be a [[Definition:Banach Algebra|Banach algebra]].
Let $B$ be a [[Definition:Closed Set|closed]] [[Definition:Unital Subalgebra|unital subalgebra]] of $A$.
Let $x \in B$.
Let $\map {\rho_A} x$ and $\map {\rho_B} x$ be the [[Definition:Resolvent Set/Unital Algebra|resolvent sets]] of $x$ in $A$ and $B$ respec... | Let $\map G A$ and $\map G B$ be the [[Definition:Group of Units|group of units]] of $A$ and $B$ respectively.
First, from [[Resolvent Set of Element in Subalgebra]] we have:
:$\map {\rho_B} x \subseteq \map {\rho_A} x$
Define $R : \map {\rho_A} x \to A$ by:
:$\map R \lambda = \paren {\lambda {\mathbf 1}_A - x}^{-... | Resolvent Set of Element in Closed Unital Subalgebra of Banach Algebra is Clopen in Resolvent Set | https://proofwiki.org/wiki/Resolvent_Set_of_Element_in_Closed_Unital_Subalgebra_of_Banach_Algebra_is_Clopen_in_Resolvent_Set | https://proofwiki.org/wiki/Resolvent_Set_of_Element_in_Closed_Unital_Subalgebra_of_Banach_Algebra_is_Clopen_in_Resolvent_Set | [
"Resolvent Sets"
] | [
"Definition:Banach Algebra",
"Definition:Closed Set",
"Definition:Unital Subalgebra",
"Definition:Resolvent Set/Unital Algebra",
"Definition:Clopen Set"
] | [
"Definition:Group of Units",
"Resolvent Set of Element in Subalgebra",
"Resolvent Mapping is Continuous/Banach Algebra",
"Definition:Continuous Mapping",
"Definition:Continuous Mapping",
"Definition:Closed Set",
"Definition:Closed Set",
"Definition:Closed Set",
"Resolvent Set of Element of Banach Al... |
proofwiki-21933 | Not All Matroids are Base-Orderable | Not all matroids are base-orderable. | {{ProofWanted}}
Category:Matroid Theory
hnkkybqnkcy878q8ufhn3f8qvipag2o | Not all [[Definition:Matroid|matroids]] are [[Definition:Base-Orderable Matroid|base-orderable]]. | {{ProofWanted}}
[[Category:Matroid Theory]]
hnkkybqnkcy878q8ufhn3f8qvipag2o | Not All Matroids are Base-Orderable | https://proofwiki.org/wiki/Not_All_Matroids_are_Base-Orderable | https://proofwiki.org/wiki/Not_All_Matroids_are_Base-Orderable | [
"Matroid Theory"
] | [
"Definition:Matroid",
"Definition:Base-Orderable Matroid"
] | [
"Category:Matroid Theory"
] |
proofwiki-21934 | Open Balls form Local Basis of Metric Space | Let $\struct {M, d}$ be a metric space.
Let $\tau$ be the topology on $M$ induced by $d$.
Let $x \in X$.
For each $\epsilon > 0$, let $\map {B_\epsilon} x$ be the open $\epsilon$-ball around $x$.
Let $\BB_x = \set {\map {B_\epsilon} x : \epsilon > 0}$.
Then $\BB_x$ is a local basis at $x$. | Let $U$ be an open neighborhood of $x$.
From the definition of an open set in $\struct {M, d}$:
:there exists $\epsilon > 0$ such that $\map {B_\epsilon} x \subseteq U$.
Hence:
:there exists $V \in \BB_x$ such that $V \subseteq U$.
So $\BB_x$ is a local basis at $x$.
{{qed}}
Category:Local Bases
Category:Metric Spac... | Let $\struct {M, d}$ be a [[Definition:Metric Space|metric space]].
Let $\tau$ be the [[Definition:Topology Induced by Metric|topology on $M$ induced by $d$]].
Let $x \in X$.
For each $\epsilon > 0$, let $\map {B_\epsilon} x$ be the [[Definition:Open Ball|open $\epsilon$-ball]] around $x$.
Let $\BB_x = \set {\ma... | Let $U$ be an [[Definition:Open Neighborhood|open neighborhood]] of $x$.
From the definition of an [[Definition:Open Set|open set]] in $\struct {M, d}$:
:there exists $\epsilon > 0$ such that $\map {B_\epsilon} x \subseteq U$.
Hence:
:there exists $V \in \BB_x$ such that $V \subseteq U$.
So $\BB_x$ is a [[Definit... | Open Balls form Local Basis of Metric Space | https://proofwiki.org/wiki/Open_Balls_form_Local_Basis_of_Metric_Space | https://proofwiki.org/wiki/Open_Balls_form_Local_Basis_of_Metric_Space | [
"Local Bases",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Open Ball",
"Definition:Local Basis"
] | [
"Definition:Open Neighborhood",
"Definition:Open Set",
"Definition:Local Basis",
"Category:Local Bases",
"Category:Metric Spaces"
] |
proofwiki-21935 | Normed Vector Space is Locally Connected | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Then $\struct {X, \norm {\, \cdot \,} }$ is locally connected. | Let $x \in X$.
Let $\epsilon > 0$.
Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball around $x$ in $\struct {X, \norm {\, \cdot \,} }$.
From Open Balls form Local Basis of Metric Space, the set $\BB_x = \set {\map {B_\epsilon} x : \epsilon > 0}$ is a local basis at $x$.
From Open Ball in Normed Vector Space is Co... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Then $\struct {X, \norm {\, \cdot \,} }$ is [[Definition:Locally Connected Space|locally connected]]. | Let $x \in X$.
Let $\epsilon > 0$.
Let $\map {B_\epsilon} x$ be the [[Definition:Open Ball|open $\epsilon$-ball]] around $x$ in $\struct {X, \norm {\, \cdot \,} }$.
From [[Open Balls form Local Basis of Metric Space]], the [[Definition:Set|set]] $\BB_x = \set {\map {B_\epsilon} x : \epsilon > 0}$ is a [[Definition... | Normed Vector Space is Locally Connected | https://proofwiki.org/wiki/Normed_Vector_Space_is_Locally_Connected | https://proofwiki.org/wiki/Normed_Vector_Space_is_Locally_Connected | [
"Locally Connected Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Locally Connected Space"
] | [
"Definition:Open Ball",
"Open Balls form Local Basis of Metric Space",
"Definition:Set",
"Definition:Local Basis",
"Open Ball in Normed Vector Space is Connected",
"Definition:Connected Set (Topology)",
"Definition:Local Basis",
"Definition:Connected Set (Topology)",
"Definition:Locally Connected Sp... |
proofwiki-21936 | Locally Connected Separable Topological Space has Countably Many Components | Let $\struct {X, \tau}$ be a locally connected separable topological space.
Let $\CC$ be the set of components of $\struct {X, \tau}$.
Then $\CC$ is countable. | From Component of Locally Connected Space is Open:
:$\CC \subseteq \tau$
From Equivalence Classes are Disjoint, we have:
:the sets in $\CC$ are pairwise disjoint.
From Collection of Pairwise Disjoint Open Sets in Separable Topological Space is Countable, it follows that:
:$\CC$ is countable.
{{qed}}
Category:Locally Co... | Let $\struct {X, \tau}$ be a [[Definition:Locally Connected Space|locally connected]] [[Definition:Separable Space|separable]] [[Definition:Topological Space|topological space]].
Let $\CC$ be the [[Definition:Set|set]] of [[Definition:Component (Topology)|components]] of $\struct {X, \tau}$.
Then $\CC$ is [[Definit... | From [[Component of Locally Connected Space is Open]]:
:$\CC \subseteq \tau$
From [[Equivalence Classes are Disjoint]], we have:
:the [[Definition:Set|sets]] in $\CC$ are [[Definition:Pairwise Disjoint|pairwise disjoint]].
From [[Collection of Pairwise Disjoint Open Sets in Separable Topological Space is Countable]],... | Locally Connected Separable Topological Space has Countably Many Components | https://proofwiki.org/wiki/Locally_Connected_Separable_Topological_Space_has_Countably_Many_Components | https://proofwiki.org/wiki/Locally_Connected_Separable_Topological_Space_has_Countably_Many_Components | [
"Locally Connected Spaces",
"Components (Topology)"
] | [
"Definition:Locally Connected Space",
"Definition:Separable Space",
"Definition:Topological Space",
"Definition:Set",
"Definition:Component (Topology)",
"Definition:Countable Set"
] | [
"Component of Locally Connected Space is Open",
"Equivalence Classes are Disjoint",
"Definition:Set",
"Definition:Pairwise Disjoint",
"Separable Space satisfies Countable Chain Condition",
"Definition:Countable Set",
"Category:Locally Connected Spaces",
"Category:Components (Topology)"
] |
proofwiki-21937 | Boundary of Spectrum of Element in Subalgebra of Unital Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.
Let $B$ be a closed unital subalgebra of $A$.
Let $x \in B$.
Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the spectra of $x$ in $A$ and $B$ respectively.
Then:
:$\partial \map {\sigma_B} x \subseteq \partial \map {\sigma_A} x$
where $\... | Let:
:$\lambda \in \partial \map {\sigma_B} x$.
From Boundary is Intersection of Closure with Closure of Complement and Spectrum of Element of Banach Algebra is Closed, we have:
:$\partial \map {\sigma_B} x = \map {\sigma_B} x \cap \map \cl {\map {\rho_B} x}$
where $\map {\rho_B} x$ is the resolvent set of $x$ in $B$.... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $B$ be a [[Definition:Closed Set|closed]] [[Definition:Unital Subalgebra|unital subalgebra]] of $A$.
Let $x \in B$.
Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the [[Definition:Spectrum (Spe... | Let:
:$\lambda \in \partial \map {\sigma_B} x$.
From [[Boundary is Intersection of Closure with Closure of Complement]] and [[Spectrum of Element of Banach Algebra is Closed]], we have:
:$\partial \map {\sigma_B} x = \map {\sigma_B} x \cap \map \cl {\map {\rho_B} x}$
where $\map {\rho_B} x$ is the [[Definition:Resolv... | Boundary of Spectrum of Element in Subalgebra of Unital Banach Algebra/Proof 2 | https://proofwiki.org/wiki/Boundary_of_Spectrum_of_Element_in_Subalgebra_of_Unital_Banach_Algebra | https://proofwiki.org/wiki/Boundary_of_Spectrum_of_Element_in_Subalgebra_of_Unital_Banach_Algebra/Proof_2 | [
"Spectra (Spectral Theory)",
"Unital Banach Algebras",
"Boundary of Spectrum of Element in Subalgebra of Unital Banach Algebra"
] | [
"Definition:Unital Banach Algebra",
"Definition:Closed Set",
"Definition:Unital Subalgebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Boundary (Topology)"
] | [
"Boundary is Intersection of Closure with Closure of Complement",
"Spectrum of Element of Banach Algebra is Closed",
"Definition:Resolvent Set",
"Definition:Resolvent Set",
"Closure of Subset in Subspace",
"Definition:Closure (Topology)",
"Definition:Topological Subspace",
"Resolvent Set of Element in... |
proofwiki-21938 | Component of Resolvent Set of Element in Unital Banach Algebra is Disjoint from or Component of Resolvent Set in Closed Subalgebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.
Let $B$ be a closed unital subalgebra of $A$.
Let $x \in B$.
Let $\map {\rho_A} x$ and $\map {\rho_B} x$ be the resolvent sets of $x$ in $A$ and $B$ respectively.
Let $U$ be a component of $\map {\rho_A} x$.
Then either:
:$U$ is a component... | From Resolvent Set of Element of Banach Algebra is Open:
:$\map {\rho_B} x$ is open in $\C$.
From Normed Vector Space is Locally Connected:
:$\C$ is locally connected.
From Open Subset of Locally Connected Space is Locally Connected:
:$\map {\rho_B} x$ is locally connected.
Hence, from Component of Locally Connected Sp... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $B$ be a [[Definition:Closed Set|closed]] [[Definition:Unital Subalgebra|unital subalgebra]] of $A$.
Let $x \in B$.
Let $\map {\rho_A} x$ and $\map {\rho_B} x$ be the [[Definition:Resolvent Set/Uni... | From [[Resolvent Set of Element of Banach Algebra is Open]]:
:$\map {\rho_B} x$ is [[Definition:Open Set|open]] in $\C$.
From [[Normed Vector Space is Locally Connected]]:
:$\C$ is [[Definition:Locally Connected Space|locally connected]].
From [[Open Subset of Locally Connected Space is Locally Connected]]:
:$\map {\... | Component of Resolvent Set of Element in Unital Banach Algebra is Disjoint from or Component of Resolvent Set in Closed Subalgebra | https://proofwiki.org/wiki/Component_of_Resolvent_Set_of_Element_in_Unital_Banach_Algebra_is_Disjoint_from_or_Component_of_Resolvent_Set_in_Closed_Subalgebra | https://proofwiki.org/wiki/Component_of_Resolvent_Set_of_Element_in_Unital_Banach_Algebra_is_Disjoint_from_or_Component_of_Resolvent_Set_in_Closed_Subalgebra | [
"Resolvent Sets",
"Unital Banach Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:Closed Set",
"Definition:Unital Subalgebra",
"Definition:Resolvent Set/Unital Algebra",
"Definition:Component (Topology)",
"Definition:Component (Topology)"
] | [
"Resolvent Set of Element of Banach Algebra is Open",
"Definition:Open Set",
"Normed Vector Space is Locally Connected",
"Definition:Locally Connected Space",
"Open Subset of Locally Connected Space is Locally Connected",
"Definition:Locally Connected Space",
"Component of Locally Connected Space is Ope... |
proofwiki-21939 | Resolvent Set of Element in Banach Algebra has same Unique Unbounded Component as Resolvent Set of Element in Closed Subalgebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.
Let $B$ be a closed unital subalgebra of $A$.
Let $x \in B$.
Let $\map {\rho_A} x$ and $\map {\rho_B} x$ be the resolvent sets of $x$ in $A$ and $B$ respectively.
Then $\map {\rho_A} x$ and $\map {\rho_B} x$ both have unique unbounded compon... | Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the spectra of $x$ in $A$ and $B$ respectively.
From Spectrum of Element of Banach Algebra is Bounded, we have:
:$\map {\sigma_A} x$ and $\map {\sigma_B} x$ are bounded.
Hence, from Complement of Bounded Set in Complex Plane has at most One Unbounded Component:
:$\map... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $B$ be a [[Definition:Closed Set|closed]] [[Definition:Unital Subalgebra|unital subalgebra]] of $A$.
Let $x \in B$.
Let $\map {\rho_A} x$ and $\map {\rho_B} x$ be the [[Definition:Resolvent Set/Uni... | Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectra]] of $x$ in $A$ and $B$ respectively.
From [[Spectrum of Element of Banach Algebra is Bounded]], we have:
:$\map {\sigma_A} x$ and $\map {\sigma_B} x$ are [[Definition:Bounded Subset of Complex Plane|... | Resolvent Set of Element in Banach Algebra has same Unique Unbounded Component as Resolvent Set of Element in Closed Subalgebra | https://proofwiki.org/wiki/Resolvent_Set_of_Element_in_Banach_Algebra_has_same_Unique_Unbounded_Component_as_Resolvent_Set_of_Element_in_Closed_Subalgebra | https://proofwiki.org/wiki/Resolvent_Set_of_Element_in_Banach_Algebra_has_same_Unique_Unbounded_Component_as_Resolvent_Set_of_Element_in_Closed_Subalgebra | [
"Resolvent Sets",
"Components (Topology)",
"Unital Banach Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:Closed Set",
"Definition:Unital Subalgebra",
"Definition:Resolvent Set/Unital Algebra",
"Definition:Bounded Metric Space/Complex/Unbounded",
"Definition:Component (Topology)",
"Definition:Component (Topology)"
] | [
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Spectrum of Element of Banach Algebra is Bounded",
"Definition:Bounded Metric Space/Complex",
"Complement of Bounded Set in Complex Plane has at most One Unbounded Component",
"Definition:Bounded Metric Space/Complex/Unbounded",
"Definition:Componen... |
proofwiki-21940 | Spectral Permanence Theorem | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.
Let $B$ be a closed unital subalgebra of $A$.
Let $x \in B$.
Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the spectra of $x$ in $A$ and $B$ respectively.
Let $\map {\rho_A} x$ and $\map {\rho_B} x$ be the resolvent sets of $x$ in $A$ a... | Let $\family {V_\alpha}_{\alpha \mathop \in J}$ be the set of bounded components of $\map {\rho_A} x$.
From Component of Resolvent Set of Element in Unital Banach Algebra is Disjoint from or Component of Resolvent Set in Closed Subalgebra, for each $\alpha \in J$ we either have:
:$V_\alpha \subseteq \map {\sigma_B} x$
... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $B$ be a [[Definition:Closed Set|closed]] [[Definition:Unital Subalgebra|unital subalgebra]] of $A$.
Let $x \in B$.
Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the [[Definition:Spectrum (Spe... | Let $\family {V_\alpha}_{\alpha \mathop \in J}$ be the [[Definition:Set|set]] of [[Definition:Bounded Subset of Complex Plane|bounded]] [[Definition:Component (Topology)|components]] of $\map {\rho_A} x$.
From [[Component of Resolvent Set of Element in Unital Banach Algebra is Disjoint from or Component of Resolvent S... | Spectral Permanence Theorem | https://proofwiki.org/wiki/Spectral_Permanence_Theorem | https://proofwiki.org/wiki/Spectral_Permanence_Theorem | [
"Spectra (Spectral Theory)",
"Unital Banach Algebras",
"Named Theorems"
] | [
"Definition:Unital Banach Algebra",
"Definition:Closed Set",
"Definition:Unital Subalgebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Resolvent Set",
"Definition:Indexing Set/Family",
"Definition:Bounded Metric Space/Complex",
"Definition:Component (Topology)",
"Definition... | [
"Definition:Set",
"Definition:Bounded Metric Space/Complex",
"Definition:Component (Topology)",
"Component of Resolvent Set of Element in Unital Banach Algebra is Disjoint from or Component of Resolvent Set in Closed Subalgebra",
"Definition:Subset",
"Spectrum of Element in Unital Subalgebra",
"Resolven... |
proofwiki-21941 | Finite Set Formed by Substitution has Larger Intersection | :$\card{R \cap S} = \card{T \cap S} + 1$ | We have:
{{begin-eqn}}
{{eqn | l = \card{R \cap S}
| r = \card{\paren{\paren{T \setminus \set y} \cup \set x} \cap S}
}}
{{eqn | r = \card{\paren{\paren{T \setminus \set y} \cap S } \cup \paren{ \set x \cap S} }
| c = Intersection Distributes over Union
}}
{{eqn | r = \card{\paren{\paren{T \setminus \set y}... | :$\card{R \cap S} = \card{T \cap S} + 1$ | We have:
{{begin-eqn}}
{{eqn | l = \card{R \cap S}
| r = \card{\paren{\paren{T \setminus \set y} \cup \set x} \cap S}
}}
{{eqn | r = \card{\paren{\paren{T \setminus \set y} \cap S } \cup \paren{ \set x \cap S} }
| c = [[Intersection Distributes over Union]]
}}
{{eqn | r = \card{\paren{\paren{T \setminus \se... | Finite Set Formed by Substitution has Larger Intersection | https://proofwiki.org/wiki/Finite_Set_Formed_by_Substitution_has_Larger_Intersection | https://proofwiki.org/wiki/Finite_Set_Formed_by_Substitution_has_Larger_Intersection | [
"Finite Sets",
"Set Intersection"
] | [] | [
"Intersection Distributes over Union",
"Set Intersection Distributes over Set Difference",
"Set Difference with Empty Set is Self",
"Cardinality of Set Union/Corollary",
"Cardinality of Singleton",
"Category:Finite Sets",
"Category:Set Intersection"
] |
proofwiki-21942 | Matrix Scalar Product Distributes over Matrix Entrywise Addition | Let $\mathbf A$ and $\mathbf B$ be matrices both of order $m \times n$.
Let $k$ be a scalar.
Then:
:$k \paren {\mathbf A + \mathbf B} = k \mathbf A + k \mathbf B$
where:
:$+$ denotes matrix entrywise addition
:$k \mathbf A$ etc. denotes matrix scalar product. | <onlyinclude>
Let $a_{i j}$ and $b_{i j}$ denote the $\tuple {i, j}$th entry in $\mathbf A$ and $\mathbf B$ respectively.
{{begin-eqn}}
{{eqn | o =
| r = k \mathbf A + k \mathbf B
| c =
}}
{{eqn | q = \forall i \in \closedint 1 m, \forall j \in \closedint 1 n
| r = k a_{i j} + k b_{i j}
| c = ... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Matrix|matrices]] both of [[Definition:Order of Matrix|order]] $m \times n$.
Let $k$ be a [[Definition:Scalar of Matrix Scalar Product|scalar]].
Then:
:$k \paren {\mathbf A + \mathbf B} = k \mathbf A + k \mathbf B$
where:
:$+$ denotes [[Definition:Matrix Entrywise Addit... | <onlyinclude>
Let $a_{i j}$ and $b_{i j}$ denote the $\tuple {i, j}$th [[Definition:Entry of Matrix|entry]] in $\mathbf A$ and $\mathbf B$ respectively.
{{begin-eqn}}
{{eqn | o =
| r = k \mathbf A + k \mathbf B
| c =
}}
{{eqn | q = \forall i \in \closedint 1 m, \forall j \in \closedint 1 n
| r = k ... | Matrix Scalar Product Distributes over Matrix Entrywise Addition | https://proofwiki.org/wiki/Matrix_Scalar_Product_Distributes_over_Matrix_Entrywise_Addition | https://proofwiki.org/wiki/Matrix_Scalar_Product_Distributes_over_Matrix_Entrywise_Addition | [
"Matrix Scalar Product",
"Matrix Entrywise Addition",
"Examples of Distributive Operations"
] | [
"Definition:Matrix",
"Definition:Matrix/Order",
"Definition:Matrix Scalar Product/Scalar",
"Definition:Matrix Entrywise Addition",
"Definition:Matrix Scalar Product"
] | [
"Definition:Matrix/Element",
"Distributive Laws/Arithmetic"
] |
proofwiki-21943 | Finite Set Formed by Substitution has Same Cardinality | :$\card R = \card T$ | We have:
{{begin-eqn}}
{{eqn | l = \card R
| r = \card{\paren{T \setminus \set y} \cup \set x}
}}
{{eqn | r = \card{T \setminus \set y} + \card{\set x}
| c = Cardinality of Pairwise Disjoint Set Union
}}
{{eqn | r = \paren{\card T - \card{\set y} } + \card{\set x}
| c = Cardinality of Set Difference w... | :$\card R = \card T$ | We have:
{{begin-eqn}}
{{eqn | l = \card R
| r = \card{\paren{T \setminus \set y} \cup \set x}
}}
{{eqn | r = \card{T \setminus \set y} + \card{\set x}
| c = [[Cardinality of Pairwise Disjoint Set Union]]
}}
{{eqn | r = \paren{\card T - \card{\set y} } + \card{\set x}
| c = [[Cardinality of Set Differ... | Finite Set Formed by Substitution has Same Cardinality | https://proofwiki.org/wiki/Finite_Set_Formed_by_Substitution_has_Same_Cardinality | https://proofwiki.org/wiki/Finite_Set_Formed_by_Substitution_has_Same_Cardinality | [
"Finite Sets",
"Cardinality"
] | [] | [
"Cardinality of Set Union/Corollary",
"Cardinality of Set Difference with Subset",
"Cardinality of Singleton",
"Category:Finite Sets",
"Category:Cardinality"
] |
proofwiki-21944 | Completion Theorem (Normed Algebra) | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$.
Then there exists a Banach algebra $\struct {\widetilde A, \widetilde {\norm {\, \cdot \,} } }$ and a isometric algebra homomorphism $\phi : A \to \widetilde A$ such that $\phi \sqbrk A$ is dense in $\widetilde A$.
Thi... | === Proof of Existence ===
Let $\struct {A^{\ast \ast}, \norm {\, \cdot \,}_{A^{\ast \ast} } }$ be the second normed dual of $\struct {A, \norm {\, \cdot \,} }$.
From Normed Dual Space is Banach Space, $\struct {A^{\ast \ast}, \norm {\, \cdot \,}_{A^{\ast \ast} } }$ is a Banach space.
Let $\phi : A \to A^{\ast \ast}$ ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $\GF$.
Then there exists a [[Definition:Banach Algebra|Banach algebra]] $\struct {\widetilde A, \widetilde {\norm {\, \cdot \,} } }$ and a [[Definition:Isometric Isomorphism|isometric]] [[Defi... | === Proof of Existence ===
Let $\struct {A^{\ast \ast}, \norm {\, \cdot \,}_{A^{\ast \ast} } }$ be the [[Definition:Second Normed Dual|second normed dual]] of $\struct {A, \norm {\, \cdot \,} }$.
From [[Normed Dual Space is Banach Space]], $\struct {A^{\ast \ast}, \norm {\, \cdot \,}_{A^{\ast \ast} } }$ is a [[Defini... | Completion Theorem (Normed Algebra) | https://proofwiki.org/wiki/Completion_Theorem_(Normed_Algebra) | https://proofwiki.org/wiki/Completion_Theorem_(Normed_Algebra) | [
"Normed Algebras",
"Completion Theorem"
] | [
"Definition:Normed Algebra",
"Definition:Banach Algebra",
"Definition:Isometric Isomorphism",
"Definition:Algebra Homomorphism",
"Definition:Everywhere Dense",
"Definition:Banach Algebra",
"Definition:Isometric Isomorphism",
"Definition:Algebra Homomorphism",
"Definition:Unital Normed Algebra"
] | [
"Definition:Second Normed Dual",
"Normed Dual Space is Banach Space",
"Definition:Banach Space",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Closure (Topology)",
"Completion Theorem (Normed Vector Space)",
"Definition:Linear Isometry",
"Definition:Everywhere Dense",
... |
proofwiki-21945 | Spectrum of Element of Banach Algebra is Non-Empty/Corollary | Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$.
Then $\map {\sigma_A} x \ne \O$. | Suppose first that $\struct {A, \norm {\, \cdot \,} }$ is unital.
From Completion Theorem (Normed Algebra), there exists a unital Banach algebra $\struct {\widetilde A, \widetilde {\norm {\, \cdot \,} } }$ and an isometric algebra homomorphism $\phi : A \to \widetilde A$ where:
:${\mathbf 1}_{\widetilde A} = \map \phi... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $x$ in $A$.
Then $\map {\sigma_A} x \ne \O$. | Suppose first that $\struct {A, \norm {\, \cdot \,} }$ is [[Definition:Unital Normed Algebra|unital]].
From [[Completion Theorem (Normed Algebra)]], there exists a [[Definition:Unital Banach Algebra|unital Banach algebra]] $\struct {\widetilde A, \widetilde {\norm {\, \cdot \,} } }$ and an [[Definition:Isometric Isom... | Spectrum of Element of Banach Algebra is Non-Empty/Corollary | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Non-Empty/Corollary | https://proofwiki.org/wiki/Spectrum_of_Element_of_Banach_Algebra_is_Non-Empty/Corollary | [
"Spectrum of Element of Banach Algebra is Non-Empty"
] | [
"Definition:Normed Algebra",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Unital Normed Algebra",
"Completion Theorem (Normed Algebra)",
"Definition:Unital Banach Algebra",
"Definition:Isometric Isomorphism",
"Definition:Algebra Homomorphism",
"Spectrum of Element of Banach Algebra is Non-Empty",
"Spectrum of Element in Unital Subalgebra",
"Definition:Group of U... |
proofwiki-21946 | Gelfand-Mazur Theorem | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital normed algebra over $\C$ where:
:$\map G A = A \setminus \set { {\mathbf 0}_A}$
where $\map G A$ is the group of units of $A$.
Then $A$ is isometrically algebra isomorphic to $\C$. | Define $\theta : \C \to A$ by:
:$\map \theta \lambda = \lambda {\mathbf 1}_A$
for each $\lambda \in \C$.
For $z, w, \lambda \in \C$ we have:
{{begin-eqn}}
{{eqn | l = \map \theta {z + \lambda w}
| r = \paren {z + \lambda w} {\mathbf 1}_A
}}
{{eqn | r = z {\mathbf 1}_A + \lambda \paren {w {\mathbf 1}_A}
}}
{{eqn | r ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Normed Algebra|unital normed algebra]] over $\C$ where:
:$\map G A = A \setminus \set { {\mathbf 0}_A}$
where $\map G A$ is the [[Definition:Group of Units|group of units]] of $A$.
Then $A$ is [[Definition:Isometry|isometrically]] [[Definition:Algebra I... | Define $\theta : \C \to A$ by:
:$\map \theta \lambda = \lambda {\mathbf 1}_A$
for each $\lambda \in \C$.
For $z, w, \lambda \in \C$ we have:
{{begin-eqn}}
{{eqn | l = \map \theta {z + \lambda w}
| r = \paren {z + \lambda w} {\mathbf 1}_A
}}
{{eqn | r = z {\mathbf 1}_A + \lambda \paren {w {\mathbf 1}_A}
}}
{{eqn | r... | Gelfand-Mazur Theorem | https://proofwiki.org/wiki/Gelfand-Mazur_Theorem | https://proofwiki.org/wiki/Gelfand-Mazur_Theorem | [
"Normed Algebras",
"Algebra Isomorphisms"
] | [
"Definition:Unital Normed Algebra",
"Definition:Group of Units",
"Definition:Isometry",
"Definition:Algebra Isomorphism"
] | [
"Definition:Linear Transformation",
"Definition:Algebra Homomorphism",
"Definition:Linear Isometry",
"Definition:Surjection",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Surjection",
"Definition:Isometric Isomorphism",
"Definition:Isometric Isomorphism",
"Category:Normed Alge... |
proofwiki-21947 | Open Subset of Locally Compact Hausdorff Space is Locally Compact Hausdorff Space | Let $\struct {X, \tau_X}$ be a locally compact Hausdorff space.
Let $Y \subseteq X$ be open.
Let $\tau_Y$ be the subspace topology on $Y$ inherited from $\struct {X, \tau_X}$.
Then $\struct {Y, \tau_Y}$ is a locally compact Hausdorff space. | From $T_2$ Property is Hereditary, $\struct {Y, \tau_Y}$ is Hausdorff.
Let $y \in Y$.
Let $\CC$ be a neighborhood basis of $y$ in $\struct {X, \tau_X}$ consisting of compact sets.
Let $U$ be a neighborhood of $y$ in $\struct {Y, \tau_Y}$.
Then there exists an open neighborhood $V$ of $y$ in $\struct {Y, \tau_Y}$ such... | Let $\struct {X, \tau_X}$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $Y \subseteq X$ be [[Definition:Open Set|open]].
Let $\tau_Y$ be the [[Definition:Subspace|subspace topology]] on $Y$ inherited from $\struct {X, \tau_X}$.
Then $\struct {Y, \tau_Y}$ is a [[Definition:... | From [[T2 Property is Hereditary|$T_2$ Property is Hereditary]], $\struct {Y, \tau_Y}$ is [[Definition:Hausdorff Space|Hausdorff]].
Let $y \in Y$.
Let $\CC$ be a [[Definition:Neighborhood Basis|neighborhood basis]] of $y$ in $\struct {X, \tau_X}$ consisting of [[Definition:Compact Topological Space|compact sets]].
... | Open Subset of Locally Compact Hausdorff Space is Locally Compact Hausdorff Space | https://proofwiki.org/wiki/Open_Subset_of_Locally_Compact_Hausdorff_Space_is_Locally_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Open_Subset_of_Locally_Compact_Hausdorff_Space_is_Locally_Compact_Hausdorff_Space | [
"Locally Compact Hausdorff Spaces"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Open Set",
"Definition:Subspace",
"Definition:Locally Compact Hausdorff Space"
] | [
"T2 Property is Hereditary",
"Definition:T2 Space",
"Definition:Neighborhood Basis",
"Definition:Compact Topological Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Neighborhood",
"Open Set in Open Subspace",
"Definition:Open Set",
"Definition:Neighborhood Basis",
"Open Set in... |
proofwiki-21948 | Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $\struct {\Phi_A, w^\ast}$ be the spectrum of $A$.
Then $\struct {\Phi_A, w^\ast}$ is a locally compact Hausdorff space. | From Weak-* Topology is Hausdorff, $\struct {\Phi_A, w^\ast}$ is a Hausdorff space.
By the definition of a character, we have:
:$\Phi_A = \set {\phi \in A^\ast : \phi \ne {\mathbf 0}_{A^\ast}, \, \map \phi x \map \phi y = \map \phi {x y} \text { for all } x, y \in A}$
hence:
:$\Phi_A \cup \set { {\mathbf 0}_{A^\ast} } ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $\struct {\Phi_A, w^\ast}$ be the [[Definition:Spectrum of Banach Algebra|spectrum]] of $A$.
Then $\struct {\Phi_A, w^\ast}$ is a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]]. | From [[Weak-* Topology is Hausdorff]], $\struct {\Phi_A, w^\ast}$ is a [[Definition:Hausdorff Space|Hausdorff space]].
By the definition of a [[Definition:Character (Banach Algebra)|character]], we have:
:$\Phi_A = \set {\phi \in A^\ast : \phi \ne {\mathbf 0}_{A^\ast}, \, \map \phi x \map \phi y = \map \phi {x y} \tex... | Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space | https://proofwiki.org/wiki/Spectrum_of_Banach_Algebra_is_Weak-*_Locally_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Spectrum_of_Banach_Algebra_is_Weak-*_Locally_Compact_Hausdorff_Space | [
"Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space",
"Locally Compact Hausdorff Spaces",
"Weak-* Topologies",
"Spectra (Banach Algebras)"
] | [
"Definition:Banach Algebra",
"Definition:Spectrum of Banach Algebra",
"Definition:Locally Compact Hausdorff Space"
] | [
"Weak-* Topology is Hausdorff",
"Definition:T2 Space",
"Definition:Character (Banach Algebra)",
"Character on Banach Algebra is Continuous",
"Definition:Closed Unit Ball",
"Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual",
"Characterizati... |
proofwiki-21949 | Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space/Corollary | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$. | Let $\phi \in \Phi_A$.
From Character on Unital Banach Algebra is Unital Algebra Homomorphism, we have that:
:the condition that $\map \phi x \map \phi y = \map \phi {x y}$ for all $x, y \in A$ implies that $\map \phi { {\mathbf 1}_A} = 1$.
That is, $\map \phi x \map \phi y = \map \phi {x y}$ for all $x, y \in A$ impl... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$. | Let $\phi \in \Phi_A$.
From [[Character on Unital Banach Algebra is Unital Algebra Homomorphism]], we have that:
:the condition that $\map \phi x \map \phi y = \map \phi {x y}$ for all $x, y \in A$ implies that $\map \phi { {\mathbf 1}_A} = 1$.
That is, $\map \phi x \map \phi y = \map \phi {x y}$ for all $x, y \in A... | Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space/Corollary | https://proofwiki.org/wiki/Spectrum_of_Banach_Algebra_is_Weak-*_Locally_Compact_Hausdorff_Space/Corollary | https://proofwiki.org/wiki/Spectrum_of_Banach_Algebra_is_Weak-*_Locally_Compact_Hausdorff_Space/Corollary | [
"Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space"
] | [
"Definition:Unital Banach Algebra"
] | [
"Character on Unital Banach Algebra is Unital Algebra Homomorphism",
"Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space",
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Category:Spectrum of Banach Algebra ... |
proofwiki-21950 | Continuous Complex-Valued Function Vanishing at Infinity is Bounded and Attains Supremum | Let $X$ be a locally compact Hausdorff space.
Let $f : X \to \C$ be a continuous function that vanishes at infinity.
Then $f$ is bounded and there exists $x_\ast \in X$ such that:
:$\ds \map f {x_\ast} = \sup_{x \mathop \in X} \cmod {\map f x}$ | From the definition of a function vanishing at infinity:
:there exists a compact set $F \subseteq X$ such that:
::$\cmod {\map f x} < 1$ for all $x \in X \setminus F$.
From Continuous Function on Compact Space is Bounded:
:$f$ is bounded on $F$.
That is, there exists a real number $M > 0$ such that:
:$\cmod {\map f x}... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $f : X \to \C$ be a [[Definition:Continuous Function|continuous function]] that [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]].
Then $f$ is [[Definition:Bounded Complex-Valued Function... | From the definition of a [[Definition:Complex-Valued Function Vanishing at Infinity|function vanishing at infinity]]:
:there exists a [[Definition:Compact Topological Space|compact set]] $F \subseteq X$ such that:
::$\cmod {\map f x} < 1$ for all $x \in X \setminus F$.
From [[Continuous Function on Compact Space is Bo... | Continuous Complex-Valued Function Vanishing at Infinity is Bounded and Attains Supremum | https://proofwiki.org/wiki/Continuous_Complex-Valued_Function_Vanishing_at_Infinity_is_Bounded_and_Attains_Supremum | https://proofwiki.org/wiki/Continuous_Complex-Valued_Function_Vanishing_at_Infinity_is_Bounded_and_Attains_Supremum | [
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Continuous Function",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Bounded Mapping/Complex-Valued"
] | [
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space",
"Continuous Function on Compact Space is Bounded",
"Definition:Bounded Mapping/Complex-Valued",
"Definition:Real Number",
"Definition:Bounded Mapping/Complex-Valued",
"Definition:Bounded Mapping/Real-Valu... |
proofwiki-21951 | Independence System Induced from Set of Subsets | Let $S$ be a finite set.
Let $\mathscr A$ be a non-empty set of subsets of $S$.
Let $\mathscr I = \set {X \subseteq S : \exists A \in \mathscr A : X \subseteq A}$.
Then $\struct {S, \mathscr I}$ is an independence system. | It is shown that $\mathscr I$ satisfies the independence system axioms:
{{:Axiom:Independence System Axioms}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Let $\mathscr I = \set {X \subseteq S : \exists A \in \mathscr A : X \subseteq A}$.
Then $\struct {S, \mathscr I}$ is an [[Definition:Indepe... | It is shown that $\mathscr I$ satisfies the [[Axiom:Independence System Axioms|independence system axioms]]:
{{:Axiom:Independence System Axioms}} | Independence System Induced from Set of Subsets | https://proofwiki.org/wiki/Independence_System_Induced_from_Set_of_Subsets | https://proofwiki.org/wiki/Independence_System_Induced_from_Set_of_Subsets | [
"Independence Systems",
"Subsets"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Definition:Independence System"
] | [
"Axiom:Independence System Axioms",
"Axiom:Independence System Axioms",
"Axiom:Independence System Axioms",
"Axiom:Independence System Axioms"
] |
proofwiki-21952 | Complex-Valued Function Vanishing in Neighborhood of Infinity Vanishes at Infinity | Let $X$ be a locally compact Hausdorff space.
Let $f : X \to \C$ be a function vanishing in a neighborhood of infinity.
Then $f$ vanishes at infinity. | Let $\epsilon > 0$.
Since $f$ vanishes in a neighborhood of infinity, there exists a compact $F \subseteq X$ such that:
:$\map f x = 0$ for $x \in X \setminus F$
In particular:
:$\cmod {\map f x} = 0 < \epsilon$ for each $x \in X \setminus F$.
Hence $f$ vanishes at infinity.
{{qed}}
Category:Complex-Valued Functions ... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $f : X \to \C$ be a [[Definition:Function|function]] [[Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity|vanishing in a neighborhood of infinity]].
Then $f$ [[Definition:Complex-Valued Function V... | Let $\epsilon > 0$.
Since $f$ [[Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity|vanishes in a neighborhood of infinity]], there exists a [[Definition:Compact Topological Space|compact]] $F \subseteq X$ such that:
:$\map f x = 0$ for $x \in X \setminus F$
In particular:
:$\cmod {\map f x} = 0... | Complex-Valued Function Vanishing in Neighborhood of Infinity Vanishes at Infinity | https://proofwiki.org/wiki/Complex-Valued_Function_Vanishing_in_Neighborhood_of_Infinity_Vanishes_at_Infinity | https://proofwiki.org/wiki/Complex-Valued_Function_Vanishing_in_Neighborhood_of_Infinity_Vanishes_at_Infinity | [
"Complex-Valued Functions Vanishing at Infinity",
"Complex-Valued Functions Vanishing in Neighborhood of Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Function",
"Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity",
"Definition:Compact Topological Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Category:Complex-Valued Functions Vanishing at Infinity",
"Category:Complex-Valued Functions Vanishing in Neighborhood of Infinity"... |
proofwiki-21953 | Zero Function Vanishes in Neighborhood of Infinity | Let $X$ be a locally compact Hausdorff space.
Define a function $f : X \to \C$ by:
:$\map f x = 0$
for each $x \in X$.
Then $f$ vanishes in a neighborhood of infinity. | From Empty Set is Compact:
:$\O \subseteq X$ is compact.
We then have:
:$\map f x = 0$ for all $x \in X \setminus \O$
Hence $f$ vanishes in a neighborhood of infinity.
{{qed}}
Category:Complex-Valued Functions Vanishing in Neighborhood of Infinity
mcdzq8whisimmco3vn1uhq0cj7i3api | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Define a [[Definition:Function|function]] $f : X \to \C$ by:
:$\map f x = 0$
for each $x \in X$.
Then $f$ [[Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity|vanishes in a neighborhood of infinity]]. | From [[Empty Set is Compact]]:
:$\O \subseteq X$ is [[Definition:Compact Topological Space|compact]].
We then have:
:$\map f x = 0$ for all $x \in X \setminus \O$
Hence $f$ [[Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity|vanishes in a neighborhood of infinity]].
{{qed}}
[[Category:Complex... | Zero Function Vanishes in Neighborhood of Infinity | https://proofwiki.org/wiki/Zero_Function_Vanishes_in_Neighborhood_of_Infinity | https://proofwiki.org/wiki/Zero_Function_Vanishes_in_Neighborhood_of_Infinity | [
"Complex-Valued Functions Vanishing in Neighborhood of Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Function",
"Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity"
] | [
"Empty Set is Compact",
"Definition:Compact Topological Space",
"Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity",
"Category:Complex-Valued Functions Vanishing in Neighborhood of Infinity"
] |
proofwiki-21954 | Linear Combination of Complex-Valued Functions Vanishing at Infinity Vanishes at Infinity | Let $X$ be a locally compact Hausdorff space.
Let $f, g : X \to \C$ be complex-valued functions vanishing at infinity.
Let $\lambda \in \C$.
Then $f + \lambda g$ vanishes at infinity. | If $\lambda = 0$, we are done immediately.
Take $\lambda \ne 0$.
Let $\epsilon > 0$.
Since $f$ vanishes at infinity, there exists a compact set $F_1 \subseteq X$ such that:
:$\cmod {\map f x} < \dfrac \epsilon 2$ for each $x \in X \setminus F_1$.
Since $g$ vanishes at infinity, there exists a compact set $F_2 \subse... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $f, g : X \to \C$ be [[Definition:Complex-Valued Function Vanishing at Infinity|complex-valued functions vanishing at infinity]].
Let $\lambda \in \C$.
Then $f + \lambda g$ [[Definition:Complex-Valued Function Vanishin... | If $\lambda = 0$, we are done immediately.
Take $\lambda \ne 0$.
Let $\epsilon > 0$.
Since $f$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]], there exists a [[Definition:Compact Topological Space|compact set]] $F_1 \subseteq X$ such that:
:$\cmod {\map f x} < \dfrac \epsilon 2$... | Linear Combination of Complex-Valued Functions Vanishing at Infinity Vanishes at Infinity | https://proofwiki.org/wiki/Linear_Combination_of_Complex-Valued_Functions_Vanishing_at_Infinity_Vanishes_at_Infinity | https://proofwiki.org/wiki/Linear_Combination_of_Complex-Valued_Functions_Vanishing_at_Infinity_Vanishes_at_Infinity | [
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space",
"Union of Two Compact Sets is Compact",
"Definition:Compact Topological Space",
"Triangle Inequality/Com... |
proofwiki-21955 | Uniform Limit of Sequence of Continuous Complex-Valued Functions Vanishing at Infinity Vanishes at Infinity | Let $X$ be a locally compact Hausdorff space.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of functions $f_n : X \to \C$ that vanishes at infinity.
Let $f : X \to \C$ be a function such that:
:$f_n - f$ is bounded for each $n \in \N$
and:
:$\ds \sup_{x \mathop \in X} \cmod {\map {f_n} x - \map f x} \to 0$ as... | Let $\epsilon > 0$.
Since:
:$\ds \sup_{x \mathop \in X} \cmod {\map {f_n} x - \map f x} \to 0$ as $n \to \infty$
we can take $N \in \N$ such that:
:$\ds \sup_{x \mathop \in X} \cmod {\map {f_N} x - \map f x} < \frac \epsilon 2$
Further since $f_N$ vanishes at infinity, there exists a compact set $F \subseteq X$ such t... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Function|functions]] $f_n : X \to \C$ that [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]].
Let... | Let $\epsilon > 0$.
Since:
:$\ds \sup_{x \mathop \in X} \cmod {\map {f_n} x - \map f x} \to 0$ as $n \to \infty$
we can take $N \in \N$ such that:
:$\ds \sup_{x \mathop \in X} \cmod {\map {f_N} x - \map f x} < \frac \epsilon 2$
Further since $f_N$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes a... | Uniform Limit of Sequence of Continuous Complex-Valued Functions Vanishing at Infinity Vanishes at Infinity | https://proofwiki.org/wiki/Uniform_Limit_of_Sequence_of_Continuous_Complex-Valued_Functions_Vanishing_at_Infinity_Vanishes_at_Infinity | https://proofwiki.org/wiki/Uniform_Limit_of_Sequence_of_Continuous_Complex-Valued_Functions_Vanishing_at_Infinity_Vanishes_at_Infinity | [
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Sequence",
"Definition:Function",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Function",
"Definition:Bounded Mapping/Complex-Valued",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space",
"Triangle Inequality/Complex Numbers",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Category:Complex-Valued Functions Vanishing at Infinity"
] |
proofwiki-21956 | Matroid Bases Satisfy Formulation 4 Base Axiom | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\mathscr B$ be the set of bases of the matroid $M$.
Then $\mathscr B$ satisfies formulation $4$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 4}} | Let $B_1, B_2 \in \mathscr B$.
Let $x \in B_1 \setminus B_2$.
From Matroid Base Union External Element has Fundamental Circuit:
:there exists a fundamental circuit $\map C {x, B_2}$ of $M$ such that $x \in \map C {x, B_2} \subseteq B_2 \cup \set x$
By definition of set intersection:
:$x \in B_1 \cap \map C {x, B_2}$
Fr... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\mathscr B$ be the set of [[Definition:Base of Matroid|bases]] of the [[Definition:Matroid|matroid]] $M$.
Then $\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 4|formulation $4$ of base axiom]]:
{{:Axiom:Base Axiom (Matroid)/F... | Let $B_1, B_2 \in \mathscr B$.
Let $x \in B_1 \setminus B_2$.
From [[Matroid Base Union External Element has Fundamental Circuit]]:
:there exists a [[Definition:Fundamental Circuit (Matroid)|fundamental circuit]] $\map C {x, B_2}$ of $M$ such that $x \in \map C {x, B_2} \subseteq B_2 \cup \set x$
By definition of [... | Matroid Bases Satisfy Formulation 4 Base Axiom | https://proofwiki.org/wiki/Matroid_Bases_Satisfy_Formulation_4_Base_Axiom | https://proofwiki.org/wiki/Matroid_Bases_Satisfy_Formulation_4_Base_Axiom | [
"Matroid Bases"
] | [
"Definition:Matroid",
"Definition:Base of Matroid",
"Definition:Matroid",
"Axiom:Base Axiom (Matroid)/Formulation 4"
] | [
"Matroid Unique Circuit Property/Corollary",
"Definition:Fundamental Circuit (Matroid)",
"Definition:Set Intersection",
"Element of Matroid Base and Circuit has Substitute",
"Set Difference with Subset is Superset of Set Difference",
"Set Difference over Subset",
"Set Difference with Union is Set Differ... |
proofwiki-21957 | Product of Complex-Valued Functions Vanishing at Infinity Vanishes at Infinity | Let $X$ be a locally compact Hausdorff space.
Let $f, g : X \to \C$ be complex-valued functions vanishing at infinity.
Then $f g$ vanishes at infinity. | Let $\epsilon > 0$.
Since $f$ vanishes at infinity, there exists a compact set $F_1 \subseteq X$ such that:
:$\cmod {\map f x} < \sqrt \epsilon$ for each $x \in X \setminus F_1$.
Since $g$ vanishes at infinity, there exists a compact set $F_2 \subseteq X$ such that:
:$\cmod {\map g x} < \sqrt \epsilon$ for each $x \in... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $f, g : X \to \C$ be [[Definition:Complex-Valued Function Vanishing at Infinity|complex-valued functions vanishing at infinity]].
Then $f g$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinit... | Let $\epsilon > 0$.
Since $f$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]], there exists a [[Definition:Compact Topological Space|compact set]] $F_1 \subseteq X$ such that:
:$\cmod {\map f x} < \sqrt \epsilon$ for each $x \in X \setminus F_1$.
Since $g$ [[Definition:Complex-Value... | Product of Complex-Valued Functions Vanishing at Infinity Vanishes at Infinity | https://proofwiki.org/wiki/Product_of_Complex-Valued_Functions_Vanishing_at_Infinity_Vanishes_at_Infinity | https://proofwiki.org/wiki/Product_of_Complex-Valued_Functions_Vanishing_at_Infinity_Vanishes_at_Infinity | [
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space",
"Union of Two Compact Sets is Compact",
"Definition:Compact Topological Space",
"Definition:Compact Topo... |
proofwiki-21958 | Mean Deviation about Mean equals Zero | Let $S$ be an object upon which an arithmetic mean and a mean deviation $\map D S$ are both defined.
Let $\map D S$ be the mean deviation about the arithmetic mean $\bar x$ of $S$.
Then:
:$\map D S = 0$ | From Sum of Deviations from Mean:
:$\ds \sum_{i \mathop = 1}^n \paren {x_i - \bar x} = 0$
Therefore:
:$\ds \map D S = \frac 1 n \sum_{i \mathop = 1}^n \paren {x_i - \bar x} = 0$
{{qed}} | Let $S$ be an [[Definition:Object|object]] upon which an [[Definition:Arithmetic Mean|arithmetic mean]] and a [[Definition:Mean Deviation|mean deviation]] $\map D S$ are both defined.
Let $\map D S$ be the [[Definition:Mean Deviation|mean deviation]] about the [[Definition:Arithmetic Mean|arithmetic mean]] $\bar x$ of... | From [[Sum of Deviations from Mean]]:
:$\ds \sum_{i \mathop = 1}^n \paren {x_i - \bar x} = 0$
Therefore:
:$\ds \map D S = \frac 1 n \sum_{i \mathop = 1}^n \paren {x_i - \bar x} = 0$
{{qed}} | Mean Deviation about Mean equals Zero | https://proofwiki.org/wiki/Mean_Deviation_about_Mean_equals_Zero | https://proofwiki.org/wiki/Mean_Deviation_about_Mean_equals_Zero | [
"Mean Deviation",
"Arithmetic Mean"
] | [
"Definition:Object",
"Definition:Arithmetic Mean",
"Definition:Mean Deviation",
"Definition:Mean Deviation",
"Definition:Arithmetic Mean"
] | [
"Sum of Deviations from Mean"
] |
proofwiki-21959 | Mean Squared Error equals Variance for Unbiased Estimator | Let $T$ be an unbiased estimator for a population parameter of a population $P$.
Then the mean squared error for $T$ equals the variance of $T$. | Let $T$ be a general estimator.
From Mean Squared Error for Biased Estimator:
:$M = \var T + \paren {\map B T}^2$
where:
:$\var T$ denotes the variance of $T$
:$\map B T$ denotes the bias on $T$.
By definition, if $T$ is unbiased:
:$\map B T = 0$
The result follows.
{{qed}} | Let $T$ be an [[Definition:Unbiased Estimator|unbiased estimator]] for a [[Definition:Population Parameter|population parameter]] of a [[Definition:Population|population]] $P$.
Then the [[Definition:Mean Squared Error|mean squared error]] for $T$ equals the [[Definition:Variance|variance]] of $T$. | Let $T$ be a general [[Definition:Estimator|estimator]].
From [[Mean Squared Error for Biased Estimator]]:
:$M = \var T + \paren {\map B T}^2$
where:
:$\var T$ denotes the [[Definition:Variance|variance]] of $T$
:$\map B T$ denotes the [[Definition:Bias|bias]] on $T$.
By definition, if $T$ is [[Definition:Unbiased E... | Mean Squared Error equals Variance for Unbiased Estimator | https://proofwiki.org/wiki/Mean_Squared_Error_equals_Variance_for_Unbiased_Estimator | https://proofwiki.org/wiki/Mean_Squared_Error_equals_Variance_for_Unbiased_Estimator | [
"Mean Squared Error",
"Variance"
] | [
"Definition:Unbiased Estimator",
"Definition:Population Parameter",
"Definition:Population",
"Definition:Mean Squared Error",
"Definition:Variance"
] | [
"Definition:Estimator",
"Mean Squared Error for Biased Estimator",
"Definition:Variance",
"Definition:Bias",
"Definition:Unbiased Estimator"
] |
proofwiki-21960 | Variance is Least Mean Square Deviation about Point | Let $S$ be an object upon which:
:the mean square deviation
:the expectation
:the variance
is defined.
For each point $x$ in $S$, let $\map M x$ denote the mean square deviation of $S$ about $x$.
Then the minimum of $\map M x$ is the is the variance of $S$.
That is, the expectation $\bar x$ of $S$ is the value of $S$ f... | === Discrete Case ===
Let $\map f {x_i}$ be the probability mass function of $x_i \in S$.
Then:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \map M x
| r = 0
| c = Interior Extremum Theorem
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \sum_{i \mathop = 1}^n \paren {x_i - x}^2 \map f {x_i}
| r = 0
... | Let $S$ be an [[Definition:Object|object]] upon which:
:the [[Definition:Mean Square Deviation|mean square deviation]]
:the [[Definition:Expectation|expectation]]
:the [[Definition:Variance|variance]]
is defined.
For each [[Definition:Element|point]] $x$ in $S$, let $\map M x$ denote the [[Definition:Mean Square Devia... | === Discrete Case ===
Let $\map f {x_i}$ be the [[Definition:Probability Mass Function|probability mass function]] of $x_i \in S$.
Then:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \map M x
| r = 0
| c = [[Interior Extremum Theorem]]
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \sum_{i \mathop = 1}^... | Variance is Least Mean Square Deviation about Point | https://proofwiki.org/wiki/Variance_is_Least_Mean_Square_Deviation_about_Point | https://proofwiki.org/wiki/Variance_is_Least_Mean_Square_Deviation_about_Point | [
"Mean Square Deviation",
"Variance"
] | [
"Definition:Object",
"Definition:Mean Square Deviation",
"Definition:Expectation",
"Definition:Variance",
"Definition:Element",
"Definition:Mean Square Deviation",
"Definition:Minimum Value of Real Function",
"Definition:Variance",
"Definition:Expectation",
"Definition:Mean Square Deviation"
] | [
"Definition:Probability Mass Function",
"Interior Extremum Theorem",
"Power Rule for Derivatives",
"Definition:Division",
"Summation of Sum of Mappings on Finite Set",
"Definition:Constant",
"Definition:Variance",
"Definition:Variance",
"Power Rule for Derivatives",
"Definition:Division",
"Defin... |
proofwiki-21961 | Dual Matroid is Matroid | Let $M = \struct {S, \mathscr I}$ be a matroid.
Then the dual $M^*$ of $M$ is a matroid. | Let $\mathscr B$ be the set of bases of the matroid $M$.
From Matroid Bases Satisfy Formulation 4 Base Axiom:
:$\mathscr B$ satisfies formulation 4 base axiom.
By formulation 5 base axiom:
:$\mathscr B$ satisfies formulation 5 base axiom.
Let $\mathscr B^* = \set{S \setminus B : B \in \mathscr B}$.
From Subsets Satisfy... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Then the [[Definition:Dual Matroid|dual]] $M^*$ of $M$ is a [[Definition:Matroid|matroid]]. | Let $\mathscr B$ be the set of [[Definition:Base of Matroid|bases]] of the [[Definition:Matroid|matroid]] $M$.
From [[Matroid Bases Satisfy Formulation 4 Base Axiom]]:
:$\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 4|formulation 4 base axiom]].
By [[Axiom:Base Axiom (Matroid)/Formulation 5|formulat... | Dual Matroid is Matroid | https://proofwiki.org/wiki/Dual_Matroid_is_Matroid | https://proofwiki.org/wiki/Dual_Matroid_is_Matroid | [
"Dual Matroids"
] | [
"Definition:Matroid",
"Definition:Dual Matroid",
"Definition:Matroid"
] | [
"Definition:Base of Matroid",
"Definition:Matroid",
"Matroid Bases Satisfy Formulation 4 Base Axiom",
"Axiom:Base Axiom (Matroid)/Formulation 4",
"Axiom:Base Axiom (Matroid)/Formulation 5",
"Axiom:Base Axiom (Matroid)/Formulation 5",
"Subsets Satisfy Formulation 5 Matroid Base Axiom Iff Complements Sati... |
proofwiki-21962 | Kasteleyn's Formula | The number of perfect covers of a chessboard of dimensions $m \times n$ is given by the formula:
:$\ds \prod_{j \mathop = 1}^{\ceiling {\frac m 2} } \prod_{k \mathop = 1}^{\ceiling {\frac n 2} } \paren {4 \cos^2 \frac {\pi j} {m + 1} + 4 \cos^2 \frac {\pi k} {n + 1} }$ | {{ProofWanted|Import proof from Kasteleyns paper}}
{{Namedfor|Pieter Willem Kasteleyn|cat = Kasteleyn}} | The number of [[Definition:Perfect Cover of Chessboard|perfect covers]] of a [[Definition:Chessboard|chessboard]] of dimensions $m \times n$ is given by the formula:
:$\ds \prod_{j \mathop = 1}^{\ceiling {\frac m 2} } \prod_{k \mathop = 1}^{\ceiling {\frac n 2} } \paren {4 \cos^2 \frac {\pi j} {m + 1} + 4 \cos^2 \frac... | {{ProofWanted|Import proof from Kasteleyns paper}}
{{Namedfor|Pieter Willem Kasteleyn|cat = Kasteleyn}} | Kasteleyn's Formula | https://proofwiki.org/wiki/Kasteleyn's_Formula | https://proofwiki.org/wiki/Kasteleyn's_Formula | [
"Chessboard Tilings",
"Chessboard Puzzles",
"Dominoes"
] | [
"Definition:Chessboard Tiling",
"Definition:Chess/Chessboard"
] | [] |
proofwiki-21963 | Perpendicular Bisector is Unique | Let $AB$ be a straight line segment in a Euclidean geometry.
Let $\LL$ be the perpendicular bisector of $AB$.
Then $\LL$ is unique. | Let $\LL_2$ be another perpendicular bisector of $AB$.
Since both $\LL$ and $\LL_2$ are perpendicular to $AB$, they are parallel by Equal Corresponding Angles implies Parallel Lines.
Since both $\LL$ and $\LL_2$ bisect $AB$, they intersect at the midpoint of $AB$.
Therefore, the distance between $\LL$ and $\LL_2$, is z... | Let $AB$ be a [[Definition:Straight Line Segment|straight line segment]] in a [[Definition:Euclidean Geometry|Euclidean geometry]].
Let $\LL$ be the [[Definition:Perpendicular Bisector|perpendicular bisector]] of $AB$.
Then $\LL$ is [[Definition:Unique|unique]]. | Let $\LL_2$ be another [[Definition:Perpendicular Bisector|perpendicular bisector]] of $AB$.
Since both $\LL$ and $\LL_2$ are [[Definition:Perpendicular|perpendicular]] to $AB$, they are [[Definition:Parallel Lines|parallel]] by [[Equal Corresponding Angles implies Parallel Lines]].
Since both $\LL$ and $\LL_2$ [[Def... | Perpendicular Bisector is Unique | https://proofwiki.org/wiki/Perpendicular_Bisector_is_Unique | https://proofwiki.org/wiki/Perpendicular_Bisector_is_Unique | [
"Perpendicular Bisectors"
] | [
"Definition:Line/Straight Line Segment",
"Definition:Euclidean Geometry",
"Definition:Perpendicular Bisector",
"Definition:Unique"
] | [
"Definition:Perpendicular Bisector",
"Definition:Right Angle/Perpendicular",
"Definition:Parallel (Geometry)/Lines",
"Equal Corresponding Angles implies Parallel Lines",
"Definition:Bisection",
"Definition:Line/Midpoint",
"Definition:Distance between Parallel Lines",
"Distance between Two Parallel Str... |
proofwiki-21964 | Complex-Valued Function on Compact Hausdorff Space Vanishes at Infinity | Let $X$ be a compact Hausdorff space.
Let $f : X \to \C$ be a function.
Then $f$ vanishes at infinity. | Let $\epsilon > 0$.
Then:
:$\cmod {\map f x} < \epsilon$ for all $x \in \O$.
By hypothesis:
:$X$ is compact.
Hence we in particular have:
:$\cmod {\map f x} < \epsilon$ for all $x \in X \setminus X$
with $X$ compact.
Hence $f$ vanishes at infinity.
{{qed}}
Category:Complex-Valued Functions Vanishing at Infinity
hn4y929... | Let $X$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $f : X \to \C$ be a [[Definition:Function|function]].
Then $f$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]]. | Let $\epsilon > 0$.
Then:
:$\cmod {\map f x} < \epsilon$ for all $x \in \O$.
By hypothesis:
:$X$ is [[Definition:Compact Topological Space|compact]].
Hence we in particular have:
:$\cmod {\map f x} < \epsilon$ for all $x \in X \setminus X$
with $X$ [[Definition:Compact Topological Space|compact]].
Hence $f$ [[Defin... | Complex-Valued Function on Compact Hausdorff Space Vanishes at Infinity | https://proofwiki.org/wiki/Complex-Valued_Function_on_Compact_Hausdorff_Space_Vanishes_at_Infinity | https://proofwiki.org/wiki/Complex-Valued_Function_on_Compact_Hausdorff_Space_Vanishes_at_Infinity | [
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Function",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Definition:Compact Topological Space",
"Definition:Compact Topological Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Category:Complex-Valued Functions Vanishing at Infinity"
] |
proofwiki-21965 | Algebra of all Mappings is Algebra | Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {A, +_A, \circ_A, \ast_A}$ be an $R$-algebra.
Let $S$ be a set.
Let $\struct {A^S, +, \circ, \ast}$ the $R$-algebra of all mappings from $S$ to $A$.
Then $\struct {A^S, +, \circ, \ast}$ is an $R$-algebra. | From Module of All Mappings is Module, $\struct {A^S, +, \circ}$ is a module.
It remains to show that $\ast$ is $R$-bilinear.
Let $f, g, h \in A^S$ and $\lambda \in R$.
We have, for $x \in S$:
{{begin-eqn}}
{{eqn | l = \map {\paren {\paren {f + \lambda \circ g} \ast h} } x
| r = \map {\paren {f + \lambda \circ g} ... | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {A, +_A, \circ_A, \ast_A}$ be an [[Definition:Algebra over Ring|$R$-algebra]].
Let $S$ be a [[Definition:Set|set]].
Let $\struct {A^S, +, \circ, \ast}$ the [[Definition:Algebra of all Mappings|$R$-algebra of all mappings... | From [[Module of All Mappings is Module]], $\struct {A^S, +, \circ}$ is a [[Definition:Module over Ring|module]].
It remains to show that $\ast$ is [[Definition:Bilinear Mapping|$R$-bilinear]].
Let $f, g, h \in A^S$ and $\lambda \in R$.
We have, for $x \in S$:
{{begin-eqn}}
{{eqn | l = \map {\paren {\paren {f + \... | Algebra of all Mappings is Algebra | https://proofwiki.org/wiki/Algebra_of_all_Mappings_is_Algebra | https://proofwiki.org/wiki/Algebra_of_all_Mappings_is_Algebra | [
"Algebras over Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Algebra over Ring",
"Definition:Set",
"Definition:Algebra of all Mappings",
"Definition:Algebra over Ring"
] | [
"Module of All Mappings is Module",
"Definition:Module over Ring",
"Definition:Bilinear Mapping",
"Definition:Algebra over Ring",
"Category:Algebras over Rings"
] |
proofwiki-21966 | Linear Combination of Bounded Mappings on Normed Vector Space is Bounded | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a normed division ring.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $R$.
Let $S$ be a set.
Let $f, g : S \to X$ be bounded mappings.
Let $\lambda \in R$.
Then $f + \lambda g$ is bounded. | Since $f$ is bounded, there exists a real number $M_1 > 0$ such that:
:$\norm {\map f x} \le M_1$ for each $x \in X$.
Since $g$ is bounded, there exists a real number $M_2 > 0$ such that:
:$\norm {\map g x} \le M_2$ for each $x \in X$.
Then from {{NormAxiomVector|3}}, we have:
:$\norm {\map f x + \lambda \map g x} \l... | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $R$.
Let $S$ be a [[Definition:Set|set]].
Let $f, g : S \to X$ be [[Definition:Bounded Mapping on Normed Ve... | Since $f$ is [[Definition:Bounded Mapping on Normed Vector Space|bounded]], there exists a [[Definition:Real Number|real number]] $M_1 > 0$ such that:
:$\norm {\map f x} \le M_1$ for each $x \in X$.
Since $g$ is [[Definition:Bounded Mapping on Normed Vector Space|bounded]], there exists a [[Definition:Real Number|rea... | Linear Combination of Bounded Mappings on Normed Vector Space is Bounded | https://proofwiki.org/wiki/Linear_Combination_of_Bounded_Mappings_on_Normed_Vector_Space_is_Bounded | https://proofwiki.org/wiki/Linear_Combination_of_Bounded_Mappings_on_Normed_Vector_Space_is_Bounded | [
"Bounded Mappings on Normed Vector Spaces"
] | [
"Definition:Normed Division Ring",
"Definition:Normed Vector Space",
"Definition:Set",
"Definition:Bounded Mapping/Normed Vector Space",
"Definition:Bounded Mapping/Normed Vector Space"
] | [
"Definition:Bounded Mapping/Normed Vector Space",
"Definition:Real Number",
"Definition:Bounded Mapping/Normed Vector Space",
"Definition:Real Number",
"Definition:Bounded Mapping/Normed Vector Space",
"Category:Bounded Mappings on Normed Vector Spaces"
] |
proofwiki-21967 | Product of Bounded Mappings on Normed Algebra is Bounded | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a normed division ring.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed algebra over $R$.
Let $S$ be a set.
Let $f, g : S \to X$ be bounded mappings.
Then $f g$ is bounded. | Since $f$ is bounded, there exists a real number $M_1 > 0$ such that:
:$\norm {\map f x} \le M_1$ for each $x \in X$.
Since $g$ is bounded, there exists a real number $M_2 > 0$ such that:
:$\norm {\map g x} \le M_2$ for each $x \in X$.
Since $\norm {\, \cdot \,}$ is a algebra norm, we have:
:$\norm {\map f x \map g x... | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $R$.
Let $S$ be a [[Definition:Set|set]].
Let $f, g : S \to X$ be [[Definition:Bounded Mapping on Normed Vector Space... | Since $f$ is [[Definition:Bounded Mapping on Normed Vector Space|bounded]], there exists a [[Definition:Real Number|real number]] $M_1 > 0$ such that:
:$\norm {\map f x} \le M_1$ for each $x \in X$.
Since $g$ is [[Definition:Bounded Mapping on Normed Vector Space|bounded]], there exists a [[Definition:Real Number|rea... | Product of Bounded Mappings on Normed Algebra is Bounded | https://proofwiki.org/wiki/Product_of_Bounded_Mappings_on_Normed_Algebra_is_Bounded | https://proofwiki.org/wiki/Product_of_Bounded_Mappings_on_Normed_Algebra_is_Bounded | [
"Bounded Mappings on Normed Vector Spaces"
] | [
"Definition:Normed Division Ring",
"Definition:Normed Algebra",
"Definition:Set",
"Definition:Bounded Mapping/Normed Vector Space",
"Definition:Bounded Mapping/Normed Vector Space"
] | [
"Definition:Bounded Mapping/Normed Vector Space",
"Definition:Real Number",
"Definition:Bounded Mapping/Normed Vector Space",
"Definition:Real Number",
"Definition:Norm/Algebra",
"Definition:Bounded Mapping/Normed Vector Space",
"Category:Bounded Mappings on Normed Vector Spaces"
] |
proofwiki-21968 | Banach Algebra of Continuous Functions on Compact Hausdorff Space is Banach Algebra | Let $X$ be a compact Hausdorff space.
Let $\map \CC X = \map \CC {X, \C}$ be the vector space of continuous functions on $X$.
Let $\ast$ be pointwise multiplication on $\C^X$.
Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\map \CC X$.
Then $\struct {\map \CC X, \ast, \norm {\, \cdot \,} }$ is a Banach alg... | We first show that $\struct {\map \CC X, \ast}$ is an algebra over $\C$.
For this, we show that $\struct {\map \CC X, \ast}$ is a subalgebra of $\C^X$.
As shown in Continuous Functions on Compact Space form Banach Space, $\map \CC X$ is a linear subspace of $\C^X$.
From Product of Continuous Functions on Topological Ri... | Let $X$ be a [[Definition:Compact Topological Subspace|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $\map \CC X = \map \CC {X, \C}$ be the [[Definition:Space of Continuous Functions on Compact Hausdorff Space|vector space of continuous functions on $X$]].
Let $\ast$ be [[Definition:Pointwise Multipli... | We first show that $\struct {\map \CC X, \ast}$ is an [[Definition:Algebra over Field|algebra over $\C$]].
For this, we show that $\struct {\map \CC X, \ast}$ is a [[Definition:Subalgebra|subalgebra]] of $\C^X$.
As shown in [[Continuous Functions on Compact Space form Banach Space]], $\map \CC X$ is a [[Definition:Li... | Banach Algebra of Continuous Functions on Compact Hausdorff Space is Banach Algebra | https://proofwiki.org/wiki/Banach_Algebra_of_Continuous_Functions_on_Compact_Hausdorff_Space_is_Banach_Algebra | https://proofwiki.org/wiki/Banach_Algebra_of_Continuous_Functions_on_Compact_Hausdorff_Space_is_Banach_Algebra | [
"Banach Algebras"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:T2 Space",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Pointwise Multiplication",
"Definition:Supremum Norm",
"Definition:Banach Algebra"
] | [
"Definition:Algebra over Field",
"Definition:Subalgebra",
"Continuous Functions on Compact Space form Banach Space",
"Definition:Linear Subspace",
"Product of Continuous Functions on Topological Ring is Continuous",
"Definition:Subalgebra",
"Definition:Algebra over Field",
"Continuous Functions on Com... |
proofwiki-21969 | Linear Combination of Complex-Valued Functions Vanishing in Neighborhood of Infinity Vanishes in Neighborhood of Infinity | Let $X$ be a locally compact Hausdorff space.
Let $f, g : X \to \C$ be complex-valued functions vanishing in a neighborhood of infinity.
Let $\lambda \in \C$.
Then $f + \lambda g$ vanishes in a neighborhood of infinity. | Since $f$ vanishes in a neighborhood of infinity, there exists a compact set $F_1 \subseteq X$ such that:
:$\map f x = 0$ for each $x \in X \setminus F_1$.
Since $g$ vanishes in a neighborhood of infinity, there exists a compact set $F_2 \subseteq X$ such that:
:$\map g x = 0$ for each $x \in X \setminus F_2$.
From Uni... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $f, g : X \to \C$ be [[Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity|complex-valued functions vanishing in a neighborhood of infinity]].
Let $\lambda \in \C$.
Then $f + \lambda g$ [[Definitio... | Since $f$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes in a neighborhood of infinity]], there exists a [[Definition:Compact Topological Space|compact set]] $F_1 \subseteq X$ such that:
:$\map f x = 0$ for each $x \in X \setminus F_1$.
Since $g$ [[Definition:Complex-Valued Function Vanishing at I... | Linear Combination of Complex-Valued Functions Vanishing in Neighborhood of Infinity Vanishes in Neighborhood of Infinity | https://proofwiki.org/wiki/Linear_Combination_of_Complex-Valued_Functions_Vanishing_in_Neighborhood_of_Infinity_Vanishes_in_Neighborhood_of_Infinity | https://proofwiki.org/wiki/Linear_Combination_of_Complex-Valued_Functions_Vanishing_in_Neighborhood_of_Infinity_Vanishes_in_Neighborhood_of_Infinity | [
"Complex-Valued Functions Vanishing in Neighborhood of Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity",
"Definition:Complex-Valued Function Vanishing in Neighborhood of Infinity"
] | [
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space",
"Union of Two Compact Sets is Compact",
"Definition:Compact Topological Space",
"Definition:Compact Topo... |
proofwiki-21970 | Space of Continuous Functions on Locally Compact Hausdorff Space Vanishing at Infinity is Banach Space | Let $X$ be a locally compact Hausdorff space.
Let $\struct {\map {\CC_0} X, \norm {\, \cdot \,}_\infty}$ be the space of continuous functions on $X$ vanishing at infinity.
Then $\struct {\map {\CC_0} X, \norm {\, \cdot \,}_\infty}$ is a Banach space over $\C$. | We first show that $\map {\CC_0} X$ is a vector space.
From Bounded Continuous Functions on Topological Space form Banach Space, it is enough to show that $\map {\CC_0} X$ is a linear subspace of $\map {\CC_b} X$, where $\map {\CC_b} X$ is the space of bounded continuous functions on $X$ valued in $\C$.
From Linear C... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $\struct {\map {\CC_0} X, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Continuous Functions on Locally Compact Hausdorff Space Vanishing at Infinity|space of continuous functions on $X$ vanishing at infinity]]... | We first show that $\map {\CC_0} X$ is a [[Definition:Vector Space|vector space]].
From [[Bounded Continuous Functions on Topological Space form Banach Space]], it is enough to show that $\map {\CC_0} X$ is a [[Definition:Linear Subspace|linear subspace]] of $\map {\CC_b} X$, where $\map {\CC_b} X$ is the [[Definitio... | Space of Continuous Functions on Locally Compact Hausdorff Space Vanishing at Infinity is Banach Space | https://proofwiki.org/wiki/Space_of_Continuous_Functions_on_Locally_Compact_Hausdorff_Space_Vanishing_at_Infinity_is_Banach_Space | https://proofwiki.org/wiki/Space_of_Continuous_Functions_on_Locally_Compact_Hausdorff_Space_Vanishing_at_Infinity_is_Banach_Space | [
"Complex-Valued Functions Vanishing at Infinity",
"Banach Spaces"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Space of Continuous Functions on Locally Compact Hausdorff Space Vanishing at Infinity",
"Definition:Banach Space"
] | [
"Definition:Vector Space",
"Bounded Continuous Functions on Topological Space form Banach Space",
"Definition:Linear Subspace",
"Definition:Space of Bounded Continuous Functions on Topological Space",
"Linear Combination of Complex-Valued Functions Vanishing at Infinity Vanishes at Infinity",
"One-Step Ve... |
proofwiki-21971 | Sequential Characterization of Closed Linear Transformation | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be Banach spaces.
Let $\map D T$ be a linear subspace of $X$.
Let $T : \map D T \to Y$ be a linear transformation.
Then $T$ is closed {{iff}}:
:for each convergent sequence $\sequence {x_n}_{n \in \N}$ in $\struct {X, \norm {\, \cdot \,}_... | Let $\struct {X \times Y, \norm \cdot_{X \times Y} }$ be the direct product $X \times Y$ equipped with the direct product norm.
From Direct Product of Banach Spaces is Banach Space, $\struct {X \times Y, \norm \cdot_{X \times Y} }$ is a Banach space.
Let:
:$\map G T = \set {\tuple {x, T x} \in X \times Y : x \in \map ... | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Banach Space|Banach spaces]].
Let $\map D T$ be a [[Definition:Linear Subspace|linear subspace]] of $X$.
Let $T : \map D T \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Then $T$ is [[Definitio... | Let $\struct {X \times Y, \norm \cdot_{X \times Y} }$ be the [[Definition:Direct Product of Vector Spaces|direct product]] $X \times Y$ equipped with the [[Definition:Direct Product Norm|direct product norm]].
From [[Direct Product of Banach Spaces is Banach Space]], $\struct {X \times Y, \norm \cdot_{X \times Y} }$ i... | Sequential Characterization of Closed Linear Transformation | https://proofwiki.org/wiki/Sequential_Characterization_of_Closed_Linear_Transformation | https://proofwiki.org/wiki/Sequential_Characterization_of_Closed_Linear_Transformation | [
"Closed Linear Transformations"
] | [
"Definition:Banach Space",
"Definition:Linear Subspace",
"Definition:Linear Transformation",
"Definition:Closed Linear Transformation",
"Definition:Convergent Sequence",
"Definition:Convergent Sequence"
] | [
"Definition:Direct Product of Vector Spaces",
"Definition:Direct Product Norm",
"Direct Product of Banach Spaces is Banach Space",
"Definition:Banach Space",
"Definition:Graph of Mapping"
] |
proofwiki-21972 | Independent Sets of Dual Matroid | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\mathscr B$ be the set of bases of the matroid $M$.
Then the independent subsets of the dual $M^*$ of $M$ is:
:$\mathscr I^* = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq \paren{S \setminus B}}$ | By definition of the dual matroid the set of bases $\mathscr B^*$ of the dual $M^*$ of $M$:
:$\mathscr B^* = \set{S \setminus B : B \in \mathscr B}$
From Characterization of Matroid Independent Sets in Terms of Bases:
:$\mathscr I^* = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq \paren{S \setminus B}}$
{... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\mathscr B$ be the set of [[Definition:Base of Matroid|bases]] of the [[Definition:Matroid|matroid]] $M$.
Then the [[Definition:Independent Subset (Matroid)|independent subsets]] of the [[Definition:Dual Matroid|dual]] $M^*$ of $M$ is:
:$\ma... | By definition of the [[Definition:Dual Matroid|dual matroid]] the [[Definition:Set|set]] of [[Definition:Base of Matroid|bases]] $\mathscr B^*$ of the [[Definition:Dual Matroid|dual]] $M^*$ of $M$:
:$\mathscr B^* = \set{S \setminus B : B \in \mathscr B}$
From [[Characterization of Matroid Independent Sets in Terms of ... | Independent Sets of Dual Matroid | https://proofwiki.org/wiki/Independent_Sets_of_Dual_Matroid | https://proofwiki.org/wiki/Independent_Sets_of_Dual_Matroid | [
"Dual Matroids",
"Matroid Independent Subsets"
] | [
"Definition:Matroid",
"Definition:Base of Matroid",
"Definition:Matroid",
"Definition:Matroid/Independent Set",
"Definition:Dual Matroid"
] | [
"Definition:Dual Matroid",
"Definition:Set",
"Definition:Base of Matroid",
"Definition:Dual Matroid",
"Characterization of Matroid Independent Sets in Terms of Bases",
"Category:Dual Matroids",
"Category:Matroid Independent Subsets"
] |
proofwiki-21973 | Characterization of Matroid Independent Sets in Terms of Bases | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\mathscr B$ be the set of bases of the matroid $M$.
Then:
:$\mathscr I = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq B}$ | Let $\mathscr I^\prime = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq B}$
From Independent Subset is Contained in Base:
:$\mathscr I \subseteq \mathscr I^\prime$
By definition of matroid base:
:$\mathscr B \subseteq \mathscr I$
By matroid axiom $\text I 2$:
:$\mathscr I^\prime \subseteq \mathscr I$
By se... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\mathscr B$ be the [[Definition:Set|set]] of [[Definition:Base of Matroid|bases]] of the [[Definition:Matroid|matroid]] $M$.
Then:
:$\mathscr I = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq B}$ | Let $\mathscr I^\prime = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq B}$
From [[Independent Subset is Contained in Base]]:
:$\mathscr I \subseteq \mathscr I^\prime$
By definition of [[Definition:Base of Matroid|matroid base]]:
:$\mathscr B \subseteq \mathscr I$
By [[Axiom:Matroid Axioms|matroid axi... | Characterization of Matroid Independent Sets in Terms of Bases | https://proofwiki.org/wiki/Characterization_of_Matroid_Independent_Sets_in_Terms_of_Bases | https://proofwiki.org/wiki/Characterization_of_Matroid_Independent_Sets_in_Terms_of_Bases | [
"Matroid Bases",
"Matroid Independent Subsets"
] | [
"Definition:Matroid",
"Definition:Set",
"Definition:Base of Matroid",
"Definition:Matroid"
] | [
"Independent Subset is Contained in Base",
"Definition:Base of Matroid",
"Axiom:Matroid Axioms",
"Definition:Set Equality",
"Category:Matroid Bases",
"Category:Matroid Independent Subsets"
] |
proofwiki-21974 | Dual of Dual Matroid Equals Matroid | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $M^*$ denote the dual of $M$.
Then:
:$\paren{M^*}^* = M$
That is, the dual $\paren{M^*}^*$ of $M^*$ is $M$. | Let $\mathscr B$ denote the set of bases of $M$.
Let $\paren{\mathscr B^*}^*$ denote the set of bases of $\paren{M^*}^*$.
By definition of dual matroid:
:
We have:
{{begin-eqn}}
{{eqn | l = \paren{\mathscr B^*}^*
| r = \set{S \setminus \paren{S \setminus B} : B \in \mathscr B}
| c = {{Defof|Dual Matroid}}
}... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $M^*$ denote the [[Definition:Dual Matroid|dual]] of $M$.
Then:
:$\paren{M^*}^* = M$
That is, the [[Definition:Dual Matroid|dual]] $\paren{M^*}^*$ of $M^*$ is $M$. | Let $\mathscr B$ denote the [[Definition:Set|set]] of [[Definition:Base of Matroid|bases]] of $M$.
Let $\paren{\mathscr B^*}^*$ denote the [[Definition:Set|set]] of [[Definition:Base of Matroid|bases]] of $\paren{M^*}^*$.
By definition of [[Definition:Dual Matroid|dual matroid]]:
:
We have:
{{begin-eqn}}
{{eqn | l =... | Dual of Dual Matroid Equals Matroid | https://proofwiki.org/wiki/Dual_of_Dual_Matroid_Equals_Matroid | https://proofwiki.org/wiki/Dual_of_Dual_Matroid_Equals_Matroid | [
"Dual Matroids"
] | [
"Definition:Matroid",
"Definition:Dual Matroid",
"Definition:Dual Matroid"
] | [
"Definition:Set",
"Definition:Base of Matroid",
"Definition:Set",
"Definition:Base of Matroid",
"Definition:Dual Matroid",
"Complement of Complement",
"Definition:Matroid/Independent Set",
"Characterization of Matroid Independent Sets in Terms of Bases",
"Characterization of Matroid Independent Sets... |
proofwiki-21975 | Trapezium Rule for Definite Integrals/Error Term | The error can be quantified as:
:$\dfrac {\paren {b - a}^3 \map {f' '} \xi} {12 n^2}$
where $\xi \in \closedint a b$. | {{Proofread}}
{{tidy}}
first, through integration by parts:
:<nowiki>$\int_{x_{i+1}}^{x_i} f(x)\ dx \\</nowiki> =\int_{0}^{h} f(t+x_i)\ dt \\ = \left [ (t-\frac{h}{2})f(t+x_i) \right ]^h_0 -\int_{0}^{h} (t-h/2) f'(t+x_i)\ dt \\ = \left [ (t-\frac{h}{2})f(t+x_i) \right ]^h_0 -\left [ \left(\frac{(t-h/2)^2}{2}-\f... | The [[Definition:Error|error]] can be quantified as:
:$\dfrac {\paren {b - a}^3 \map {f' '} \xi} {12 n^2}$
where $\xi \in \closedint a b$. | {{Proofread}}
{{tidy}}
first, through integration by parts:
:<nowiki>$\int_{x_{i+1}}^{x_i} f(x)\ dx \\</nowiki> =\int_{0}^{h} f(t+x_i)\ dt \\ = \left [ (t-\frac{h}{2})f(t+x_i) \right ]^h_0 -\int_{0}^{h} (t-h/2) f'(t+x_i)\ dt \\ = \left [ (t-\frac{h}{2})f(t+x_i) \right ]^h_0 -\left [ \left(\frac{(t-h/2)^2}{2}-\f... | Trapezium Rule for Definite Integrals/Error Term | https://proofwiki.org/wiki/Trapezium_Rule_for_Definite_Integrals/Error_Term | https://proofwiki.org/wiki/Trapezium_Rule_for_Definite_Integrals/Error_Term | [
"Trapezium Rule for Definite Integrals"
] | [
"Definition:Error"
] | [] |
proofwiki-21976 | Possibility Operator in terms of Necessity Operator | The possibility operator $\pos$ can be expressed in terms of the necessity operator $\nec$ thus:
:$\pos P \iff \lnot \nec \lnot P$ | {{begin-eqn}}
{{eqn | q = \forall P
| o =
| r = \pos P
| c =
}}
{{eqn | o = \leadstoandfrom
| r = \exists w: \map P w
| c = {{Defof|Possibility Operator}}
}}
{{eqn | o = \leadstoandfrom
| r = \lnot \forall w: \lnot \map P w
| c = Assertion of Existence
}}
{{eqn | o = \leadsto... | The [[Definition:Possibility Operator|possibility operator]] $\pos$ can be expressed in terms of the [[Definition:Necessity Operator|necessity operator]] $\nec$ thus:
:$\pos P \iff \lnot \nec \lnot P$ | {{begin-eqn}}
{{eqn | q = \forall P
| o =
| r = \pos P
| c =
}}
{{eqn | o = \leadstoandfrom
| r = \exists w: \map P w
| c = {{Defof|Possibility Operator}}
}}
{{eqn | o = \leadstoandfrom
| r = \lnot \forall w: \lnot \map P w
| c = [[Assertion of Existence]]
}}
{{eqn | o = \lea... | Possibility Operator in terms of Necessity Operator | https://proofwiki.org/wiki/Possibility_Operator_in_terms_of_Necessity_Operator | https://proofwiki.org/wiki/Possibility_Operator_in_terms_of_Necessity_Operator | [
"Possibility Operator",
"Necessity Operator"
] | [
"Definition:Modal Operator/Possibility",
"Definition:Modal Operator/Necessity"
] | [
"De Morgan's Laws (Predicate Logic)/Assertion of Existence"
] |
proofwiki-21977 | Natural Numbers under Multiplication form Monoid | The set of natural numbers under multiplication $\struct {\N, \times}$ is a monoid. | Taking the monoid axioms in turn: | The [[Definition:Set|set]] of [[Definition:Natural Number|natural numbers]] under [[Definition:Natural Number Multiplication|multiplication]] $\struct {\N, \times}$ is a [[Definition:Monoid|monoid]]. | Taking the [[Axiom:Monoid Axioms|monoid axioms]] in turn: | Natural Numbers under Multiplication form Monoid | https://proofwiki.org/wiki/Natural_Numbers_under_Multiplication_form_Monoid | https://proofwiki.org/wiki/Natural_Numbers_under_Multiplication_form_Monoid | [
"Natural Number Multiplication",
"Examples of Monoids"
] | [
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Multiplication/Natural Numbers",
"Definition:Monoid"
] | [
"Axiom:Monoid Axioms",
"Axiom:Monoid Axioms"
] |
proofwiki-21978 | Subsets Satisfy Formulation 1 Matroid Base Axiom Iff Complements Satisfy Formulation 5 | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
Let $\mathscr B^* = \set{S \setminus B : B \in \mathscr B}$.
Then:
:$\mathscr B$ satisfies formulation $1$ base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}}
{{iff}}
:$\mathscr B^*$ satisfies formulation $5$ base axiom:
{{begin-axiom... | === Necessary Condition ===
Let $\mathscr B$ satisfies formulation $1$ base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}}
Let $B^*_1, B^*_2 \in \mathscr B^*$.
By definition of $\mathscr B^*$:
:$\exists B_1, B_2 \in \mathscr B$:
::$B^*_1 = S \setminus B_1$
::$B^*_2 = S \setminus B_2$
We have:
{{begin-eqn}}
{{eqn... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Let $\mathscr B^* = \set{S \setminus B : B \in \mathscr B}$.
Then:
:$\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 1|formu... | === Necessary Condition ===
Let $\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 1|formulation $1$ base axiom]]:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}}
Let $B^*_1, B^*_2 \in \mathscr B^*$.
By definition of $\mathscr B^*$:
:$\exists B_1, B_2 \in \mathscr B$:
::$B^*_1 = S \setminus B_1$
::$B^*_... | Subsets Satisfy Formulation 1 Matroid Base Axiom Iff Complements Satisfy Formulation 5 | https://proofwiki.org/wiki/Subsets_Satisfy_Formulation_1_Matroid_Base_Axiom_Iff_Complements_Satisfy_Formulation_5 | https://proofwiki.org/wiki/Subsets_Satisfy_Formulation_1_Matroid_Base_Axiom_Iff_Complements_Satisfy_Formulation_5 | [
"Matroid Bases"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Axiom:Base Axiom (Matroid)/Formulation 5"
] | [
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Set Difference of Complements",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Intersection with Subset is Subset",
"Set Difference with Set Difference is Union of Set Difference with Intersection",
"Set Difference with Set Difference is Union of Set Difference w... |
proofwiki-21979 | Subsets Satisfy Formulation 5 Matroid Base Axiom Iff Complements Satisfy Formulation 1 | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
Let $\mathscr B^* = \set{S \setminus B : B \in \mathscr B}$.
Then:
:$\mathscr B$ satisfies formulation $5$ base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 5}}
{{iff}}
:$\mathscr B^*$ satisfies formulation $1$ base axiom:
{{begin-axiom... | Let $\mathscr B^{**} = \set{S \setminus B^* : B^* \in \mathscr B^*}$.
Then:
:$\mathscr B^{**} = \set{S \setminus \paren{S \setminus B} : B \in \mathscr B}$
From Relative Complement of Relative Complement:
:$\mathscr B^{**} = \mathscr B$
From Subsets Satisfy Formulation 1 Matroid Base Axiom Iff Complements Satisfy Formu... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Let $\mathscr B^* = \set{S \setminus B : B \in \mathscr B}$.
Then:
:$\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 5|formu... | Let $\mathscr B^{**} = \set{S \setminus B^* : B^* \in \mathscr B^*}$.
Then:
:$\mathscr B^{**} = \set{S \setminus \paren{S \setminus B} : B \in \mathscr B}$
From [[Relative Complement of Relative Complement]]:
:$\mathscr B^{**} = \mathscr B$
From [[Subsets Satisfy Formulation 1 Matroid Base Axiom Iff Complements Sat... | Subsets Satisfy Formulation 5 Matroid Base Axiom Iff Complements Satisfy Formulation 1 | https://proofwiki.org/wiki/Subsets_Satisfy_Formulation_5_Matroid_Base_Axiom_Iff_Complements_Satisfy_Formulation_1 | https://proofwiki.org/wiki/Subsets_Satisfy_Formulation_5_Matroid_Base_Axiom_Iff_Complements_Satisfy_Formulation_1 | [
"Matroid Bases"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Axiom:Base Axiom (Matroid)/Formulation 5",
"Axiom:Base Axiom (Matroid)/Formulation 1"
] | [
"Relative Complement of Relative Complement",
"Subsets Satisfy Formulation 1 Matroid Base Axiom Iff Complements Satisfy Formulation 5",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Axiom:Base Axiom (Matroid)/Formulation 5",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Axiom:Base Axiom (Matroid)/Formulatio... |
proofwiki-21980 | Matroid is Uniquely Defined by Bases | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\mathscr B$ be the set of bases of the matroid $M$.
Then:
:$M$ is the only matroid on $S$ whose set of bases is $\mathscr B$. | Let $M^\prime = \struct {S, \mathscr I^\prime}$ whose set of bases is $\mathscr B$.
From Characterization of Matroid Independent Sets in Terms of Bases:
:$\mathscr I^\prime = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq B}$
Similarly:
:$\mathscr I = \set{X \subseteq S : \exists B \in \mathscr B : X \subs... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\mathscr B$ be the [[Definition:Set|set]] of [[Definition:Base of Matroid|bases]] of the [[Definition:Matroid|matroid]] $M$.
Then:
:$M$ is the only [[Definition:Matroid|matroid]] on $S$ whose [[Definition:Set|set]] of [[Definition:Base of Ma... | Let $M^\prime = \struct {S, \mathscr I^\prime}$ whose [[Definition:Set|set]] of [[Definition:Base of Matroid|bases]] is $\mathscr B$.
From [[Characterization of Matroid Independent Sets in Terms of Bases]]:
:$\mathscr I^\prime = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq B}$
Similarly:
:$\mathscr I ... | Matroid is Uniquely Defined by Bases | https://proofwiki.org/wiki/Matroid_is_Uniquely_Defined_by_Bases | https://proofwiki.org/wiki/Matroid_is_Uniquely_Defined_by_Bases | [
"Matroid Theory",
"Matroid Bases"
] | [
"Definition:Matroid",
"Definition:Set",
"Definition:Base of Matroid",
"Definition:Matroid",
"Definition:Matroid",
"Definition:Set",
"Definition:Base of Matroid"
] | [
"Definition:Set",
"Definition:Base of Matroid",
"Characterization of Matroid Independent Sets in Terms of Bases",
"Category:Matroid Theory",
"Category:Matroid Bases"
] |
proofwiki-21981 | Set Difference of Matroid Dependent Set with Independent Set is Non-empty | Let $I$ be an independent subset of $M$.
Let $D$ be a dependent subset of $M$.
Then:
:$D \setminus I \ne \O$ | From Independent Subset Contains No Dependent Subset:
:$D \nsubseteq I$
By definition of subset:
:$\exists x \in D : x \notin I$
By definition of set difference:
:$\exists x \in D \setminus I$
The result follows.
{{qed}}
Category:Matroid Dependent Subsets
Category:Matroid Independent Subsets
Category:Set Difference of ... | Let $I$ be an [[Definition:Independent Subset (Matroid)|independent subset]] of $M$.
Let $D$ be a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$.
Then:
:$D \setminus I \ne \O$ | From [[Independent Subset Contains No Dependent Subset]]:
:$D \nsubseteq I$
By definition of [[Definition:Subset|subset]]:
:$\exists x \in D : x \notin I$
By definition of [[Definition:Set Difference|set difference]]:
:$\exists x \in D \setminus I$
The result follows.
{{qed}}
[[Category:Matroid Dependent Subsets]]
... | Set Difference of Matroid Dependent Set with Independent Set is Non-empty | https://proofwiki.org/wiki/Set_Difference_of_Matroid_Dependent_Set_with_Independent_Set_is_Non-empty | https://proofwiki.org/wiki/Set_Difference_of_Matroid_Dependent_Set_with_Independent_Set_is_Non-empty | [
"Matroid Dependent Subsets",
"Matroid Independent Subsets",
"Set Difference of Matroid Dependent Set with Independent Set is Non-empty"
] | [
"Definition:Matroid/Independent Set",
"Definition:Matroid/Dependent Set"
] | [
"Independent Subset Contains No Dependent Subset",
"Definition:Subset",
"Definition:Set Difference",
"Category:Matroid Dependent Subsets",
"Category:Matroid Independent Subsets",
"Category:Set Difference of Matroid Dependent Set with Independent Set is Non-empty"
] |
proofwiki-21982 | Set Difference of Matroid Dependent Set with Independent Set is Non-empty/Corollary 2 | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $I$ be an independent subset of $M$.
Let $C$ be a circuit of $M$.
Then:
:$C \setminus I \ne \O$ | By definition of matroid circuit:
:$C$ is a dependent subset of $M$
From Set Difference of Matroid Dependent Set with Independent Set is Non-empty:
:$C \setminus I \ne \O$
{{qed}}
Category:Set Difference of Matroid Dependent Set with Independent Set is Non-empty
fnwiu0dvcy6jl0dfd54742113jtc4u9 | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $I$ be an [[Definition:Independent Subset (Matroid)|independent subset]] of $M$.
Let $C$ be a [[Definition:Circuit (Matroid)|circuit]] of $M$.
Then:
:$C \setminus I \ne \O$ | By definition of [[Definition:Circuit (Matroid)|matroid circuit]]:
:$C$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$
From [[Set Difference of Matroid Dependent Set with Independent Set is Non-empty]]:
:$C \setminus I \ne \O$
{{qed}}
[[Category:Set Difference of Matroid Dependent Set with Ind... | Set Difference of Matroid Dependent Set with Independent Set is Non-empty/Corollary 2 | https://proofwiki.org/wiki/Set_Difference_of_Matroid_Dependent_Set_with_Independent_Set_is_Non-empty/Corollary_2 | https://proofwiki.org/wiki/Set_Difference_of_Matroid_Dependent_Set_with_Independent_Set_is_Non-empty/Corollary_2 | [
"Set Difference of Matroid Dependent Set with Independent Set is Non-empty"
] | [
"Definition:Matroid",
"Definition:Matroid/Independent Set",
"Definition:Circuit (Matroid)"
] | [
"Definition:Circuit (Matroid)",
"Definition:Matroid/Dependent Set",
"Set Difference of Matroid Dependent Set with Independent Set is Non-empty",
"Category:Set Difference of Matroid Dependent Set with Independent Set is Non-empty"
] |
proofwiki-21983 | Set Difference of Matroid Dependent Set with Independent Set is Non-empty/Corollary 1 | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $D$ be a dependent subset of $M$.
Let $B$ be a base of $M$.
Then:
:$D \setminus B \ne \O$ | By definition of matroid base:
:$B$ is an independent subset of $M$
From Set Difference of Matroid Dependent Set with Independent Set is Non-empty:
:$D \setminus B \ne \O$
{{qed}}
Category:Set Difference of Matroid Dependent Set with Independent Set is Non-empty
ahqrj958aua1frqgfr4903ogol8nk6z | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $D$ be a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$.
Let $B$ be a [[Definition:Base of Matroid|base]] of $M$.
Then:
:$D \setminus B \ne \O$ | By definition of [[Definition:Base of Matroid|matroid base]]:
:$B$ is an [[Definition:Independent Subset (Matroid)|independent subset]] of $M$
From [[Set Difference of Matroid Dependent Set with Independent Set is Non-empty]]:
:$D \setminus B \ne \O$
{{qed}}
[[Category:Set Difference of Matroid Dependent Set with Ind... | Set Difference of Matroid Dependent Set with Independent Set is Non-empty/Corollary 1 | https://proofwiki.org/wiki/Set_Difference_of_Matroid_Dependent_Set_with_Independent_Set_is_Non-empty/Corollary_1 | https://proofwiki.org/wiki/Set_Difference_of_Matroid_Dependent_Set_with_Independent_Set_is_Non-empty/Corollary_1 | [
"Set Difference of Matroid Dependent Set with Independent Set is Non-empty"
] | [
"Definition:Matroid",
"Definition:Matroid/Dependent Set",
"Definition:Base of Matroid"
] | [
"Definition:Base of Matroid",
"Definition:Matroid/Independent Set",
"Set Difference of Matroid Dependent Set with Independent Set is Non-empty",
"Category:Set Difference of Matroid Dependent Set with Independent Set is Non-empty"
] |
proofwiki-21984 | Set Difference of Matroid Dependent Set with Independent Set is Non-empty/Corollary 3 | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $B$ be an base of $M$.
Let $C$ be a circuit of $M$.
Then:
:$C \setminus B \ne \O$ | By definition of matroid base:
:$B$ is an independent subset of $M$
By definition of matroid circuit:
:$C$ is a dependent subset of $M$
From Set Difference of Matroid Dependent Set with Independent Set is Non-empty:
:$C \setminus B \ne \O$
{{qed}}
Category:Set Difference of Matroid Dependent Set with Independent Set is... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $B$ be an [[Definition:Base of Matroid|base]] of $M$.
Let $C$ be a [[Definition:Circuit (Matroid)|circuit]] of $M$.
Then:
:$C \setminus B \ne \O$ | By definition of [[Definition:Base of Matroid|matroid base]]:
:$B$ is an [[Definition:Independent Subset (Matroid)|independent subset]] of $M$
By definition of [[Definition:Circuit (Matroid)|matroid circuit]]:
:$C$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$
From [[Set Difference of Matroid... | Set Difference of Matroid Dependent Set with Independent Set is Non-empty/Corollary 3 | https://proofwiki.org/wiki/Set_Difference_of_Matroid_Dependent_Set_with_Independent_Set_is_Non-empty/Corollary_3 | https://proofwiki.org/wiki/Set_Difference_of_Matroid_Dependent_Set_with_Independent_Set_is_Non-empty/Corollary_3 | [
"Set Difference of Matroid Dependent Set with Independent Set is Non-empty"
] | [
"Definition:Matroid",
"Definition:Base of Matroid",
"Definition:Circuit (Matroid)"
] | [
"Definition:Base of Matroid",
"Definition:Matroid/Independent Set",
"Definition:Circuit (Matroid)",
"Definition:Matroid/Dependent Set",
"Set Difference of Matroid Dependent Set with Independent Set is Non-empty",
"Category:Set Difference of Matroid Dependent Set with Independent Set is Non-empty"
] |
proofwiki-21985 | Preimage of Subset under Identity Mapping | Let $S$ be a set.
Let $\iota_S: S \to S$ be the identity mapping on $S$.
Let $T \subseteq S$.
Then:
:$\iota_S^{-1} \sqbrk T = T$
where $\iota_S^{-1} \sqbrk T$ is the preimage of $T$ under $\iota_S$. | By definition of identity mapping:
:$\iota_S: S \to S: \forall x \in S: \map {\iota_S} x = x$
Let $i_S: S \to S$ be the inclusion mapping of $S$ into $S$.
By definition of inclusion mapping:
:$i_S: S \to S: \forall x \in S: \map {i_S} x = x$
From Equality of Mappings:
:$\iota_S = i_S$
From Preimage of Subset under Incl... | Let $S$ be a [[Definition:Set|set]].
Let $\iota_S: S \to S$ be the [[Definition:Identity Mapping|identity mapping]] on $S$.
Let $T \subseteq S$.
Then:
:$\iota_S^{-1} \sqbrk T = T$
where $\iota_S^{-1} \sqbrk T$ is the [[Definition:Preimage of Subset under Mapping|preimage]] of $T$ under $\iota_S$. | By definition of [[Definition:Identity Mapping|identity mapping]]:
:$\iota_S: S \to S: \forall x \in S: \map {\iota_S} x = x$
Let $i_S: S \to S$ be the [[Definition:Inclusion Mapping|inclusion mapping]] of $S$ into $S$.
By definition of [[Definition:Inclusion Mapping|inclusion mapping]]:
:$i_S: S \to S: \forall x \i... | Preimage of Subset under Identity Mapping | https://proofwiki.org/wiki/Preimage_of_Subset_under_Identity_Mapping | https://proofwiki.org/wiki/Preimage_of_Subset_under_Identity_Mapping | [
"Identity Mappings",
"Subsets"
] | [
"Definition:Set",
"Definition:Identity Mapping",
"Definition:Preimage/Mapping/Subset"
] | [
"Definition:Identity Mapping",
"Definition:Inclusion Mapping",
"Definition:Inclusion Mapping",
"Equality of Mappings",
"Preimage of Subset under Inclusion Mapping",
"Intersection with Subset is Subset",
"Category:Identity Mappings",
"Category:Subsets"
] |
proofwiki-21986 | Extension Theorem for Positive Linear Functional defined on Cofinal Linear Subspace | Let $\struct {X, \succeq}$ be a preordered vector space over $\R$.
Let $Y$ be a linear subspace of $X$ that is cofinal in $\struct {X, \succeq}$.
Let $f_0 : Y \to \R$ be a positive linear functional.
Then there exists a positive linear functional $f : X \to \R$ such that:
:$\map f y = \map {f_0} y$ for each $y \in Y$. | Let:
:$P = \set {x \in X : x \succeq {\mathbf 0}_X}$
Let:
:$Z = Y + P - P = \set {y + z - w : y \in Y, \, z, w \in P}$
be the Minkowski sum of $Y$, $P$ and $-P$. | Let $\struct {X, \succeq}$ be a [[Definition:Preordered Vector Space|preordered vector space]] over $\R$.
Let $Y$ be a [[Definition:Linear Subspace|linear subspace]] of $X$ that is [[Definition:Cofinal Subset/Preordered Vector Space|cofinal]] in $\struct {X, \succeq}$.
Let $f_0 : Y \to \R$ be a [[Definition:Positive ... | Let:
:$P = \set {x \in X : x \succeq {\mathbf 0}_X}$
Let:
:$Z = Y + P - P = \set {y + z - w : y \in Y, \, z, w \in P}$
be the [[Definition:Minkowski Sum|Minkowski sum]] of $Y$, $P$ and $-P$. | Extension Theorem for Positive Linear Functional defined on Cofinal Linear Subspace | https://proofwiki.org/wiki/Extension_Theorem_for_Positive_Linear_Functional_defined_on_Cofinal_Linear_Subspace | https://proofwiki.org/wiki/Extension_Theorem_for_Positive_Linear_Functional_defined_on_Cofinal_Linear_Subspace | [
"Positive Linear Functionals",
"Preordered Vector Spaces"
] | [
"Definition:Preordered Vector Space",
"Definition:Linear Subspace",
"Definition:Cofinal Subset/Preordered Vector Space",
"Definition:Positive Linear Functional",
"Definition:Positive Linear Functional"
] | [
"Definition:Minkowski Sum",
"Definition:Minkowski Sum"
] |
proofwiki-21987 | Extension Theorem for Positive Linear Functional defined on Cofinal Linear Subspace/Lemma 2 | :$q$ is a sublinear functional. | Let $x, z \in Z$.
Note that if $y, w \in Y$ have $y \succeq x$ and $w \succeq z$, we have $y + w \succeq x + z$.
We are then able to deduce that:
:$\set {\map {f_0} {y + w} : y, w \in Y \text { and } y \succeq x, \, w \succeq z} \subseteq \set {\map {f_0} y : y \in Y \text { and } y \succeq x + z}$
From Infimum of Sub... | :$q$ is a [[Definition:Sublinear Functional|sublinear functional]]. | Let $x, z \in Z$.
Note that if $y, w \in Y$ have $y \succeq x$ and $w \succeq z$, we have $y + w \succeq x + z$.
We are then able to deduce that:
:$\set {\map {f_0} {y + w} : y, w \in Y \text { and } y \succeq x, \, w \succeq z} \subseteq \set {\map {f_0} y : y \in Y \text { and } y \succeq x + z}$
From [[Infimum o... | Extension Theorem for Positive Linear Functional defined on Cofinal Linear Subspace/Lemma 2 | https://proofwiki.org/wiki/Extension_Theorem_for_Positive_Linear_Functional_defined_on_Cofinal_Linear_Subspace/Lemma_2 | https://proofwiki.org/wiki/Extension_Theorem_for_Positive_Linear_Functional_defined_on_Cofinal_Linear_Subspace/Lemma_2 | [] | [
"Definition:Sublinear Functional"
] | [
"Infimum of Subset",
"Definition:Linear Functional",
"Definition:Subadditive Function",
"Definition:Homogeneous Function/Positive Homogeneity",
"Definition:Preordered Vector Space",
"Definition:Infimum of Set/Real Numbers",
"Multiple of Infimum",
"Definition:Sublinear Functional"
] |
proofwiki-21988 | Proper Modular Ideal of Algebra is Contained in Maximal Ideal | Let $R$ be a ring.
Let $\struct {A, \ast}$ be an $R$-algebra.
Let $\struct {J, \ast}$ be a proper modular ideal of $\struct {A, \ast}$.
Then there exists a maximal ideal $M$ of $\struct {A, \ast}$ such that $J \subseteq M$. | Let $u \in A$ be such that:
:$a - u a \in J$ and $a - a u \in J$
for each $a \in A$.
Let $S$ be the set of ideals $I$ of $\struct {A, \ast}$ such that:
:$u \not \in I$ and $J \subseteq I$.
Consider the partially ordered set $\struct {S, \subseteq}$ where $\subseteq$ is the subset relation.
We use Zorn's Lemma on $S$.
L... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {A, \ast}$ be an [[Definition:Algebra over Ring|$R$-algebra]].
Let $\struct {J, \ast}$ be a [[Definition:Proper Ideal of Algebra|proper]] [[Definition:Modular Ideal of Algebra|modular ideal]] of $\struct {A, \ast}$.
Then there exists a [[Defini... | Let $u \in A$ be such that:
:$a - u a \in J$ and $a - a u \in J$
for each $a \in A$.
Let $S$ be the [[Definition:Set|set]] of [[Definition:Ideal of Algebra|ideals]] $I$ of $\struct {A, \ast}$ such that:
:$u \not \in I$ and $J \subseteq I$.
Consider the [[Definition:Partially Ordered Set|partially ordered set]] $\stru... | Proper Modular Ideal of Algebra is Contained in Maximal Ideal | https://proofwiki.org/wiki/Proper_Modular_Ideal_of_Algebra_is_Contained_in_Maximal_Ideal | https://proofwiki.org/wiki/Proper_Modular_Ideal_of_Algebra_is_Contained_in_Maximal_Ideal | [
"Modular Ideals of Algebras",
"Maximal Ideals of Algebras"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Algebra over Ring",
"Definition:Ideal of Algebra/Proper Ideal",
"Definition:Modular Ideal of Algebra",
"Definition:Maximal Ideal of Algebra"
] | [
"Definition:Set",
"Definition:Ideal of Algebra",
"Definition:Partially Ordered Set",
"Definition:Subset Relation",
"Zorn's Lemma",
"Definition:Chain (Order Theory)",
"Definition:Upper Bound of Set",
"Union of Chain of Ideals is Ideal/Algebra over Ring",
"Definition:Ideal of Algebra",
"Set Union Pr... |
proofwiki-21989 | Union of Chain of Submodules is Submodule | Let $R$ be a ring.
Let $M$ be an $R$-module.
Let $\NN$ be a chain of submodules with respect to inclusion.
Let:
:$N = \bigcup \NN$
Then $N$ is a submodule of $M$. | We use the Submodule Test. | Let $R$ be a [[Definition:Ring|ring]].
Let $M$ be an [[Definition:Module over Ring|$R$-module]].
Let $\NN$ be a [[Definition:Chain (Order Theory)|chain]] of [[Definition:Submodule|submodules]] with respect to [[Definition:Set Inclusion|inclusion]].
Let:
:$N = \bigcup \NN$
Then $N$ is a [[Definition:Submodule|submo... | We use the [[Submodule Test]]. | Union of Chain of Submodules is Submodule | https://proofwiki.org/wiki/Union_of_Chain_of_Submodules_is_Submodule | https://proofwiki.org/wiki/Union_of_Chain_of_Submodules_is_Submodule | [
"Submodules"
] | [
"Definition:Ring",
"Definition:Module over Ring",
"Definition:Chain (Order Theory)",
"Definition:Submodule",
"Definition:Subset",
"Definition:Submodule"
] | [
"Submodule Test",
"Submodule Test"
] |
proofwiki-21990 | Quotient Algebra is Algebra | Let $R$ be a ring.
Let $A$ be a $R$-algebra.
Let $I$ be an ideal of $A$.
Let $A/I$ be the quotient module of $A$ modulo $I$.
For $a, b \in A$, define $\ast : \paren {A/I}^2 \to A/I$ by:
:$\paren {a + I} \ast \paren {b + I} = a b + I$
Then $\ast$ is well-defined and $\struct {A/I, \ast}$ is an $R$-algebra. | === $\ast$ is Well-Defined ===
It suffices to show that if:
:$a - a' \in I$
and:
:$b - b' \in I$
then:
:$ab - a' b' \in I$
Suppose that:
:$a - a' = i_1 \in I$
and:
:$b - b' = i_2 \in I$
Then we have:
:$a' b' = \paren {a + i_1} \paren {b + i_2} = a b + i_1 b + a i_2 + i_1 i_2$
From the definition of an ideal, we have $... | Let $R$ be a [[Definition:Ring|ring]].
Let $A$ be a [[Definition:Algebra over Ring|$R$-algebra]].
Let $I$ be an [[Definition:Ideal of Algebra|ideal]] of $A$.
Let $A/I$ be the [[Definition:Quotient Module|quotient module of $A$ modulo $I$]].
For $a, b \in A$, define $\ast : \paren {A/I}^2 \to A/I$ by:
:$\paren {a +... | === $\ast$ is Well-Defined ===
It suffices to show that if:
:$a - a' \in I$
and:
:$b - b' \in I$
then:
:$ab - a' b' \in I$
Suppose that:
:$a - a' = i_1 \in I$
and:
:$b - b' = i_2 \in I$
Then we have:
:$a' b' = \paren {a + i_1} \paren {b + i_2} = a b + i_1 b + a i_2 + i_1 i_2$
From the definition of an [[Definition... | Quotient Algebra is Algebra | https://proofwiki.org/wiki/Quotient_Algebra_is_Algebra | https://proofwiki.org/wiki/Quotient_Algebra_is_Algebra | [
"Quotient Algebras"
] | [
"Definition:Ring",
"Definition:Algebra over Ring",
"Definition:Ideal of Algebra",
"Definition:Quotient Module",
"Definition:Algebra over Ring"
] | [
"Definition:Ideal of Algebra"
] |
proofwiki-21991 | Quotient Algebra is Unital iff Quotienting Ideal is Modular | Let $R$ be a ring.
Let $A$ be a $R$-algebra.
Let $I$ be an proper ideal of $A$.
Let $A/I$ be the quotient algebra of $A$ modulo $I$.
Then $A/I$ is unital {{iff}} $I$ is modular. | === Necessary Condition ===
Suppose that $A/I$ is unital.
That is, there exists $u \in I$ such that:
:$\paren {a + I} \paren {u + I} = \paren {u + I} \paren {a + I} = a + I$
for each $a \in A$.
That is:
:$a u + I = u a + I = a + I$
Hence, we have:
:$a - a u \in I$
and:
:$a - u a \in I$
for each $a \in A$.
So $I$ is a m... | Let $R$ be a [[Definition:Ring|ring]].
Let $A$ be a [[Definition:Algebra over Ring|$R$-algebra]].
Let $I$ be an [[Definition:Proper Ideal of Algebra|proper ideal]] of $A$.
Let $A/I$ be the [[Definition:Quotient Algebra|quotient algebra of $A$ modulo $I$]].
Then $A/I$ is [[Definition:Unital Algebra|unital]] {{iff}... | === Necessary Condition ===
Suppose that $A/I$ is [[Definition:Unital Algebra|unital]].
That is, there exists $u \in I$ such that:
:$\paren {a + I} \paren {u + I} = \paren {u + I} \paren {a + I} = a + I$
for each $a \in A$.
That is:
:$a u + I = u a + I = a + I$
Hence, we have:
:$a - a u \in I$
and:
:$a - u a \in I$... | Quotient Algebra is Unital iff Quotienting Ideal is Modular | https://proofwiki.org/wiki/Quotient_Algebra_is_Unital_iff_Quotienting_Ideal_is_Modular | https://proofwiki.org/wiki/Quotient_Algebra_is_Unital_iff_Quotienting_Ideal_is_Modular | [
"Quotient Algebras",
"Modular Ideals of Algebras"
] | [
"Definition:Ring",
"Definition:Algebra over Ring",
"Definition:Ideal of Algebra/Proper Ideal",
"Definition:Quotient Algebra",
"Definition:Unital Algebra",
"Definition:Modular Ideal of Algebra"
] | [
"Definition:Unital Algebra",
"Definition:Modular Ideal of Algebra"
] |
proofwiki-21992 | Ideal of Unital Algebra is Modular | Let $R$ be a ring.
Let $A$ be a unital $R$-algebra.
Let $I$ be an ideal of $A$.
Then $I$ is a modular ideal of $A$. | Let ${\mathbf 1}_A$ be the identity element of $A$.
Let ${\mathbf 0}_A$ be the zero vector of $A$.
Then we have:
:$a {\mathbf 1}_A = {\mathbf 1}_A a$
for each $a \in A$.
Hence we have:
:$a - {\mathbf 1}_A a = {\mathbf 0}_A$
and:
:$a - a {\mathbf 1}_A = {\mathbf 0}_A$
Because $I$ is a submodule of $A$:
:${\mathbf 0}_A ... | Let $R$ be a [[Definition:Ring|ring]].
Let $A$ be a [[Definition:Unital Algebra|unital $R$-algebra]].
Let $I$ be an [[Definition:Ideal of Algebra|ideal]] of $A$.
Then $I$ is a [[Definition:Modular Ideal of Algebra|modular ideal of $A$]]. | Let ${\mathbf 1}_A$ be the [[Definition:Identity Element|identity element]] of $A$.
Let ${\mathbf 0}_A$ be the [[Definition:Zero Vector|zero vector]] of $A$.
Then we have:
:$a {\mathbf 1}_A = {\mathbf 1}_A a$
for each $a \in A$.
Hence we have:
:$a - {\mathbf 1}_A a = {\mathbf 0}_A$
and:
:$a - a {\mathbf 1}_A = {\ma... | Ideal of Unital Algebra is Modular | https://proofwiki.org/wiki/Ideal_of_Unital_Algebra_is_Modular | https://proofwiki.org/wiki/Ideal_of_Unital_Algebra_is_Modular | [
"Algebras",
"Modular Ideals of Algebras",
"Unital Algebras"
] | [
"Definition:Ring",
"Definition:Unital Algebra",
"Definition:Ideal of Algebra",
"Definition:Modular Ideal of Algebra"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Zero Vector",
"Definition:Submodule",
"Definition:Modular Ideal of Algebra"
] |
proofwiki-21993 | Quotient Normed Algebra is Normed Algebra | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a normed division ring.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra.
Let $I$ be a closed ideal of $A$.
Let $A/I$ be the quotient algebra of $A$ modulo $I$.
Let $\norm {\, \cdot \,}_{A/I}$ be the quotient norm.
Then $\struct {A/I, \norm {\, \cdot \,}_{A/I} }$ ... | Since $I$ is a closed linear subspace of $A$, Quotient Norm is Norm shows that $\struct {A/I, \norm {\, \cdot \,}_{A/I} }$ is a normed vector space.
It remains to show that for $a, b \in A$ we have:
:$\norm {a b + I}_{A/I} \le \norm {a + I}_{A/I} \norm {b + I}_{A/I}$
Let $\epsilon > 0$.
By the definition of the quotien... | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]].
Let $I$ be a [[Definition:Closed Set|closed]] [[Definition:Ideal of Algebra|ideal]] of $A$.
Let $A/I$ be the [[Definition... | Since $I$ is a [[Definition:Closed Linear Subspace|closed linear subspace]] of $A$, [[Quotient Norm is Norm]] shows that $\struct {A/I, \norm {\, \cdot \,}_{A/I} }$ is a [[Definition:Normed Vector Space|normed vector space]].
It remains to show that for $a, b \in A$ we have:
:$\norm {a b + I}_{A/I} \le \norm {a + I}_{... | Quotient Normed Algebra is Normed Algebra | https://proofwiki.org/wiki/Quotient_Normed_Algebra_is_Normed_Algebra | https://proofwiki.org/wiki/Quotient_Normed_Algebra_is_Normed_Algebra | [
"Normed Algebras"
] | [
"Definition:Normed Division Ring",
"Definition:Normed Algebra",
"Definition:Closed Set",
"Definition:Ideal of Algebra",
"Definition:Quotient Algebra",
"Definition:Quotient Norm",
"Definition:Normed Algebra"
] | [
"Definition:Closed Linear Subspace",
"Quotient Norm is Norm",
"Definition:Normed Vector Space",
"Definition:Quotient Norm",
"Definition:Ideal of Algebra",
"Definition:Norm/Algebra",
"Definition:Normed Algebra"
] |
proofwiki-21994 | Norm of Identity Element in Normed Algebra Lower Bounded by One | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a normed division ring.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $R$ with identity element ${\mathbf 1}_A \ne {\mathbf 0}_A$.
Then $1 \le \norm { {\mathbf 1}_A}$. | By the definition of a algebra norm, we have:
:$\norm { {\mathbf 1}_A} = \norm { {\mathbf 1}_A^2} \le \norm { {\mathbf 1}_A}^2$
From {{NormAxiomVector|1}}, we have $\norm { {\mathbf 1}_A} \ne 0$, we have:
:$1 \le \norm { {\mathbf 1}_A}$
{{qed}}
Category:Normed Algebras
7d7on612eusvxfg1spba7j2g7nt27tg | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $R$ with [[Definition:Identity Element|identity element]] ${\mathbf 1}_A \ne {\mathbf 0}_A$.
Then $1 \le \norm { {\ma... | By the definition of a [[Definition:Norm on Algebra|algebra norm]], we have:
:$\norm { {\mathbf 1}_A} = \norm { {\mathbf 1}_A^2} \le \norm { {\mathbf 1}_A}^2$
From {{NormAxiomVector|1}}, we have $\norm { {\mathbf 1}_A} \ne 0$, we have:
:$1 \le \norm { {\mathbf 1}_A}$
{{qed}}
[[Category:Normed Algebras]]
7d7on612eusvx... | Norm of Identity Element in Normed Algebra Lower Bounded by One | https://proofwiki.org/wiki/Norm_of_Identity_Element_in_Normed_Algebra_Lower_Bounded_by_One | https://proofwiki.org/wiki/Norm_of_Identity_Element_in_Normed_Algebra_Lower_Bounded_by_One | [
"Normed Algebras"
] | [
"Definition:Normed Division Ring",
"Definition:Normed Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Norm/Algebra",
"Category:Normed Algebras"
] |
proofwiki-21995 | Quotient Normed Algebra of Unital Normed Algebra is Unital Normed Algebra | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a normed division ring.
Let $\struct {A, \norm {\, \cdot \,} }$ be a unital normed algebra with identity element ${\mathbf 1}_A$.
Let $I$ be a closed proper ideal of $A$.
Let $\struct {A/I, \norm {\, \cdot \,}_{A/I} }$ be the quotient normed algebra of $A$ modulo $I$.
Then:... | From Norm of Identity Element in Normed Algebra Lower Bounded by One, we have $1 \le \norm { {\mathbf 1}_A + I}_{A/I}$.
Since $I$ is a linear subspace of $A$, we have ${\mathbf 0}_A \in I$.
Hence we have:
:$\ds \inf_{i \mathop \in I} \norm { {\mathbf 1}_A + i} \le \norm { {\mathbf 1}_A + {\mathbf 0}_A} = \norm { {\mat... | Let $\struct {R, \norm {\, \cdot \,}_R}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Normed Algebra|unital normed algebra]] with [[Definition:Identity Element|identity element]] ${\mathbf 1}_A$.
Let $I$ be a [[Definition:Closed Set|c... | From [[Norm of Identity Element in Normed Algebra Lower Bounded by One]], we have $1 \le \norm { {\mathbf 1}_A + I}_{A/I}$.
Since $I$ is a [[Definition:Linear Subspace|linear subspace]] of $A$, we have ${\mathbf 0}_A \in I$.
Hence we have:
:$\ds \inf_{i \mathop \in I} \norm { {\mathbf 1}_A + i} \le \norm { {\mathbf ... | Quotient Normed Algebra of Unital Normed Algebra is Unital Normed Algebra | https://proofwiki.org/wiki/Quotient_Normed_Algebra_of_Unital_Normed_Algebra_is_Unital_Normed_Algebra | https://proofwiki.org/wiki/Quotient_Normed_Algebra_of_Unital_Normed_Algebra_is_Unital_Normed_Algebra | [
"Quotient Normed Algebras",
"Unital Normed Algebras"
] | [
"Definition:Normed Division Ring",
"Definition:Unital Normed Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Closed Set",
"Definition:Ideal of Algebra/Proper Ideal",
"Definition:Quotient Normed Algebra"
] | [
"Norm of Identity Element in Normed Algebra Lower Bounded by One",
"Definition:Linear Subspace",
"Definition:Unital Normed Algebra",
"Category:Quotient Normed Algebras",
"Category:Unital Normed Algebras"
] |
proofwiki-21996 | Maximal Ideal of Unital Commutative Banach Algebra is Kernel of Character | Let $\struct {A, \norm {\, \cdot \,} }$ be a commutative unital Banach algebra over $\C$.
Let $M$ be a maximal ideal of $A$.
Then there exists a character $\phi$ such that:
:$\ker \phi = M$ | From Maximal Ideal in Unital Banach Algebra is Closed, $M$ is closed.
Let $A/M$ be the quotient algebra of $A$ modulo $M$.
From Quotient Normed Algebra of Unital Normed Algebra is Unital Normed Algebra, $A/M$ is a unital normed algebra.
From Quotient Algebra of Commutative Algebra is Commutative, $A/M$ is a commutativ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $M$ be a [[Definition:Maximal Ideal of Algebra|maximal ideal]] of $A$.
Then there exists a [[Definition:Character (Banach Algebra)... | From [[Maximal Ideal in Unital Banach Algebra is Closed]], $M$ is [[Definition:Closed Set|closed]].
Let $A/M$ be the [[Definition:Quotient Algebra|quotient algebra of $A$ modulo $M$]].
From [[Quotient Normed Algebra of Unital Normed Algebra is Unital Normed Algebra]], $A/M$ is a [[Definition:Unital Normed Algebra|uni... | Maximal Ideal of Unital Commutative Banach Algebra is Kernel of Character | https://proofwiki.org/wiki/Maximal_Ideal_of_Unital_Commutative_Banach_Algebra_is_Kernel_of_Character | https://proofwiki.org/wiki/Maximal_Ideal_of_Unital_Commutative_Banach_Algebra_is_Kernel_of_Character | [
"Commutative Banach Algebras",
"Unital Banach Algebras",
"Characters (Banach Algebras)",
"Commutative Banach Algebras"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unital Banach Algebra",
"Definition:Maximal Ideal of Algebra",
"Definition:Character (Banach Algebra)"
] | [
"Maximal Ideal in Unital Banach Algebra is Closed",
"Definition:Closed Set",
"Definition:Quotient Algebra",
"Quotient Normed Algebra of Unital Normed Algebra is Unital Normed Algebra",
"Definition:Unital Normed Algebra",
"Quotient Algebra of Commutative Algebra is Commutative",
"Definition:Commutative A... |
proofwiki-21997 | Element of Unital Commutative Algebra Invertible iff not Contained in Maximal Ideal | Let $R$ be a ring.
Let $A$ be an unital commutative $R$-algebra.
Let $a \in A$.
Then $a$ is invertible {{iff}} there does not exist a maximal ideal $M$ with $a \in M$. | === Necessary Condition ===
We prove the contrapositive.
Then, by Proof by Contraposition we will done.
We will prove that if $a$ is in a maximal ideal then $a$ is not invertible.
Let $M$ be a maximal ideal with $a \in M$.
{{AimForCont}} there exists $b \in A$ with $a b = {\mathbf 1}_A$.
Since $M$ is an ideal, we hav... | Let $R$ be a [[Definition:Ring|ring]].
Let $A$ be an [[Definition:Unital Algebra over Ring|unital]] [[Definition:Commutative Algebra (Abstract Algebra)|commutative $R$-algebra]].
Let $a \in A$.
Then $a$ is [[Definition:Invertible Element|invertible]] {{iff}} there does not exist a [[Definition:Maximal Ideal of Alge... | === Necessary Condition ===
We prove the [[Definition:Contrapositive|contrapositive]].
Then, by [[Proof by Contraposition]] we will done.
We will prove that if $a$ is in a [[Definition:Maximal Ideal of Algebra|maximal ideal]] then $a$ is not [[Definition:Invertible Element|invertible]].
Let $M$ be a [[Definition:M... | Element of Unital Commutative Algebra Invertible iff not Contained in Maximal Ideal | https://proofwiki.org/wiki/Element_of_Unital_Commutative_Algebra_Invertible_iff_not_Contained_in_Maximal_Ideal | https://proofwiki.org/wiki/Element_of_Unital_Commutative_Algebra_Invertible_iff_not_Contained_in_Maximal_Ideal | [
"Commutative Algebras",
"Maximal Ideals of Algebras"
] | [
"Definition:Ring",
"Definition:Unital Algebra",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Invertible Element",
"Definition:Maximal Ideal of Algebra"
] | [
"Definition:Contrapositive Statement",
"Proof by Contraposition",
"Definition:Maximal Ideal of Algebra",
"Definition:Invertible Element",
"Definition:Maximal Ideal of Algebra",
"Definition:Ideal of Algebra",
"Definition:Ideal of Algebra/Proper Ideal",
"Definition:Invertible Element",
"Definition:Max... |
proofwiki-21998 | Characterization of Maximal Ideal of Unital Commutative Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a commutative unital Banach algebra over $\C$.
Let $M \subseteq A$.
Then $M$ is maximal {{iff}} there exists a character $\phi$ with $M = \ker \phi$. | === Necessary Condition ===
This is Maximal Ideal of Unital Commutative Banach Algebra is Kernel of Character.
{{qed|lemma}} | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $M \subseteq A$.
Then $M$ is [[Definition:Maximal Ideal of Algebra|maximal]] {{iff}} there exists a [[Definition:Character (Banach ... | === Necessary Condition ===
This is [[Maximal Ideal of Unital Commutative Banach Algebra is Kernel of Character]].
{{qed|lemma}} | Characterization of Maximal Ideal of Unital Commutative Banach Algebra | https://proofwiki.org/wiki/Characterization_of_Maximal_Ideal_of_Unital_Commutative_Banach_Algebra | https://proofwiki.org/wiki/Characterization_of_Maximal_Ideal_of_Unital_Commutative_Banach_Algebra | [
"Maximal Ideals of Algebras",
"Commutative Banach Algebras"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unital Banach Algebra",
"Definition:Maximal Ideal of Algebra",
"Definition:Character (Banach Algebra)"
] | [
"Maximal Ideal of Unital Commutative Banach Algebra is Kernel of Character"
] |
proofwiki-21999 | Element of Unital Commutative Banach Algebra is Invertible iff not in Kernel of Character | Let $\struct {A, \norm {\, \cdot \,} }$ be a commutative unital Banach algebra over $\C$.
Let $a \in A$.
Then $x$ is invertible {{iff}}:
:for every character $\phi$ of $A$, we have $a \not \in \ker \phi$. | From Element of Unital Commutative Algebra Invertible iff not Contained in Maximal Ideal, we have that $a$ is invertible {{iff}}:
:for every maximal ideal $M$ of $A$, we have $a \not \in M$.
From Characterization of Maximal Ideal of Unital Commutative Banach Algebra, we have that:
:$M$ is a maximal ideal of $A$ {{iff}... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $a \in A$.
Then $x$ is [[Definition:Invertible Element|invertible]] {{iff}}:
:for every [[Definition:Character (Banach Algebra)|cha... | From [[Element of Unital Commutative Algebra Invertible iff not Contained in Maximal Ideal]], we have that $a$ is [[Definition:Invertible Element|invertible]] {{iff}}:
:for every [[Definition:Maximal Ideal of Algebra|maximal ideal]] $M$ of $A$, we have $a \not \in M$.
From [[Characterization of Maximal Ideal of Unita... | Element of Unital Commutative Banach Algebra is Invertible iff not in Kernel of Character | https://proofwiki.org/wiki/Element_of_Unital_Commutative_Banach_Algebra_is_Invertible_iff_not_in_Kernel_of_Character | https://proofwiki.org/wiki/Element_of_Unital_Commutative_Banach_Algebra_is_Invertible_iff_not_in_Kernel_of_Character | [
"Commutative Banach Algebras"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unital Banach Algebra",
"Definition:Invertible Element",
"Definition:Character (Banach Algebra)"
] | [
"Element of Unital Commutative Algebra Invertible iff not Contained in Maximal Ideal",
"Definition:Invertible Element",
"Definition:Maximal Ideal of Algebra",
"Characterization of Maximal Ideal of Unital Commutative Banach Algebra",
"Definition:Maximal Ideal of Algebra",
"Definition:Character (Banach Alge... |
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