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-19400 | Minimal Polynomial Exists | Let $L / K$ be a field extension.
Let $\alpha \in L$ be algebraic over $K$.
Then there exists a minimal polynomial $f \in K \sqbrk x$ for $\alpha$ over $K$. | According to definition 2 of minimal polynomial, we ought to find $f \in K \sqbrk x$ such that:
:$f \in K \sqbrk x$ is an irreducible, monic polynomial such that $\map f \alpha = 0$
Since $\alpha$ is algebraic over $K$, there is some $f \in K \sqbrk x$ such that $\map f \alpha = 0$.
By Polynomial Forms over Field form ... | Let $L / K$ be a [[Definition:Field Extension|field extension]].
Let $\alpha \in L$ be [[Definition:Algebraic Element of Field Extension|algebraic]] over $K$.
Then there exists a [[Definition:Minimal Polynomial|minimal polynomial]] $f \in K \sqbrk x$ for $\alpha$ over $K$. | According to [[Definition:Minimal Polynomial/Definition 2|definition 2 of minimal polynomial]], we ought to find $f \in K \sqbrk x$ such that:
:$f \in K \sqbrk x$ is an [[Definition:Irreducible Polynomial|irreducible]], [[Definition:Monic Polynomial|monic polynomial]] such that $\map f \alpha = 0$
Since $\alpha$ is ... | Minimal Polynomial Exists/Proof 2 | https://proofwiki.org/wiki/Minimal_Polynomial_Exists | https://proofwiki.org/wiki/Minimal_Polynomial_Exists/Proof_2 | [
"Minimal Polynomial Exists",
"Minimal Polynomials"
] | [
"Definition:Field Extension",
"Definition:Algebraic Element of Field Extension",
"Definition:Minimal Polynomial"
] | [
"Definition:Minimal Polynomial/Definition 2",
"Definition:Irreducible Polynomial",
"Definition:Monic Polynomial",
"Definition:Algebraic Element of Field Extension",
"Polynomial Forms over Field form Principal Ideal Domain/Corollary 3",
"Definition:Complete Factorization",
"Definition:Irreducible Polynom... |
proofwiki-19401 | Congruence Modulo Principal Ideal of P-adic Integers | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $\Z_p$ be the $p$-adic integers.
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map B a$ denote the open ball of center $a$ of radius $\epsilon$.
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map { {B_\epsilon}^-} ... | === Condition (1) is Equivalent to Condition (3) ===
We have:
{{begin-eqn}}
{{eqn | l = x \equiv y \pmod{p^{k+1} \Z_p}
| o = \iff
| r = x - y \in p^{k+1} \Z_p
| c = {{Defof|Congruence Modulo Ideal}}
}}
{{eqn | o = \iff
| r = x + p^{k+1} \Z_p = y + p^{k+1} \Z_p
| c = Cosets are Equal iff Pr... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map B a$ denote the [[Definition:Open... | === Condition (1) is Equivalent to Condition (3) ===
We have:
{{begin-eqn}}
{{eqn | l = x \equiv y \pmod{p^{k+1} \Z_p}
| o = \iff
| r = x - y \in p^{k+1} \Z_p
| c = {{Defof|Congruence Modulo Ideal}}
}}
{{eqn | o = \iff
| r = x + p^{k+1} \Z_p = y + p^{k+1} \Z_p
| c = [[Cosets are Equal iff ... | Congruence Modulo Principal Ideal of P-adic Integers | https://proofwiki.org/wiki/Congruence_Modulo_Principal_Ideal_of_P-adic_Integers | https://proofwiki.org/wiki/Congruence_Modulo_Principal_Ideal_of_P-adic_Integers | [
"P-adic Integers"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:P-adic Integer",
"Definition:Open Ball/P-adic Numbers",
"Definition:Open Ball/P-adic Numbers/Center",
"Definition:Open Ball/P-adic Numbers/Radius",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Closed Ball/P-adi... | [
"Cosets are Equal iff Product with Inverse in Subgroup"
] |
proofwiki-19402 | Congruence Modulo Equivalence for Integers in P-adic Integers | Let $\Z_p$ be the $p$-adic integers for some prime $p$.
For any $a, b \in \Z_p$ and $n \in \N$, let $x \equiv y \pmod{p^n \Z_p}$ denote congruence modulo the principal ideal $p^n\Z_p$.
For any integers $a, b \in \Z$ and $n \in \N$, let $x \equiv y \pmod{p^n}$ denote congruence modulo integer $p^n$.
Let $x, y \in \Z$ be... | === Lemma ===
{{:Congruence Modulo Equivalence for Integers in P-adic Integers/Lemma 1}}{{qed|lemma}}
We have:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| o = \equiv
| r = y
| rr= \pmod {p^k \Z_p}
}}
{{eqn | ll = \leadstoandfrom
| l = x - y
| o = \in
| r = p^k \Z_p
| c = {{Defo... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
For any $a, b \in \Z_p$ and $n \in \N$, let $x \equiv y \pmod{p^n \Z_p}$ denote [[Definition:Congruence Modulo Ideal|congruence]] modulo the [[Definition:Principal Ideal|principal ideal]] $p^n\Z_p$.
For a... | === [[Congruence Modulo Equivalence for Integers in P-adic Integers/Lemma 1|Lemma]] ===
{{:Congruence Modulo Equivalence for Integers in P-adic Integers/Lemma 1}}{{qed|lemma}}
We have:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| o = \equiv
| r = y
| rr= \pmod {p^k \Z_p}
}}
{{eqn | ll = \leadstoandfr... | Congruence Modulo Equivalence for Integers in P-adic Integers | https://proofwiki.org/wiki/Congruence_Modulo_Equivalence_for_Integers_in_P-adic_Integers | https://proofwiki.org/wiki/Congruence_Modulo_Equivalence_for_Integers_in_P-adic_Integers | [
"P-adic Integers",
"Congruence Modulo Equivalence for Integers in P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Congruence Modulo Ideal",
"Definition:Principal Ideal",
"Definition:Integer",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Integer"
] | [
"Congruence Modulo Equivalence for Integers in P-adic Integers/Lemma 1",
"Congruence Modulo Equivalence for Integers in P-adic Integers/Lemma 1",
"Category:P-adic Integers",
"Category:Congruence Modulo Equivalence for Integers in P-adic Integers"
] |
proofwiki-19403 | Congruence Modulo Equivalence for Integers in P-adic Integers/Lemma 1 | :$\forall a \in \Z: \dfrac a {p^k} \in \Z_p \iff \dfrac a {p^k} \in \Z$ | === Necessary Condition ===
Let $a \in \Z$.
We have:
{{begin-eqn}}
{{eqn | l = \dfrac a {p^k}
| o = \in
| r = \Z_p
}}
{{eqn | ll = \leadsto
| l = \dfrac a {p^k}
| o = \in
| r = \Z_p \cap \Q
}}
{{eqn | ll = \leadsto
| q = \exists c, d \in \Z : p \nmid d
| l = \dfrac a {p^k}
... | :$\forall a \in \Z: \dfrac a {p^k} \in \Z_p \iff \dfrac a {p^k} \in \Z$ | === Necessary Condition ===
Let $a \in \Z$.
We have:
{{begin-eqn}}
{{eqn | l = \dfrac a {p^k}
| o = \in
| r = \Z_p
}}
{{eqn | ll = \leadsto
| l = \dfrac a {p^k}
| o = \in
| r = \Z_p \cap \Q
}}
{{eqn | ll = \leadsto
| q = \exists c, d \in \Z : p \nmid d
| l = \dfrac a {p^k}
... | Congruence Modulo Equivalence for Integers in P-adic Integers/Lemma 1 | https://proofwiki.org/wiki/Congruence_Modulo_Equivalence_for_Integers_in_P-adic_Integers/Lemma_1 | https://proofwiki.org/wiki/Congruence_Modulo_Equivalence_for_Integers_in_P-adic_Integers/Lemma_1 | [
"Congruence Modulo Equivalence for Integers in P-adic Integers"
] | [] | [
"Characterization of Rational P-adic Integer",
"Euclid's Lemma"
] |
proofwiki-19404 | Skewness of Binomial Distribution | Let $X$ be a discrete random variable with a Binomial distribution with parameter $n$ and $p$ for some $n \in \N$ and $0 \le p \le 1$.
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac {1 - 2 p} {\sqrt {n p q} }$
where $q = 1 - p$. | From Skewness in terms of Non-Central Moments:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where $\mu$ is the mean of $X$, and $\sigma$ the standard deviation.
We have, by Expectation of Binomial Distribution:
:$\mu = n p$
By Variance of Binomial Distribution, we also have:
:$\var X = \s... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with a [[Definition:Binomial Distribution|Binomial distribution with parameter $n$ and $p$]] for some $n \in \N$ and $0 \le p \le 1$.
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac {1 - 2 p} {\... | From [[Skewness in terms of Non-Central Moments]]:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where $\mu$ is the [[Definition:Expectation|mean]] of $X$, and $\sigma$ the [[Definition:Standard Deviation|standard deviation]].
We have, by [[Expectation of Binomial Distribution]]:
:$\mu ... | Skewness of Binomial Distribution | https://proofwiki.org/wiki/Skewness_of_Binomial_Distribution | https://proofwiki.org/wiki/Skewness_of_Binomial_Distribution | [
"Binomial Distribution",
"Skewness"
] | [
"Definition:Random Variable/Discrete",
"Definition:Binomial Distribution",
"Definition:Skewness"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Binomial Distribution",
"Variance of Binomial Distribution",
"Definition:Moment Generating Function",
"Moment Generating Function of Binomial Distribution",
"Moment in terms of Moment... |
proofwiki-19405 | Rational Number can be Expressed as Simple Finite Continued Fraction | Let $q \in \Q$ be a rational number.
Then $q$ can be expressed as a simple finite continued fraction. | Let $q = \dfrac a b$ be a rational number expressed in canonical form.
That is $b > 0$ and $a \perp b = 1$.
By the Euclidean Algorithm, we have:
{{begin-eqn}}
{{eqn | l = a
| r = q_1 b + r_1,
| rr= 0 < r_1 < b
| c = or $\dfrac a b = q_1 + \dfrac {r_1} b$
}}
{{eqn | l = b
| r = q_2 r_1 + r_2,
... | Let $q \in \Q$ be a [[Definition:Rational Number|rational number]].
Then $q$ can be expressed as a [[Definition:Simple Finite Continued Fraction|simple finite continued fraction]]. | Let $q = \dfrac a b$ be a [[Definition:Rational Number|rational number]] expressed in [[Definition:Canonical Form of Rational Number|canonical form]].
That is $b > 0$ and $a \perp b = 1$.
By the [[Euclidean Algorithm]], we have:
{{begin-eqn}}
{{eqn | l = a
| r = q_1 b + r_1,
| rr= 0 < r_1 < b
| c =... | Rational Number can be Expressed as Simple Finite Continued Fraction | https://proofwiki.org/wiki/Rational_Number_can_be_Expressed_as_Simple_Finite_Continued_Fraction | https://proofwiki.org/wiki/Rational_Number_can_be_Expressed_as_Simple_Finite_Continued_Fraction | [
"Simple Continued Fractions",
"Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Simple Continued Fraction/Finite"
] | [
"Definition:Rational Number",
"Definition:Rational Number/Canonical Form",
"Euclidean Algorithm",
"Definition:Simple Continued Fraction/Finite"
] |
proofwiki-19406 | Principal Right Ideal is Right Ideal | Let $\struct {R, +, \circ}$ be a ring with unity.
Let $a \in R$.
Let $aR$ be the principal right ideal of $R$ generated by $a$.
Then $aR$ is an right ideal of $R$. | We establish that $aR$ is an right ideal of $R$, by verifying the conditions of Test for Right Ideal.
$aR \ne \O$, as $a \circ 1_R = a \in aR$.
Let $x, y \in ar$.
Then:
{{begin-eqn}}
{{eqn | q = \exists r, s \in R
| l = x
| r = a \circ r, y = a \circ s
| c =
}}
{{eqn | ll= \leadsto
| l = x + \p... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $a \in R$.
Let $aR$ be the [[Definition:Principal Right Ideal of Ring|principal right ideal]] of $R$ generated by $a$.
Then $aR$ is an [[Definition:Right Ideal of Ring|right ideal]] of $R$. | We establish that $aR$ is an [[Definition:Right Ideal of Ring|right ideal]] of $R$, by verifying the conditions of [[Test for Right Ideal]].
$aR \ne \O$, as $a \circ 1_R = a \in aR$.
Let $x, y \in ar$.
Then:
{{begin-eqn}}
{{eqn | q = \exists r, s \in R
| l = x
| r = a \circ r, y = a \circ s
| c... | Principal Right Ideal is Right Ideal | https://proofwiki.org/wiki/Principal_Right_Ideal_is_Right_Ideal | https://proofwiki.org/wiki/Principal_Right_Ideal_is_Right_Ideal | [
"Ideal Theory"
] | [
"Definition:Ring with Unity",
"Definition:Principal Right Ideal of Ring",
"Definition:Ideal of Ring/Right Ideal"
] | [
"Definition:Ideal of Ring/Right Ideal",
"Test for Right Ideal",
"Product with Ring Negative",
"Test for Right Ideal",
"Definition:Ideal of Ring/Right Ideal",
"Category:Ideal Theory"
] |
proofwiki-19407 | Principal Left Ideal is Left Ideal | Let $\struct {R, +, \circ}$ be a ring with unity.
Let $a \in R$.
Let $Ra$ be the principal left ideal of $R$ generated by $a$.
Then $Ra$ is an left ideal of $R$. | We establish that $Ra$ is an left ideal of $R$, by verifying the conditions of Test for Left Ideal.
$Ra \ne \O$, as $1_R \circ a = a \in Ra$.
Let $x, y \in Ra$.
Then:
{{begin-eqn}}
{{eqn | q = \exists r, s \in R
| l = x
| r = r \circ a, y = s \circ a
| c =
}}
{{eqn | ll= \leadsto
| l = x + \par... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $a \in R$.
Let $Ra$ be the [[Definition:Principal Left Ideal of Ring|principal left ideal]] of $R$ generated by $a$.
Then $Ra$ is an [[Definition:Left Ideal of Ring|left ideal]] of $R$. | We establish that $Ra$ is an [[Definition:Left Ideal of Ring|left ideal]] of $R$, by verifying the conditions of [[Test for Left Ideal]].
$Ra \ne \O$, as $1_R \circ a = a \in Ra$.
Let $x, y \in Ra$.
Then:
{{begin-eqn}}
{{eqn | q = \exists r, s \in R
| l = x
| r = r \circ a, y = s \circ a
| c = ... | Principal Left Ideal is Left Ideal | https://proofwiki.org/wiki/Principal_Left_Ideal_is_Left_Ideal | https://proofwiki.org/wiki/Principal_Left_Ideal_is_Left_Ideal | [
"Ideal Theory"
] | [
"Definition:Ring with Unity",
"Definition:Principal Left Ideal of Ring",
"Definition:Ideal of Ring/Left Ideal"
] | [
"Definition:Ideal of Ring/Left Ideal",
"Test for Left Ideal",
"Product with Ring Negative",
"Test for Left Ideal",
"Definition:Ideal of Ring/Left Ideal"
] |
proofwiki-19408 | Principal Ideal of Commutative Ring | Let $\struct {R, +, \circ}$ be a commutative ring with unity.
Let $a \in R$.
Let $Ra$ be the principal left ideal of $R$ generated by $a$.
Let $aR$ be the principal right ideal of $R$ generated by $a$.
Let $\ideal a$ be the principal ideal of $R$ generated by $a$.
Then $Ra = \ideal a = aR$. | By definition of principal left ideal:
:$Ra = \set{r \circ a: r \in R}$
By definition of commutative ring with unity and center of ring:
:$a$ is in the center of $R$
From Principal Ideal from Element in Center of Ring:
:$\ideal a = R \circ a = \set{r \circ a: r \in R}$
Hence:
:$Ra = \ideal a$
We have:
{{begin-eqn}}
{{e... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $a \in R$.
Let $Ra$ be the [[Definition:Principal Left Ideal of Ring|principal left ideal]] of $R$ generated by $a$.
Let $aR$ be the [[Definition:Principal Right Ideal of Ring|principal right ideal]] of $R$ ... | By definition of [[Definition:Principal Left Ideal of Ring|principal left ideal]]:
:$Ra = \set{r \circ a: r \in R}$
By definition of [[Definition:Commutative Ring with Unity|commutative ring with unity]] and [[Definition:Center of Ring|center of ring]]:
:$a$ is in the [[Definition:Center of Ring|center]] of $R$
From ... | Principal Ideal of Commutative Ring | https://proofwiki.org/wiki/Principal_Ideal_of_Commutative_Ring | https://proofwiki.org/wiki/Principal_Ideal_of_Commutative_Ring | [
"Ideal Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Principal Left Ideal of Ring",
"Definition:Principal Right Ideal of Ring",
"Definition:Principal Ideal of Ring"
] | [
"Definition:Principal Left Ideal of Ring",
"Definition:Commutative and Unitary Ring",
"Definition:Center (Abstract Algebra)/Ring",
"Definition:Center (Abstract Algebra)/Ring",
"Principal Ideal from Element in Center of Ring",
"Definition:Commutative/Operation",
"Definition:Ring (Abstract Algebra)/Produc... |
proofwiki-19409 | Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous | Linear transformations between finite-dimensional normed vector spaces are continuous. | We have that Norms on Finite-Dimensional Real Vector Space are Equivalent.
Choose the Euclidean norm.
Let $X = \struct {\R^n, \norm {\, \cdot \,}_2}$ and $Y = \struct {\R^m, \norm {\, \cdot \,}_2}$ be normed vector spaces.
Let the matrix $A \in \R^{m \times n}$ be given by:
:$A = \begin {bmatrix}
a_{1 1} & \cdots & a_{... | [[Definition:Set of All Linear Transformations/Vector Space|Linear transformations]] between [[Definition:Finite Dimensional Vector Space|finite-dimensional]] [[Definition:Normed Vector Space|normed vector spaces]] are [[Definition:Continuous Linear Transformation Space|continuous]]. | We have that [[Norms on Finite-Dimensional Real Vector Space are Equivalent]].
Choose the [[Definition:Euclidean Norm|Euclidean norm]].
Let $X = \struct {\R^n, \norm {\, \cdot \,}_2}$ and $Y = \struct {\R^m, \norm {\, \cdot \,}_2}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let the [[Definition:Matr... | Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous | https://proofwiki.org/wiki/Linear_Transformations_between_Finite-Dimensional_Normed_Vector_Spaces_are_Continuous | https://proofwiki.org/wiki/Linear_Transformations_between_Finite-Dimensional_Normed_Vector_Spaces_are_Continuous | [
"Operator Theory",
"Continuous Mappings",
"Linear Transformations"
] | [
"Definition:Set of All Linear Transformations/Vector Space",
"Definition:Dimension of Vector Space/Finite",
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space"
] | [
"Norms on Finite-Dimensional Real Vector Space are Equivalent",
"Definition:Euclidean Norm",
"Definition:Normed Vector Space",
"Definition:Matrix",
"Set of Linear Transformations is Isomorphic to Matrix Space",
"Definition:Linear Transformation/Vector Space",
"Cauchy's Inequality",
"Continuity of Line... |
proofwiki-19410 | Equivalence of Formulations of Axiom of Unions | In the context of class theory, the following formulations of the '''{{axiom-link|Unions}}''' are equivalent:
=== Formulation 1 ===
{{:Axiom:Axiom of Unions (Set Theory)}}
=== Formulation 2 ===
{{:Axiom:Axiom of Unions (Class Theory)}} | It is assumed throughout that the {{axiom-link|Extensionality}} and the {{axiom-link|Specification}} both hold.
=== Formulation $1$ implies Formulation $2$ ===
Let formulation $1$ be axiomatic:
{{:Axiom:Axiom of Unions (Set Theory)}}
Thus it is posited that for a given set of sets $A$ the union of $A$ exists:
:$x := \b... | In the context of [[Definition:Class Theory|class theory]], the following formulations of the '''{{axiom-link|Unions}}''' are [[Definition:Logical Equivalence|equivalent]]:
=== [[Axiom:Axiom of Unions (Set Theory)|Formulation 1]] ===
{{:Axiom:Axiom of Unions (Set Theory)}}
=== [[Axiom:Axiom of Unions (Class Theory)|Fo... | It is assumed throughout that the {{axiom-link|Extensionality}} and the {{axiom-link|Specification}} both hold.
=== Formulation $1$ implies Formulation $2$ ===
Let [[Axiom:Axiom of Unions (Set Theory)|formulation $1$]] be [[Definition:Axiom|axiomatic]]:
{{:Axiom:Axiom of Unions (Set Theory)}}
Thus it is posited tha... | Equivalence of Formulations of Axiom of Unions/Proof | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Unions | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Unions/Proof | [
"Axiom of Unions",
"Definition Equivalences",
"Equivalence of Formulations of Axiom of Unions"
] | [
"Definition:Class Theory",
"Definition:Logical Equivalence",
"Axiom:Axiom of Unions/Set Theory",
"Axiom:Axiom of Unions/Class Theory"
] | [
"Axiom:Axiom of Unions/Set Theory",
"Definition:Axiom",
"Definition:Set of Sets",
"Definition:Set Union/Set of Sets",
"Definition:Set",
"Axiom:Axiom of Unions/Class Theory",
"Definition:Axiom",
"Definition:Set",
"Axiom:Axiom of Unions/Class Theory",
"Axiom:Axiom of Unions/Set Theory",
"Class Uni... |
proofwiki-19411 | Intersection of Non-Empty Class is Set/Corollary | Let $x$ be a non-empty set.
Let $\bigcap x$ denote the intersection of $x$.
Then $\bigcap x$ is a set. | It is assumed that $x$ is an element of a basic universe.
Hence from the {{axiom-link|Transitivity}}, every set is a class.
Hence Intersection of Non-Empty Class is Set applies directly.
{{qed}} | Let $x$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]].
Let $\bigcap x$ denote the [[Definition:Intersection of Set of Sets|intersection]] of $x$.
Then $\bigcap x$ is a [[Definition:Set|set]]. | It is assumed that $x$ is an [[Definition:Element|element]] of a [[Definition:Basic Universe|basic universe]].
Hence from the {{axiom-link|Transitivity}}, every [[Definition:Set|set]] is a [[Definition:Class (Class Theory)|class]].
Hence [[Intersection of Non-Empty Class is Set]] applies directly.
{{qed}} | Intersection of Non-Empty Class is Set/Corollary | https://proofwiki.org/wiki/Intersection_of_Non-Empty_Class_is_Set/Corollary | https://proofwiki.org/wiki/Intersection_of_Non-Empty_Class_is_Set/Corollary | [
"Intersection of Non-Empty Class is Set"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Set Intersection/Set of Sets",
"Definition:Set"
] | [
"Definition:Element",
"Definition:Basic Universe",
"Definition:Set",
"Definition:Class (Class Theory)",
"Intersection of Non-Empty Class is Set"
] |
proofwiki-19412 | Class Union Distributes over Class Intersection | Let $A$, $B$ and $C$ be classes.
Then:
:$A \cup \paren {B \cap C} = \paren {A \cup B} \cap \paren {A \cup C}$
where:
:$A \cup B$ denotes class union
:$B \cap C$ denotes class intersection. | {{begin-eqn}}
{{eqn | o =
| r = x \in A \cup \paren {B \cap C}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \lor \paren {x \in B \land x \in C}
| c = {{Defof|Class Union}} and {{Defof|Class Intersection}}
}}
{{eqn | o = \leadstoandfrom
| r = \paren {x \in A \lor x \in B} \land \paren {x \in A \... | Let $A$, $B$ and $C$ be [[Definition:Class (Class Theory)|classes]].
Then:
:$A \cup \paren {B \cap C} = \paren {A \cup B} \cap \paren {A \cup C}$
where:
:$A \cup B$ denotes [[Definition:Class Union|class union]]
:$B \cap C$ denotes [[Definition:Class Intersection|class intersection]]. | {{begin-eqn}}
{{eqn | o =
| r = x \in A \cup \paren {B \cap C}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \lor \paren {x \in B \land x \in C}
| c = {{Defof|Class Union}} and {{Defof|Class Intersection}}
}}
{{eqn | o = \leadstoandfrom
| r = \paren {x \in A \lor x \in B} \land \paren {x \in A \... | Class Union Distributes over Class Intersection | https://proofwiki.org/wiki/Class_Union_Distributes_over_Class_Intersection | https://proofwiki.org/wiki/Class_Union_Distributes_over_Class_Intersection | [
"Class Intersection",
"Class Union",
"Examples of Distributive Operations"
] | [
"Definition:Class (Class Theory)",
"Definition:Class Union",
"Definition:Class Intersection"
] | [
"Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive"
] |
proofwiki-19413 | Class Intersection Distributes over Class Union | Let $A$, $B$ and $C$ be classes.
Then:
:$A \cap \paren {B \cup C} = \paren {A \cap B} \cup \paren {A \cap C}$
where:
:$A \cap B$ denotes class intersection
:$B \cup C$ denotes class union. | {{begin-eqn}}
{{eqn | o =
| r = x \in A \cap \paren {B \cup C}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \land \paren {x \in B \lor x \in C}
| c = {{Defof|Class Union}} and {{Defof|Class Intersection}}
}}
{{eqn | o = \leadstoandfrom
| r = \paren {x \in A \land x \in B} \lor \paren {x \in A \... | Let $A$, $B$ and $C$ be [[Definition:Class (Class Theory)|classes]].
Then:
:$A \cap \paren {B \cup C} = \paren {A \cap B} \cup \paren {A \cap C}$
where:
:$A \cap B$ denotes [[Definition:Class Intersection|class intersection]]
:$B \cup C$ denotes [[Definition:Class Union|class union]]. | {{begin-eqn}}
{{eqn | o =
| r = x \in A \cap \paren {B \cup C}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \land \paren {x \in B \lor x \in C}
| c = {{Defof|Class Union}} and {{Defof|Class Intersection}}
}}
{{eqn | o = \leadstoandfrom
| r = \paren {x \in A \land x \in B} \lor \paren {x \in A \... | Class Intersection Distributes over Class Union | https://proofwiki.org/wiki/Class_Intersection_Distributes_over_Class_Union | https://proofwiki.org/wiki/Class_Intersection_Distributes_over_Class_Union | [
"Class Intersection",
"Class Union",
"Examples of Distributive Operations"
] | [
"Definition:Class (Class Theory)",
"Definition:Class Intersection",
"Definition:Class Union"
] | [
"Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive"
] |
proofwiki-19414 | Theory of Structure is Complete | Let $\AA$ be a structure for a signature for predicate logic $\LL$.
Let $\map {\operatorname{Th}} \AA$ be the theory of $\AA$.
Then $\map {\operatorname{Th}} \AA$ is complete. | By definition of $\map {\operatorname{Th}} \AA$ be the theory of $\AA$:
:$\AA \models \map {\operatorname{Th}} \AA$
so that $\map {\operatorname{Th}} \AA$ is consistent.
Now let $\mathbf A$ be an $\LL$-sentence.
Let $\map {\operatorname{val}_\AA} {\mathbf A}$ be the value of $\mathbf A$ in $\AA$.
Then either $\map {\op... | Let $\AA$ be a [[Definition:Structure for Predicate Logic|structure]] for a [[Definition:Signature for Predicate Logic|signature for predicate logic]] $\LL$.
Let $\map {\operatorname{Th}} \AA$ be the [[Definition:Theory of Structure|theory]] of $\AA$.
Then $\map {\operatorname{Th}} \AA$ is [[Definition:Complete Theo... | By definition of $\map {\operatorname{Th}} \AA$ be the [[Definition:Theory of Structure|theory]] of $\AA$:
:$\AA \models \map {\operatorname{Th}} \AA$
so that $\map {\operatorname{Th}} \AA$ is [[Definition:Consistent Set of Formulas|consistent]].
Now let $\mathbf A$ be an [[Definition:Sentence|$\LL$-sentence]].
Le... | Theory of Structure is Complete | https://proofwiki.org/wiki/Theory_of_Structure_is_Complete | https://proofwiki.org/wiki/Theory_of_Structure_is_Complete | [
"Model Theory for Predicate Logic"
] | [
"Definition:Structure for Predicate Logic",
"Definition:Signature (Logic)/Predicate Logic",
"Definition:Theory of Structure",
"Definition:Complete Theory"
] | [
"Definition:Theory of Structure",
"Definition:Consistent (Logic)/Set of Formulas",
"Definition:Classes of WFFs/Sentence",
"Definition:Value of Formula under Assignment/Sentence",
"Definition:Value of Formula under Assignment",
"Definition:Model (Predicate Logic)",
"Definition:Theory of Structure",
"El... |
proofwiki-19415 | Class Difference with Class Difference | :$A \setminus \paren {A \setminus B} = A \cap B$ | {{begin-eqn}}
{{eqn | o =
| r = x \in A \setminus \paren {A \setminus B}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \land x \notin \paren {A \setminus B}
| c = {{Defof|Class Difference}}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \land \lnot \paren {x \in A \land x \notin B}
| c = {{D... | :$A \setminus \paren {A \setminus B} = A \cap B$ | {{begin-eqn}}
{{eqn | o =
| r = x \in A \setminus \paren {A \setminus B}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \land x \notin \paren {A \setminus B}
| c = {{Defof|Class Difference}}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \land \lnot \paren {x \in A \land x \notin B}
| c = {{D... | Class Difference with Class Difference | https://proofwiki.org/wiki/Class_Difference_with_Class_Difference | https://proofwiki.org/wiki/Class_Difference_with_Class_Difference | [
"Class Intersection",
"Class Difference"
] | [] | [
"De Morgan's Laws (Logic)",
"Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive",
"Disjunction with Contradiction"
] |
proofwiki-19416 | Class Difference of B with Class Difference of A with B | :$B \setminus \paren {A \setminus B} = B$ | {{begin-eqn}}
{{eqn | o =
| r = x \in B \setminus \paren {A \setminus B}
}}
{{eqn | o = \leadstoandfrom
| r = x \in B \land x \notin \paren {A \setminus B}
| c = {{Defof|Class Difference}}
}}
{{eqn | o = \leadstoandfrom
| r = x \in B \land \paren {x \notin A \lor x \in B}
| c = De Morgan'... | :$B \setminus \paren {A \setminus B} = B$ | {{begin-eqn}}
{{eqn | o =
| r = x \in B \setminus \paren {A \setminus B}
}}
{{eqn | o = \leadstoandfrom
| r = x \in B \land x \notin \paren {A \setminus B}
| c = {{Defof|Class Difference}}
}}
{{eqn | o = \leadstoandfrom
| r = x \in B \land \paren {x \notin A \lor x \in B}
| c = [[De Morga... | Class Difference of B with Class Difference of A with B | https://proofwiki.org/wiki/Class_Difference_of_B_with_Class_Difference_of_A_with_B | https://proofwiki.org/wiki/Class_Difference_of_B_with_Class_Difference_of_A_with_B | [
"Class Difference"
] | [] | [
"De Morgan's Laws",
"Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive",
"Rule of Idempotence/Conjunction",
"Absorption Laws (Logic)/Disjunction Absorbs Conjunction"
] |
proofwiki-19417 | Intersection with Subclass is Subclass | :$A \subseteq B \iff A \cap B = A$ | Let $A \cap B = A$.
Then by the definition of class equality:
:$A \subseteq A \cap B$
Thus:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = A
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = A \cap B
| c = {{Defof|Subclass}}: $A \subseteq A \cap B$
}}
{{eqn | ll= \leadst... | :$A \subseteq B \iff A \cap B = A$ | Let $A \cap B = A$.
Then by the definition of [[Definition:Class Equality|class equality]]:
:$A \subseteq A \cap B$
Thus:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = A
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = A \cap B
| c = {{Defof|Subclass}}: $A \subseteq... | Intersection with Subclass is Subclass | https://proofwiki.org/wiki/Intersection_with_Subclass_is_Subclass | https://proofwiki.org/wiki/Intersection_with_Subclass_is_Subclass | [
"Subclasses",
"Class Intersection"
] | [] | [
"Definition:Class Equality",
"Definition:Class Equality"
] |
proofwiki-19418 | Union with Superclass is Superclass | :$A \subseteq B \iff A \cup B = B$ | Let $A \cup B = B$.
Then by definition of class equality:
:$A \cup B \subseteq B$
Thus:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = A
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = A
| c = Rule of Addition
}}
{{eqn | lo= \lor
| l = x
| o = \in
| r... | :$A \subseteq B \iff A \cup B = B$ | Let $A \cup B = B$.
Then by definition of [[Definition:Class Equality|class equality]]:
:$A \cup B \subseteq B$
Thus:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = A
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = A
| c = [[Rule of Addition]]
}}
{{eqn | lo= \lor
... | Union with Superclass is Superclass | https://proofwiki.org/wiki/Union_with_Superclass_is_Superclass | https://proofwiki.org/wiki/Union_with_Superclass_is_Superclass | [
"Subclasses",
"Class Union"
] | [] | [
"Definition:Class Equality",
"Rule of Addition",
"Rule of Addition",
"Set is Subset of Union",
"Rule of Addition",
"Definition:Class Equality"
] |
proofwiki-19419 | Class Difference with Class Difference with Subclass | Let $A$ and $B$ be classes.
Let $B \subseteq A$.
Then:
:$A \setminus \paren {A \setminus B} = B$ | From Class Difference with Class Difference:
:$A \setminus \paren {A \setminus B} = A \cap B$
for all classes $A$ and $B$.
From Intersection with Subclass is Subclass:
:$A \subseteq B \iff A \cap B = A$
The result follows.
{{qed}} | Let $A$ and $B$ be [[Definition:Class (Class Theory)|classes]].
Let $B \subseteq A$.
Then:
:$A \setminus \paren {A \setminus B} = B$ | From [[Class Difference with Class Difference]]:
:$A \setminus \paren {A \setminus B} = A \cap B$
for all [[Definition:Class (Class Theory)|classes]] $A$ and $B$.
From [[Intersection with Subclass is Subclass]]:
:$A \subseteq B \iff A \cap B = A$
The result follows.
{{qed}} | Class Difference with Class Difference with Subclass | https://proofwiki.org/wiki/Class_Difference_with_Class_Difference_with_Subclass | https://proofwiki.org/wiki/Class_Difference_with_Class_Difference_with_Subclass | [
"Subclasses",
"Set Difference"
] | [
"Definition:Class (Class Theory)"
] | [
"Class Difference with Class Difference",
"Definition:Class (Class Theory)",
"Intersection with Subclass is Subclass"
] |
proofwiki-19420 | Equivalence of Formulations of Axiom of Powers | In the context of class theory, the following formulations of the '''{{axiom-link|Powers}}''' are equivalent:
=== Formulation 1 ===
{{:Axiom:Axiom of Powers (Set Theory)}}
=== Formulation 2 ===
{{:Axiom:Axiom of Powers (Class Theory)}} | It is assumed throughout that the {{axiom-link|Extensionality}} and the {{axiom-link|Specification}} both hold. | In the context of [[Definition:Class Theory|class theory]], the following formulations of the '''{{axiom-link|Powers}}''' are [[Definition:Logical Equivalence|equivalent]]:
=== [[Axiom:Axiom of Powers (Set Theory)|Formulation 1]] ===
{{:Axiom:Axiom of Powers (Set Theory)}}
=== [[Axiom:Axiom of Powers (Class Theory)|Fo... | It is assumed throughout that the {{axiom-link|Extensionality}} and the {{axiom-link|Specification}} both hold. | Equivalence of Formulations of Axiom of Powers | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Powers | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Powers | [
"Axiom of Powers",
"Definition Equivalences"
] | [
"Definition:Class Theory",
"Definition:Logical Equivalence",
"Axiom:Axiom of Powers/Set Theory",
"Axiom:Axiom of Powers/Class Theory"
] | [] |
proofwiki-19421 | Linear Integral Bounded Operator is Continuous | Let $I = \closedint 0 1$ be a closed real interval.
Let $A : I \times I \to \R$ be a real function such that:
:$\ds \int_0^1 \int_0^1 \paren {\map A {t, \tau} }^2 \rd t \rd \tau < \infty$
{{Research|For now "bounded" means above. Need to check if this meaning is standard}}
where $\times$ denotes the cartesian product.
... | We have that Riemann Integral Operator is Linear Mapping.
{{Research|Probably this should be replaced with Lebesgue integral. The source does not say anything about compatibility of Riemann integral and Lebesgue space}}
Hence, $T_A$ is a linear transformation.
Furthermore:
{{begin-eqn}}
{{eqn | l = \norm {T_A \mathbf x... | Let $I = \closedint 0 1$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $A : I \times I \to \R$ be a [[Definition:Real Function of Two Variables|real function]] such that:
:$\ds \int_0^1 \int_0^1 \paren {\map A {t, \tau} }^2 \rd t \rd \tau < \infty$
{{Research|For now "bounded" means above. Need ... | We have that [[Riemann Integral Operator is Linear Mapping]].
{{Research|Probably this should be replaced with Lebesgue integral. The source does not say anything about compatibility of Riemann integral and Lebesgue space}}
Hence, $T_A$ is a [[Definition:Set of All Linear Transformations|linear transformation]].
Fur... | Linear Integral Bounded Operator is Continuous | https://proofwiki.org/wiki/Linear_Integral_Bounded_Operator_is_Continuous | https://proofwiki.org/wiki/Linear_Integral_Bounded_Operator_is_Continuous | [
"Operator Theory",
"Continuous Mappings",
"Linear Transformations"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Function/Two Variables",
"Definition:Cartesian Product",
"Definition:Integral Operator",
"Definition:Lebesgue Space",
"Definition:Continuous Linear Transformation Space"
] | [
"Riemann Integral Operator is Linear Mapping",
"Definition:Set of All Linear Transformations",
"Cauchy-Bunyakovsky-Schwarz Inequality/Definite Integrals",
"Continuity of Linear Transformation/Normed Vector Space"
] |
proofwiki-19422 | Subset is Element of Power Set | :$y \in \powerset x \iff y \subseteq x$ | By definition of power set, $\powerset x$ is the set of subsets of $x$.
Hence the result, by definition of subset and power set.
{{qed}} | :$y \in \powerset x \iff y \subseteq x$ | By definition of [[Definition:Power Set|power set]], $\powerset x$ is the [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $x$.
Hence the result, by definition of [[Definition:Subset|subset]] and [[Definition:Power Set|power set]].
{{qed}} | Subset is Element of Power Set | https://proofwiki.org/wiki/Subset_is_Element_of_Power_Set | https://proofwiki.org/wiki/Subset_is_Element_of_Power_Set | [
"Power Set",
"Subsets"
] | [] | [
"Definition:Power Set",
"Definition:Set",
"Definition:Subset",
"Definition:Subset",
"Definition:Power Set"
] |
proofwiki-19423 | Class Union Exists and is Unique | Let $V$ be a basic universe.
Let $A \subseteq V$ be a class.
Let $\bigcup A$ denote the union of $A$.
Then $\bigcup A$ is guaranteed to exist and is unique. | By the Axiom of Specification the union of $A$ can be created:
:$\bigcup A := \set {x: \exists y: x \in y \land y \in A}$
Hence $\bigcup A$ exists.
Let $B$ and $C$ both be unions of $A$.
From the definition of union:
:$\forall A$:
::$x \in B \iff \exists y \in A: x \in y$
::$x \in C \iff \exists y \in A: x \in y$
From ... | Let $V$ be a [[Definition:Basic Universe|basic universe]].
Let $A \subseteq V$ be a [[Definition:Class (Class Theory)|class]].
Let $\bigcup A$ denote the [[Definition:Union of Class|union]] of $A$.
Then $\bigcup A$ is guaranteed to exist and is [[Definition:Unique|unique]]. | By the [[Axiom:Axiom of Specification (Classes)|Axiom of Specification]] the [[Definition:Union of Class|union]] of $A$ can be created:
:$\bigcup A := \set {x: \exists y: x \in y \land y \in A}$
Hence $\bigcup A$ exists.
Let $B$ and $C$ both be [[Definition:Union of Class|unions]] of $A$.
From the definition of [[... | Class Union Exists and is Unique | https://proofwiki.org/wiki/Class_Union_Exists_and_is_Unique | https://proofwiki.org/wiki/Class_Union_Exists_and_is_Unique | [
"Class Union"
] | [
"Definition:Basic Universe",
"Definition:Class (Class Theory)",
"Definition:Class Union/General Definition",
"Definition:Unique"
] | [
"Axiom:Axiom of Specification/Class Theory",
"Definition:Class Union/General Definition",
"Definition:Class Union/General Definition",
"Definition:Class Union/General Definition",
"Biconditional is Commutative",
"Biconditional is Transitive",
"Axiom:Axiom of Extension/Class Theory",
"Definition:Class ... |
proofwiki-19424 | Trace of Product of Matrices | Let $\struct {R, +, \circ}$ be a commutative ring.
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$.
Let $\mathbf B = \sqbrk b_{n m}$ be an $n \times m$ matrix over $R$.
Then:
:$\map \tr {\mathbf A \mathbf B} = \map \tr {\mathbf B \mathbf A}$
where $\map \tr {\mathbf A}$ denotes the trace of $\mathbf... | Let $\mathbf A \mathbf B = \mathbf C = \sqbrk c_m$.
Let $\mathbf B \mathbf A = \mathbf D = \sqbrk d_n$.
Then by definition of matrix products:
{{begin-eqn}}
{{eqn | q = \forall i, j \in \closedint 1 m
| l = c_{i j}
| r = \sum_{k \mathop = 1}^n a_{i k} \circ b_{k j}
}}
{{eqn | q = \forall i, j \in \closedint... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\mathbf A = \sqbrk a_{m n}$ be an [[Definition:Matrix|$m \times n$ matrix]] over $R$.
Let $\mathbf B = \sqbrk b_{n m}$ be an [[Definition:Matrix|$n \times m$ matrix]] over $R$.
Then:
:$\map \tr {\mathbf A \mathbf B} = \map \tr {\... | Let $\mathbf A \mathbf B = \mathbf C = \sqbrk c_m$.
Let $\mathbf B \mathbf A = \mathbf D = \sqbrk d_n$.
Then by [[Definition:Matrix Product (Conventional)|definition of matrix products]]:
{{begin-eqn}}
{{eqn | q = \forall i, j \in \closedint 1 m
| l = c_{i j}
| r = \sum_{k \mathop = 1}^n a_{i k} \circ b... | Trace of Product of Matrices | https://proofwiki.org/wiki/Trace_of_Product_of_Matrices | https://proofwiki.org/wiki/Trace_of_Product_of_Matrices | [
"Traces of Matrices",
"Conventional Matrix Multiplication"
] | [
"Definition:Commutative Ring",
"Definition:Matrix",
"Definition:Matrix",
"Definition:Trace (Linear Algebra)/Matrix"
] | [
"Definition:Matrix Product (Conventional)",
"Exchange of Order of Indexed Summations",
"Definition:Commutative/Operation",
"Category:Traces of Matrices",
"Category:Conventional Matrix Multiplication"
] |
proofwiki-19425 | Self-Adjoint Densely-Defined Linear Operator has Empty Residual Spectrum | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $\struct {\map D T, T}$ be a self-adjoint densely defined linear operator.
Then the residual spectrum $\map {\sigma_r} T$ is empty. | Let $\struct {\map D {T^\ast}, T^\ast}$ be the adjoint of $\struct {\map D T, T}$.
Since $\struct {\map D T, T}$ is self-adjoint, we have:
:$\struct {\map D {T^\ast}, T^\ast} = \struct {\map D T, T}$
Suppose that $\map {\sigma_r} T$ is non-empty.
Then there exists $\lambda \in \map {\sigma_r} T$.
That is, there exist... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]].
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]].
Then the [[Definition:Residual Spectru... | Let $\struct {\map D {T^\ast}, T^\ast}$ be the [[Definition:Adjoint of Densely-Defined Linear Operator|adjoint]] of $\struct {\map D T, T}$.
Since $\struct {\map D T, T}$ is [[Definition:Self-Adjoint Densely-Defined Linear Operator|self-adjoint]], we have:
:$\struct {\map D {T^\ast}, T^\ast} = \struct {\map D T, T}... | Self-Adjoint Densely-Defined Linear Operator has Empty Residual Spectrum | https://proofwiki.org/wiki/Self-Adjoint_Densely-Defined_Linear_Operator_has_Empty_Residual_Spectrum | https://proofwiki.org/wiki/Self-Adjoint_Densely-Defined_Linear_Operator_has_Empty_Residual_Spectrum | [
"Residual Spectrums (Densely-Defined Linear Operators)",
"Residual Spectra (Densely-Defined Linear Operators)",
"Self-Adjoint Densely-Defined Linear Operators",
"Residual Spectra (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Self-Adjoint Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator",
"Definition:Residual Spectrum of Densely-Defined Linear Operator",
"Definition:Empty Set"
] | [
"Definition:Adjoint of Densely-Defined Linear Operator",
"Definition:Self-Adjoint Densely-Defined Linear Operator",
"Definition:Empty Set",
"Definition:Injection",
"Definition:Everywhere Dense",
"Linear Subspace Dense iff Zero Orthocomplement",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Compleme... |
proofwiki-19426 | Equivalence of Formulations of Axiom of Empty Set for Classes | In the context of class theory, the following formulations of the '''{{axiom-link|the Empty Set}}''' are equivalent:
=== Formulation 1 ===
{{:Axiom:Axiom of the Empty Set/Set Theory/Formulation 2}}
=== Formulation 2 ===
{{:Axiom:Axiom of the Empty Set (Class Theory)}} | It is assumed throughout that the {{axiom-link|Extension|Classes}} and the {{axiom-link|Specification|Classes}} both hold. | In the context of [[Definition:Class Theory|class theory]], the following formulations of the '''{{axiom-link|the Empty Set}}''' are [[Definition:Logical Equivalence|equivalent]]:
=== [[Axiom:Axiom of the Empty Set/Set Theory/Formulation 2|Formulation 1]] ===
{{:Axiom:Axiom of the Empty Set/Set Theory/Formulation 2}}
... | It is assumed throughout that the {{axiom-link|Extension|Classes}} and the {{axiom-link|Specification|Classes}} both hold. | Equivalence of Formulations of Axiom of Empty Set for Classes | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Empty_Set_for_Classes | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Empty_Set_for_Classes | [
"Axiom of the Empty Set",
"Definition Equivalences"
] | [
"Definition:Class Theory",
"Definition:Logical Equivalence",
"Axiom:Axiom of the Empty Set/Set Theory/Formulation 2",
"Axiom:Axiom of the Empty Set/Class Theory"
] | [] |
proofwiki-19427 | Successor Mapping is Progressing | Let $V$ be a basic universe.
Let $s: V \to V$ denote the '''successor mapping''' on $V$:
:$\forall x \in V: \map s x := x \cup \set x$
Then $s$ is a progressing mapping. | Recall
By Set is Subset of Union:
:$x \subseteq x \cup \set x$
That is:
:$x \subseteq \map s x$
Thus $s$ is by definition a progressing mapping.
{{qed}} | Let $V$ be a [[Definition:Basic Universe|basic universe]].
Let $s: V \to V$ denote the '''[[Definition:Successor Mapping|successor mapping]]''' on $V$:
:$\forall x \in V: \map s x := x \cup \set x$
Then $s$ is a [[Definition:Progressing Mapping|progressing mapping]]. | Recall
By [[Set is Subset of Union]]:
:$x \subseteq x \cup \set x$
That is:
:$x \subseteq \map s x$
Thus $s$ is by definition a [[Definition:Progressing Mapping|progressing mapping]].
{{qed}} | Successor Mapping is Progressing | https://proofwiki.org/wiki/Successor_Mapping_is_Progressing | https://proofwiki.org/wiki/Successor_Mapping_is_Progressing | [
"Successor Mapping",
"Progressing Mappings"
] | [
"Definition:Basic Universe",
"Definition:Successor Mapping",
"Definition:Progressing Mapping"
] | [
"Set is Subset of Union",
"Definition:Progressing Mapping"
] |
proofwiki-19428 | Closed Sets of Right Order Space on Real Numbers | Let $T = \struct {\R, \tau}$ be the right order space on $\R$.
Then $H \subseteq S$ is closed in $T$ {{iff}}:
:$H = \O$ or $\R$
or
:$H = \hointl {-\infty} a$ for some $a \in \R$. | By definition of the right order space on $\R$, $U \subseteq S$ is open in $T$ {{iff}}:
:$U = \O$ or $\R$
or
:$U = \openint a \infty$ for some $a \in \R$.
Note that:
:$\R \setminus \O = \R$
:$\R \setminus \R = \O$
:$\R \setminus \openint a \infty = \hointl {-\infty} a$
The result follows from the definition of closed s... | Let $T = \struct {\R, \tau}$ be the [[Definition:Right Order Topology on Real Numbers|right order space on $\R$]].
Then $H \subseteq S$ is [[Definition:Closed Set (Topology)|closed]] in $T$ {{iff}}:
:$H = \O$ or $\R$
or
:$H = \hointl {-\infty} a$ for some $a \in \R$. | By definition of the [[Definition:Right Order Topology on Real Numbers|right order space on $\R$]], $U \subseteq S$ is [[Definition:Open Set (Topology)|open]] in $T$ {{iff}}:
:$U = \O$ or $\R$
or
:$U = \openint a \infty$ for some $a \in \R$.
Note that:
:$\R \setminus \O = \R$
:$\R \setminus \R = \O$
:$\R \setminus \o... | Closed Sets of Right Order Space on Real Numbers | https://proofwiki.org/wiki/Closed_Sets_of_Right_Order_Space_on_Real_Numbers | https://proofwiki.org/wiki/Closed_Sets_of_Right_Order_Space_on_Real_Numbers | [
"Right Order Topologies",
"Examples of Closed Sets"
] | [
"Definition:Right Order Topology on Real Numbers",
"Definition:Closed Set/Topology"
] | [
"Definition:Right Order Topology on Real Numbers",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Category:Right Order Topologies",
"Category:Examples of Closed Sets"
] |
proofwiki-19429 | Right Order Topology on Real Numbers is Topology | Let $\tau$ be the '''right order topology on $\R$'''.
Then $\tau$ forms a topology on $\R$.
That is:
:$T = \struct {\R, \tau}$ is a topological space. | Write $\O = \openint \infty \infty$ and $\R = \openint {-\infty} \infty$.
Then $\tau$ can be written as $\set {\openint j \infty: j \in \overline \R}$.
First we note that:
:$m \le n \implies \openint n \infty \subseteq \openint m \infty$
By definition we have that:
:$\O \in \tau$
Then each of the open set axioms is exa... | Let $\tau$ be the '''[[Definition:Right Order Topology on Real Numbers|right order topology on $\R$]]'''.
Then $\tau$ forms a [[Definition:Topology|topology]] on $\R$.
That is:
:$T = \struct {\R, \tau}$ is a [[Definition:Topological Space|topological space]]. | Write $\O = \openint \infty \infty$ and $\R = \openint {-\infty} \infty$.
Then $\tau$ can be written as $\set {\openint j \infty: j \in \overline \R}$.
First we note that:
:$m \le n \implies \openint n \infty \subseteq \openint m \infty$
By definition we have that:
:$\O \in \tau$
Then each of the [[Axiom:Open Set ... | Right Order Topology on Real Numbers is Topology | https://proofwiki.org/wiki/Right_Order_Topology_on_Real_Numbers_is_Topology | https://proofwiki.org/wiki/Right_Order_Topology_on_Real_Numbers_is_Topology | [
"Right Order Topologies"
] | [
"Definition:Right Order Topology on Real Numbers",
"Definition:Topology",
"Definition:Topological Space"
] | [
"Axiom:Open Set Axioms",
"Axiom:Open Set Axioms"
] |
proofwiki-19430 | Image of Weakly Convergent Sequence under Compact Linear Transformation is Convergent | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.
Let $T : X \to Y$ be a compact linear transformation.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a weakly convergent sequence with:
:$x_n \weakconv x$
Then:
:$T x_n \to T x$
in the strong sense. | Let $X^\ast$ and $Y^\ast$ be the normed dual spaces of $X$ and $Y$ respectively.
Let $f \in Y^\ast$.
Then $f \circ T \in X^\ast$.
Since $x_n \weakconv x$, we have:
:$\map f {T x_n} \to \map f {T x}$ as $n \to \infty$.
So we have:
:$T x_n \weakconv T x$ as $n \to \infty$.
{{AimForCont}} that $\sequence {T x_n}_{n \i... | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $T : X \to Y$ be a [[Definition:Compact Linear Transformation|compact linear transformation]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Weak Convergence (Normed Vector Sp... | Let $X^\ast$ and $Y^\ast$ be the [[Definition:Normed Dual Space|normed dual spaces]] of $X$ and $Y$ respectively.
Let $f \in Y^\ast$.
Then $f \circ T \in X^\ast$.
Since $x_n \weakconv x$, we have:
:$\map f {T x_n} \to \map f {T x}$ as $n \to \infty$.
So we have:
:$T x_n \weakconv T x$ as $n \to \infty$.
{{Ai... | Image of Weakly Convergent Sequence under Compact Linear Transformation is Convergent | https://proofwiki.org/wiki/Image_of_Weakly_Convergent_Sequence_under_Compact_Linear_Transformation_is_Convergent | https://proofwiki.org/wiki/Image_of_Weakly_Convergent_Sequence_under_Compact_Linear_Transformation_is_Convergent | [
"Weak Convergence (Normed Vector Spaces)",
"Compact Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Compact Linear Transformation",
"Definition:Weak Convergence (Normed Vector Space)",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Definition:Normed Dual Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Subsequence",
"Weakly Convergent Sequence in Normed Vector Space is Bounded",
"Definition:Bounded Sequence/Normed Vector Space",
"Definition:Compact Linear Transformation",
"Definition:Subsequence",
"Conv... |
proofwiki-19431 | Point Spectrum of Symmetric Densely-Defined Linear Operator is Real | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\struct {\map D T, T}$ be a symmetric densely-defined linear operator.
Let $\map {\sigma_p} T$ be the point spectrum of $\struct {\map D T, T}$.
Then:
:$\map {\sigma_p} T \subseteq \R$ | If $\map {\sigma_p} T = \O$, the result is immediate.
Let $\lambda \in \map {\sigma_p} T$.
Then, from Point Spectrum of Densely-Defined Linear Operator consists of its Eigenvalues, there exists $x \in \HH \setminus \set 0$ such that:
:$T x = \lambda x$
Then, we have:
:$\innerprod {T x} x = \innerprod {\lambda x} x =... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\struct {\map D T, T}$ be a [[Definition:Symmetric Densely-Defined Linear Operator|symmetric]] [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]].
Let $\map {\sigma_p} T$ be the [[D... | If $\map {\sigma_p} T = \O$, the result is immediate.
Let $\lambda \in \map {\sigma_p} T$.
Then, from [[Point Spectrum of Densely-Defined Linear Operator consists of its Eigenvalues]], there exists $x \in \HH \setminus \set 0$ such that:
:$T x = \lambda x$
Then, we have:
:$\innerprod {T x} x = \innerprod {\lamb... | Point Spectrum of Symmetric Densely-Defined Linear Operator is Real | https://proofwiki.org/wiki/Point_Spectrum_of_Symmetric_Densely-Defined_Linear_Operator_is_Real | https://proofwiki.org/wiki/Point_Spectrum_of_Symmetric_Densely-Defined_Linear_Operator_is_Real | [
"Point Spectrums (Densely-Defined Linear Operators)",
"Point Spectra (Densely-Defined Linear Operators)",
"Symmetric Densely-Defined Linear Operators",
"Point Spectra (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Symmetric Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator",
"Definition:Point Spectrum of Densely-Defined Linear Operator"
] | [
"Point Spectrum of Densely-Defined Linear Operator consists of its Eigenvalues",
"Inner Product is Sesquilinear",
"Complex Number equals Conjugate iff Wholly Real",
"Definition:Subset"
] |
proofwiki-19432 | Banach Space is Reflexive iff Normed Dual is Reflexive | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a Banach space over $\Bbb F$.
Let $X^\ast$ be the normed dual space of $X$.
Then:
:$X$ is reflexive {{iff}} $X^\ast$ is reflexive. | Let $X^{\ast \ast}$ be the second normed dual of $X$.
Let $X^{\ast \ast \ast}$ be the normed dual of $X^{\ast \ast}$. | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\Bbb F$.
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Then:
:$X$ is [[Definition:Reflexive Space|reflexive]] {{iff}} $X^\ast$ is [[Definition:Reflexive Space|reflexive]]. | Let $X^{\ast \ast}$ be the [[Definition:Second Normed Dual|second normed dual]] of $X$.
Let $X^{\ast \ast \ast}$ be the [[Definition:Normed Dual Space|normed dual]] of $X^{\ast \ast}$. | Banach Space is Reflexive iff Normed Dual is Reflexive | https://proofwiki.org/wiki/Banach_Space_is_Reflexive_iff_Normed_Dual_is_Reflexive | https://proofwiki.org/wiki/Banach_Space_is_Reflexive_iff_Normed_Dual_is_Reflexive | [
"Reflexive Spaces",
"Normed Dual Spaces"
] | [
"Definition:Banach Space",
"Definition:Normed Dual Space",
"Definition:Reflexive Space",
"Definition:Reflexive Space"
] | [
"Definition:Second Normed Dual",
"Definition:Normed Dual Space"
] |
proofwiki-19433 | Resolvent Set of Bounded Linear Operator equal to Resolvent Set as Densely-Defined Linear Operator | Let $\HH$ be a Hilbert space over $\C$.
Let $T : \HH \to \HH$ be a bounded linear operator.
Let $\map {\rho_1} T$ be the resolvent set of $T$ as a bounded linear operator.
Let $\map {\rho_2} T$ be the resolvent set of $T$ as a densely-defined linear operator $\struct {\HH, T}$.
Then:
:$\map {\rho_1} T = \map {\rho_2}... | Let $\lambda \in \map {\rho_1} T$.
Then $T - \lambda I$ is invertible in the sense of a bounded linear transformation.
That is, $T - \lambda I$ is bijective and $\paren {T - \lambda I}^{-1}$ is bounded.
From Underlying Set of Topological Space is Everywhere Dense, we have that $\HH$ is everywhere dense in $\HH$.
So, $... | Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $T : \HH \to \HH$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\map {\rho_1} T$ be the [[Definition:Resolvent Set of Bounded Linear Operator|resolvent set]] of $T$ as a [[Definition:Bounded Linear Operator|bounded l... | Let $\lambda \in \map {\rho_1} T$.
Then $T - \lambda I$ is [[Definition:Invertible Bounded Linear Transformation|invertible in the sense of a bounded linear transformation]].
That is, $T - \lambda I$ is [[Definition:Bijection|bijective]] and $\paren {T - \lambda I}^{-1}$ is [[Definition:Bounded Linear Transformation|... | Resolvent Set of Bounded Linear Operator equal to Resolvent Set as Densely-Defined Linear Operator | https://proofwiki.org/wiki/Resolvent_Set_of_Bounded_Linear_Operator_equal_to_Resolvent_Set_as_Densely-Defined_Linear_Operator | https://proofwiki.org/wiki/Resolvent_Set_of_Bounded_Linear_Operator_equal_to_Resolvent_Set_as_Densely-Defined_Linear_Operator | [
"Resolvent Sets (Bounded Linear Operators)",
"Resolvent Sets (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Operator",
"Definition:Resolvent Set/Bounded Linear Operator",
"Definition:Bounded Linear Operator",
"Definition:Resolvent Set/Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator"
] | [
"Definition:Invertible Bounded Linear Transformation",
"Definition:Bijection",
"Definition:Bounded Linear Transformation",
"Underlying Set of Topological Space is Everywhere Dense",
"Definition:Everywhere Dense",
"Definition:Injective",
"Definition:Everywhere Dense",
"Definition:Bounded Linear Transfo... |
proofwiki-19434 | Partition of Spectrum of Densely-Defined Linear Operator | Let $\HH$ be a Hilbert space over $\C$.
Let $\struct {\map D T, T}$ be a densely-defined linear operator.
Let $\map \sigma T$ be the spectrum of $T$.
Then:
:$\map \sigma T = \map {\sigma_p} T \cup \map {\sigma_s} T \cup \map {\sigma_r} T$
where:
:$\map {\sigma_p} T$ is the point spectrum of $T$
:$\map {\sigma_s} T$ ... | Let $\lambda \in \map \sigma T$.
Then, from the definition of the resolvent set of $T$, at least one of the following is false:
:$(1) \quad$ $T - \lambda I$ is injective
:$(2) \quad$ $\map {\paren {T - \lambda I} } {\map D T}$ is everywhere dense in $\HH$
:$(3) \quad$ $\paren {T - \lambda I}^{-1}$ is bounded.
Note tha... | Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\struct {\map D T, T}$ be a [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]].
Let $\map \sigma T$ be the [[Definition:Spectrum of Densely-Defined Linear Operator|spectrum]] of $T$.
Then:
:$\map \sigma T = \ma... | Let $\lambda \in \map \sigma T$.
Then, from the definition of the [[Definition:Resolvent Set of Densely-Defined Linear Operator|resolvent set]] of $T$, at least one of the following is false:
:$(1) \quad$ $T - \lambda I$ is [[Definition:Injective|injective]]
:$(2) \quad$ $\map {\paren {T - \lambda I} } {\map D T}$ i... | Partition of Spectrum of Densely-Defined Linear Operator | https://proofwiki.org/wiki/Partition_of_Spectrum_of_Densely-Defined_Linear_Operator | https://proofwiki.org/wiki/Partition_of_Spectrum_of_Densely-Defined_Linear_Operator | [
"Spectra (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Densely-Defined Linear Operator",
"Definition:Spectrum (Spectral Theory)/Densely-Defined Linear Operator",
"Definition:Point Spectrum of Densely-Defined Linear Operator",
"Definition:Continuous Spectrum of Densely-Defined Linear Operator",
"Definition:Residual Spect... | [
"Definition:Resolvent Set/Densely-Defined Linear Operator",
"Definition:Injective",
"Definition:Everywhere Dense",
"Definition:Bounded Linear Transformation",
"Definition:Point Spectrum of Densely-Defined Linear Operator",
"Definition:Continuous Spectrum of Densely-Defined Linear Operator",
"Definition:... |
proofwiki-19435 | Spectrum of Bounded Linear Operator equal to Spectrum as Densely-Defined Linear Operator | Let $\HH$ be a Hilbert space over $\C$.
Let $T : \HH \to \HH$ be a bounded linear operator.
Let $\map {\sigma_1} T$ be the spectrum of $T$ as a bounded linear operator.
Let $\map {\sigma_2} T$ be the spectrum of $T$ as a densely-defined linear operator $\struct {\HH, T}$.
Then:
:$\map {\sigma_1} T = \map {\sigma_2} T... | From the definition of the spectrum of $T$ as a bounded linear operator, we have:
:$\map {\sigma_1} T = \C \setminus \map {\rho_1} T$
where $\map {\rho_1} T$ is the resolvent set of $T$ as a bounded linear operator.
From the definition of the spectrum of $T$ as a densely-defined linear operator $\struct {\HH, T}$, we ... | Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $T : \HH \to \HH$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\map {\sigma_1} T$ be the [[Definition:Spectrum of Bounded Linear Operator|spectrum]] of $T$ as a [[Definition:Bounded Linear Operator|bounded linear op... | From the definition of the [[Definition:Spectrum of Bounded Linear Operator|spectrum]] of $T$ as a [[Definition:Bounded Linear Operator|bounded linear operator]], we have:
:$\map {\sigma_1} T = \C \setminus \map {\rho_1} T$
where $\map {\rho_1} T$ is the [[Definition:Resolvent Set of Bounded Linear Operator|resolven... | Spectrum of Bounded Linear Operator equal to Spectrum as Densely-Defined Linear Operator | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_equal_to_Spectrum_as_Densely-Defined_Linear_Operator | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_equal_to_Spectrum_as_Densely-Defined_Linear_Operator | [
"Spectra (Bounded Linear Operators)",
"Spectra (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Operator",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Definition:Bounded Linear Operator",
"Definition:Spectrum (Spectral Theory)/Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator"
] | [
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Definition:Bounded Linear Operator",
"Definition:Resolvent Set/Bounded Linear Operator",
"Definition:Bounded Linear Operator",
"Definition:Spectrum (Spectral Theory)/Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Opera... |
proofwiki-19436 | Spectrum of Self-Adjoint Densely-Defined Linear Operator is Real and Closed | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\struct {\map D T, T}$ be a self-adjoint densely-defined linear operator.
Let $\map \sigma T$ be the spectrum of $\struct {\map D T, T}$.
Then $\map \sigma T$ is a closed subset of $\C$ and:
:$\map \sigma T \subseteq \R$ | Let $\lambda \in \map \sigma T$.
We show that $\lambda \in \R$.
From Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue, there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\map D T$ with:
:$\paren {T - \lambda I} x_n \to 0$
with $\norm {x_n} = 1$ for... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
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 $\map \sigma T$ be the [... | Let $\lambda \in \map \sigma T$.
We show that $\lambda \in \R$.
From [[Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue]], there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}_{n \mathop \in \N}$ in $\map D T$ with:
:$\paren {T - \lambda I} x_n \t... | Spectrum of Self-Adjoint Densely-Defined Linear Operator is Real and Closed | https://proofwiki.org/wiki/Spectrum_of_Self-Adjoint_Densely-Defined_Linear_Operator_is_Real_and_Closed | https://proofwiki.org/wiki/Spectrum_of_Self-Adjoint_Densely-Defined_Linear_Operator_is_Real_and_Closed | [
"Spectra (Densely-Defined Linear Operators)",
"Self-Adjoint Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Self-Adjoint Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator",
"Definition:Spectrum (Spectral Theory)/Densely-Defined Linear Operator",
"Definition:Closed Set/Complex Analysis"
] | [
"Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue",
"Definition:Sequence",
"Cauchy-Bunyakovsky-Schwarz Inequality",
"Modulus of Limit/Normed Vector Space",
"Combination Theorem for Sequences/Complex/Sum Rule",
"Convergence of Complex Conjugate of Co... |
proofwiki-19437 | Characterization of Integer Polynomial has Root in P-adic Integers | Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\map F X \in \Z \sqbrk X$ be a polynomial with integer coefficients.
Let $a \in \Z_p$.
Then:
:$\map F a = 0$
{{iff}}
:there exists a sequence $\sequence{a_n}$ of integers:
::$(1)\quad\ds\lim_{n \mathop \to \infty} {a_n} = a$
::$(2)\quad\map F {a_n} \equiv 0... | From Characterization of Polynomial has Root in P-adic Integers:
:$\map F a = 0$
{{iff}}
:there exists a sequence $\sequence{a_n}$ of integers:
::$(1)\quad\ds\lim_{n \mathop \to \infty} {a_n} = a$
::$(2)\quad\map F {a_n} \equiv 0 \mod {p^{n+1}\Z_p}$
By definition of a polynomial with integer coefficients:
:$\forall n... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\map F X \in \Z \sqbrk X$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]] with [[Definition:Integer|integer]] [[Definition:Coefficient of Polynomial|coefficients]].
Let $a \in \Z_p$.
T... | From [[Characterization of Polynomial has Root in P-adic Integers]]:
:$\map F a = 0$
{{iff}}
:there exists a [[Definition:Sequence|sequence]] $\sequence{a_n}$ of [[Definition:Integer|integers]]:
::$(1)\quad\ds\lim_{n \mathop \to \infty} {a_n} = a$
::$(2)\quad\map F {a_n} \equiv 0 \mod {p^{n+1}\Z_p}$
By definition of... | Characterization of Integer Polynomial has Root in P-adic Integers | https://proofwiki.org/wiki/Characterization_of_Integer_Polynomial_has_Root_in_P-adic_Integers | https://proofwiki.org/wiki/Characterization_of_Integer_Polynomial_has_Root_in_P-adic_Integers | [
"P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Polynomial over Ring",
"Definition:Integer",
"Definition:Coefficient of Polynomial",
"Definition:Sequence",
"Definition:Integer"
] | [
"Characterization of Polynomial has Root in P-adic Integers",
"Definition:Sequence",
"Definition:Integer",
"Definition:Polynomial over Ring",
"Definition:Integer",
"Definition:Coefficient of Polynomial",
"Congruence Modulo Equivalence for Integers in P-adic Integers",
"Definition:Sequence",
"Definit... |
proofwiki-19438 | Characterization of Integer has Square Root in P-adic Integers | Let $\Z_p$ be the $p$-adic integers for some prime $p \ne 2$.
Let $a \in Z$ be an integer such that $p \nmid a$.
Then:
:$\exists x \in \Z_p : x^2 = a$
{{iff}}
:$a$ is a quadratic residue of $p$. | Let $F \in \Z \sqbrk X$ be the polynomial:
:$\map F X = X^2 - a$
By definition of formal derivative of $F$ is:
:$\map {F'} X = 2X$ | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p \ne 2$.
Let $a \in Z$ be an [[Definition:Integer|integer]] such that $p \nmid a$.
Then:
:$\exists x \in \Z_p : x^2 = a$
{{iff}}
:$a$ is a [[Definition:Quadratic Residue|quadratic residue]] of $p$. | Let $F \in \Z \sqbrk X$ be the [[Definition:Polynomial over Ring|polynomial]]:
:$\map F X = X^2 - a$
By definition of [[Definition:Formal Derivative of Polynomial|formal derivative]] of $F$ is:
:$\map {F'} X = 2X$ | Characterization of Integer has Square Root in P-adic Integers | https://proofwiki.org/wiki/Characterization_of_Integer_has_Square_Root_in_P-adic_Integers | https://proofwiki.org/wiki/Characterization_of_Integer_has_Square_Root_in_P-adic_Integers | [
"P-adic Integers",
"Characterization of Integer has Square Root in P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Integer",
"Definition:Quadratic Residue"
] | [
"Definition:Polynomial over Ring",
"Definition:Formal Derivative of Polynomial"
] |
proofwiki-19439 | Characterization of P-adic Unit has Square Root in P-adic Units | Let $\Z_p$ be the $p$-adic integers for some odd prime $p$.
Let $Z_p^\times$ be the set of $p$-adic units.
Let $u \in Z_p^\times$ be a $p$-adic unit.
Let $u = c_0 + c_1 p + c_2 p^2 + \ldots$ be the $p$-adic expansion of $u$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$\exists x \in \Z_p^\times : x^2 {{=}} u$}}
{{item|(2):... | From Partial Sum Congruent to $p$-adic Integer Modulo Power of $p$:
:$u \equiv c_0 \pmod {p \Z_p}$ | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Odd Prime|odd prime]] $p$.
Let $Z_p^\times$ be the [[Definition:Set|set]] of [[Definition:P-adic Unit|$p$-adic units]].
Let $u \in Z_p^\times$ be a [[Definition:P-adic Unit|$p$-adic unit]].
Let $u = c_0 + c_1 p + c_2 p^2 + \ldot... | From [[Partial Sum Congruent to P-adic Integer Modulo Power of p|Partial Sum Congruent to $p$-adic Integer Modulo Power of $p$]]:
:$u \equiv c_0 \pmod {p \Z_p}$ | Characterization of P-adic Unit has Square Root in P-adic Units | https://proofwiki.org/wiki/Characterization_of_P-adic_Unit_has_Square_Root_in_P-adic_Units | https://proofwiki.org/wiki/Characterization_of_P-adic_Unit_has_Square_Root_in_P-adic_Units | [
"P-adic Units",
"Characterization of P-adic Unit has Square Root in P-adic Units"
] | [
"Definition:P-adic Integer",
"Definition:Odd Prime",
"Definition:Set",
"Definition:P-adic Unit",
"Definition:P-adic Unit",
"Definition:P-adic Expansion",
"Definition:Quadratic Residue"
] | [
"Partial Sum Congruent to P-adic Integer Modulo Power of p"
] |
proofwiki-19440 | Quotient Group of Quadratic Residues Modulo p of P-adic Units | Let $\Q_p$ be the $p$-adic numbers for some prime $p \ne 2$.
Let $\Q_p^\times$ denote the set of invertible elements of $\Q_p$.
Let $\paren{\Q_p^\times}^2 = \set{a^2 : a \in \Q_p^\times}$
Then the multiplicative quotient group $\Q_p^\times \mathop/ \paren{\Q_p^\times}^2$ has order $4$:
:$\exists c \in \Q_p^\times \setm... | By definition of field:
:$\Q_p^\times = \Q_p \setminus \set{0}$ is an abelian group
From Group of Units is Group:
:$\struct{\Q_p^\times, \times}$ is a subgroup of $\struct{\Q_p^*, \times}$
From Power of Elements is Subgroup:
:$\struct{\paren{\Q_p^\times}^2, \times}$ is a subgroup of $\struct{\Q_p^\times, \times}$
By de... | Let $\Q_p$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p \ne 2$.
Let $\Q_p^\times$ denote the [[Definition:Set|set]] of [[Definition:invertible|invertible]] [[Definition:Element|elements]] of $\Q_p$.
Let $\paren{\Q_p^\times}^2 = \set{a^2 : a \in \Q... | By definition of [[Definition:Field (Abstract Algebra)|field]]:
:$\Q_p^\times = \Q_p \setminus \set{0}$ is an [[Definition:Abelian Group|abelian group]]
From [[Group of Units is Group]]:
:$\struct{\Q_p^\times, \times}$ is a [[Definition:Subgroup|subgroup]] of $\struct{\Q_p^*, \times}$
From [[Power of Elements is Subg... | Quotient Group of Quadratic Residues Modulo p of P-adic Units | https://proofwiki.org/wiki/Quotient_Group_of_Quadratic_Residues_Modulo_p_of_P-adic_Units | https://proofwiki.org/wiki/Quotient_Group_of_Quadratic_Residues_Modulo_p_of_P-adic_Units | [
"P-adic Units"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:Set",
"Definition:invertible",
"Definition:Element",
"Definition:Multiplicative Group",
"Definition:Quotient Group",
"Definition:Order of Structure",
"Definition:Transversal (Group Theory)/Left Transversal"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Abelian Group",
"Group of Units is Group",
"Definition:Subgroup",
"Power of Elements is Subgroup",
"Definition:Subgroup",
"Definition:Quotient Group",
"Definition:Quotient Group"
] |
proofwiki-19441 | Bounded Linear Transformation to Banach Space has Unique Extension to Closure of Domain | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a normed vector space over $\Bbb F$.
Let $\map D {T_0}$ be a linear subspace of $X$.
Let $\map D T = \paren {\map D T}^-$.
Let $\map \BB {\map D {T_0}, Y}$ be the space of bounded linear transformations on $X$.
Let $\norm {\, \cdot \,}_{\map \BB {\map... | === Existence ===
Since $T_0$ is bounded, there exists a real number $M > 0$ such that:
:$\norm {T_0 x}_Y \le M \norm x_X$ for all $x \in \map D {T_0}$.
Let $x \in \map D T \setminus \map D {T_0}$.
From Point in Closure of Subset of Metric Space iff Limit of Sequence, there exists a sequence $\sequence {x_n}_{n \matho... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $\map D {T_0}$ be a [[Definition:Linear Subspace|linear subspace]] of $X$.
Let $\map D T = \paren {\map D T}^-$.
Let $\map \BB {\map D {T_0}, Y}$ be the [[Definition:Space... | === Existence ===
Since $T_0$ is [[Definition:Bounded Linear Transformation|bounded]], there exists a [[Definition:Real Number|real number]] $M > 0$ such that:
:$\norm {T_0 x}_Y \le M \norm x_X$ for all $x \in \map D {T_0}$.
Let $x \in \map D T \setminus \map D {T_0}$.
From [[Point in Closure of Subset of Metric S... | Bounded Linear Transformation to Banach Space has Unique Extension to Closure of Domain | https://proofwiki.org/wiki/Bounded_Linear_Transformation_to_Banach_Space_has_Unique_Extension_to_Closure_of_Domain | https://proofwiki.org/wiki/Bounded_Linear_Transformation_to_Banach_Space_has_Unique_Extension_to_Closure_of_Domain | [
"Bounded Linear Transformations",
"Bounded Linear Transformation to Banach Space has Unique Extension to Closure of Domain",
"Banach Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Subspace",
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Banach Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm/Bounded Linear Transformation",
"Defi... | [
"Definition:Bounded Linear Transformation",
"Definition:Real Number",
"Point in Closure of Subset of Metric Space iff Limit of Sequence",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Convergent Sequence is Cauchy Sequence/N... |
proofwiki-19442 | Existence and Uniqueness of Adjoint of Densely-Defined Linear Operator | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\Bbb F$.
Let $\struct {\map D T, T}$ be a densely defined linear operator on $\HH$.
For each $y \in \HH$, define the linear functional $f_x : \map D T \to \Bbb F$ by:
:$\map {f_y} x = \innerprod {T x} y$ for each $x \i... | We first check that $\map D {T^\ast}$ is a linear subspace of $\HH$, so that the question whether $T^\ast$ be a linear transformation is well-posed.
Let $u, v \in \map D {T^\ast}$ and $\alpha \in \Bbb F$.
Then there exists real numbers $M_1, M_2 > 0$ such that:
:$\cmod {\map {f_u} x} \le M_1 \norm x$
and:
:$\cmod {\m... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\Bbb F$.
Let $\struct {\map D T, T}$ be a [[Definition:Densely-Defined Linear Operator|densely defined linear operator]] on $\HH$.
For each $y \in \HH$, define the [[Definition:Linear Fu... | We first check that $\map D {T^\ast}$ is a [[Definition:Linear Subspace|linear subspace]] of $\HH$, so that the question whether $T^\ast$ be a [[Definition:Linear Transformation|linear transformation]] is well-posed.
Let $u, v \in \map D {T^\ast}$ and $\alpha \in \Bbb F$.
Then there exists [[Definition:Real Number|r... | Existence and Uniqueness of Adjoint of Densely-Defined Linear Operator | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Adjoint_of_Densely-Defined_Linear_Operator | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Adjoint_of_Densely-Defined_Linear_Operator | [
"Adjoints (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Densely-Defined Linear Operator",
"Definition:Linear Functional",
"Definition:Linear Transformation"
] | [
"Definition:Linear Subspace",
"Definition:Linear Transformation",
"Definition:Real Number",
"Triangle Inequality",
"Complex Modulus equals Complex Modulus of Conjugate",
"Definition:Bounded Linear Functional",
"One-Step Vector Subspace Test",
"Definition:Linear Subspace",
"Definition:Bounded Linear ... |
proofwiki-19443 | Point Spectrum of Densely-Defined Linear Operator consists of its Eigenvalues | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\struct {\map D T, T}$ be a densely-defined linear operator.
Let $\map {\sigma_p} T$ be the point spectrum of $T$.
Then $\lambda \in \map {\sigma_p} T$ {{iff}} $\lambda$ is an eigenvalue of $T$. | We have that $\lambda \in \map {\sigma_p} T$ {{iff}}:
:$T - \lambda I$ is not injective.
That is, {{iff}} there exists $x \in \map D T \setminus \set 0$ such that:
:$\paren {T - \lambda I} x = \map {\paren {T - \lambda I} } 0 = 0$
So $\lambda \in \map {\sigma_p} T$ {{iff}} there exists $x \in \map D T \setminus \set 0... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\struct {\map D T, T}$ be a [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]].
Let $\map {\sigma_p} T$ be the [[Definition:Point Spectrum of Densely-Defined Linear Operator|point s... | We have that $\lambda \in \map {\sigma_p} T$ {{iff}}:
:$T - \lambda I$ is not [[Definition:Injection|injective]].
That is, {{iff}} there exists $x \in \map D T \setminus \set 0$ such that:
:$\paren {T - \lambda I} x = \map {\paren {T - \lambda I} } 0 = 0$
So $\lambda \in \map {\sigma_p} T$ {{iff}} there exists $x ... | Point Spectrum of Densely-Defined Linear Operator consists of its Eigenvalues | https://proofwiki.org/wiki/Point_Spectrum_of_Densely-Defined_Linear_Operator_consists_of_its_Eigenvalues | https://proofwiki.org/wiki/Point_Spectrum_of_Densely-Defined_Linear_Operator_consists_of_its_Eigenvalues | [
"Point Spectrums (Densely-Defined Linear Operators)",
"Point Spectra (Densely-Defined Linear Operators)",
"Point Spectra (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Densely-Defined Linear Operator",
"Definition:Point Spectrum of Densely-Defined Linear Operator",
"Definition:Point Spectrum of Densely-Defined Linear Operator/Eigenvalue"
] | [
"Definition:Injection",
"Definition:Point Spectrum of Densely-Defined Linear Operator/Eigenvalue",
"Category:Point Spectra (Densely-Defined Linear Operators)"
] |
proofwiki-19444 | Eigenvalue of Densely-Defined Linear Operator is Approximate Eigenvalue | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\struct {\map D T, T}$ be a densely-defined linear operator.
Let $\lambda$ be an eigenvalue of $T$.
Then $\lambda$ is an approximate eigenvalue of $T$. | Since $\lambda$ is an eigenvalue of $T$, there exists $x \in \map D T \setminus \set 0$ such that:
:$\paren {T - \lambda I} x = 0$
Then setting:
:$\ds x_n = \frac x {\norm x}$
we have:
:$\paren {T - \lambda I} x_n = 0$
for each $n \in \N$, while $\norm {x_n} = 1$.
Then, we have:
:$\paren {T - \lambda I} x_n \to 0$
S... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\struct {\map D T, T}$ be a [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]].
Let $\lambda$ be an [[Definition:Eigenvalue of Densely-Defined Linear Operator|eigenvalue]] of $T$.
... | Since $\lambda$ is an [[Definition:Eigenvalue of Densely-Defined Linear Operator|eigenvalue]] of $T$, there exists $x \in \map D T \setminus \set 0$ such that:
:$\paren {T - \lambda I} x = 0$
Then setting:
:$\ds x_n = \frac x {\norm x}$
we have:
:$\paren {T - \lambda I} x_n = 0$
for each $n \in \N$, while $\nor... | Eigenvalue of Densely-Defined Linear Operator is Approximate Eigenvalue | https://proofwiki.org/wiki/Eigenvalue_of_Densely-Defined_Linear_Operator_is_Approximate_Eigenvalue | https://proofwiki.org/wiki/Eigenvalue_of_Densely-Defined_Linear_Operator_is_Approximate_Eigenvalue | [
"Approximate Eigenvalues (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Densely-Defined Linear Operator",
"Definition:Point Spectrum of Densely-Defined Linear Operator/Eigenvalue",
"Definition:Approximate Eigenvalue/Densely-Defined Linear Operator"
] | [
"Definition:Point Spectrum of Densely-Defined Linear Operator/Eigenvalue",
"Definition:Approximate Eigenvalue/Densely-Defined Linear Operator",
"Category:Approximate Eigenvalues (Densely-Defined Linear Operators)"
] |
proofwiki-19445 | Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\struct {\map D T, T}$ be a self-adjoint densely-defined linear operator.
Let $\map \sigma T$ be the spectrum of $\struct {\map D T, T}$.
Let $\map {\sigma_r} T$ be the residual spectrum of $\struct {\map D T, T}$.
Let $\lambda \in \map \si... | Let $\lambda \in \map \sigma T$.
From Partition of Spectrum of Densely-Defined Linear Operator, $\lambda$ is contained in either the point spectrum or continuous spectrum of $T$.
If $\lambda$ is contained in the point spectrum, we have the result from Eigenvalue of Densely-Defined Linear Operator is Approximate Eigenva... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
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 $\map \sigma T$ be the [... | Let $\lambda \in \map \sigma T$.
From [[Partition of Spectrum of Densely-Defined Linear Operator]], $\lambda$ is contained in either the [[Definition:Point Spectrum of Densely-Defined Linear Operator|point spectrum]] or [[Definition:Continuous Spectrum of Densely-Defined Linear Operator|continuous spectrum]] of $T$.
... | Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue | https://proofwiki.org/wiki/Element_of_Spectrum_of_Densely-Defined_Linear_Operator_not_in_Residual_Spectrum_is_Approximate_Eigenvalue | https://proofwiki.org/wiki/Element_of_Spectrum_of_Densely-Defined_Linear_Operator_not_in_Residual_Spectrum_is_Approximate_Eigenvalue | [
"Approximate Eigenvalues (Densely-Defined Linear Operators)",
"Spectra (Densely-Defined Linear Operators)",
"Element of Spectrum of Densely-Defined Linear Operator not in Residual Spectrum is Approximate Eigenvalue"
] | [
"Definition:Hilbert Space",
"Definition:Self-Adjoint Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator",
"Definition:Spectrum (Spectral Theory)/Densely-Defined Linear Operator",
"Definition:Residual Spectrum of Densely-Defined Linear Operator",
"Definition:Approximate Eigenvalu... | [
"Partition of Spectrum of Densely-Defined Linear Operator",
"Definition:Point Spectrum of Densely-Defined Linear Operator",
"Definition:Continuous Spectrum of Densely-Defined Linear Operator",
"Definition:Point Spectrum of Densely-Defined Linear Operator",
"Eigenvalue of Densely-Defined Linear Operator is A... |
proofwiki-19446 | Separability of Normed Vector Space preserved under Isometric Isomorphism | Let $\struct {X, \norm \cdot_X}$ be a separable normed vector space.
Let $\struct {Y, \norm \cdot_Y}$ be a normed vector space that is isometrically isomorphic to $\struct {X, \norm \cdot_X}$.
Let $T : X \to Y$ be an isometric isomorphism between $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$.
Then $\str... | Let $\mathcal S = \set {x_n : n \in \N}$ be a countable everywhere dense subset of $X$.
We show that $\map T {\mathcal S} = \set {T x_n : n \in \N}$ is a countable everywhere dense subset of $Y$.
Let $y \in Y$ and $\epsilon > 0$.
Since $T$ is a bijection, there exists $x \in X$ such that $y = T x$.
Since $\mathcal S$... | Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Separable Space|separable]] [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {Y, \norm \cdot_Y}$ be a [[Definition:Normed Vector Space|normed vector space]] that is [[Definition:Isometric Isomorphism on Normed Vector Space|isometrically isomorphic]... | Let $\mathcal S = \set {x_n : n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense]] [[Definition:Subset|subset]] of $X$.
We show that $\map T {\mathcal S} = \set {T x_n : n \in \N}$ is a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere d... | Separability of Normed Vector Space preserved under Isometric Isomorphism | https://proofwiki.org/wiki/Separability_of_Normed_Vector_Space_preserved_under_Isometric_Isomorphism | https://proofwiki.org/wiki/Separability_of_Normed_Vector_Space_preserved_under_Isometric_Isomorphism | [
"Isometric Isomorphisms (Normed Vector Spaces)",
"Separable Spaces"
] | [
"Definition:Separable Space",
"Definition:Normed Vector Space",
"Definition:Normed Vector Space",
"Definition:Isometric Isomorphism/Normed Vector Space",
"Definition:Isometric Isomorphism/Normed Vector Space",
"Definition:Separable Space"
] | [
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Subset",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Subset",
"Definition:Bijection",
"Definition:Everywhere Dense",
"Definition:Linear Isometry",
"Definition:Everywhere Dense",
"Image of Countable ... |
proofwiki-19447 | Propositional Tautology is Tautology in Predicate Logic | Let $\LL_1$ be the language of predicate logic.
Let $\LL_0$ be the language of propositional logic.
Let the basic WFFs of $\LL_1$ be the vocabulary of $\LL_0$.
Let $\mathbf A$ be a $\mathrm{BI}$-tautology of $\LL_0$, by considering the basic subformulas of $\mathbf A$ as part of the vocabulary of $\LL_0$.
Then $\mathbf... | We proceed by the Principle of Structural Induction on the bottom-up specification of $\LL_1$, applied to $\mathbf A$.
Define $\map {v_F} {\mathbf A} = F$ for all basic WFFs $\mathbf A$ as a boolean interpretation for $\LL_0$.
Consider the case $\mathbf A$ is formed by either $\paren{ \mathbf W ~ \PP_n }$ or $\paren{ \... | Let $\LL_1$ be the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $\LL_0$ be the [[Definition:Language of Propositional Logic|language of propositional logic]].
Let the [[Definition:Basic WFF of Predicate Logic|basic WFFs]] of $\LL_1$ be the [[Definition:Vocabulary of Propositional Logic|... | We proceed by the [[Principle of Structural Induction]] on the [[Definition:Bottom-Up Specification of Predicate Logic|bottom-up specification]] of $\LL_1$, applied to $\mathbf A$.
Define $\map {v_F} {\mathbf A} = F$ for all [[Definition:Basic WFF of Predicate Logic|basic WFFs]] $\mathbf A$ as a [[Definition:Boolean I... | Propositional Tautology is Tautology in Predicate Logic | https://proofwiki.org/wiki/Propositional_Tautology_is_Tautology_in_Predicate_Logic | https://proofwiki.org/wiki/Propositional_Tautology_is_Tautology_in_Predicate_Logic | [
"Boolean Interpretations",
"Model Theory for Predicate Logic"
] | [
"Definition:Language of Predicate Logic",
"Definition:Language of Propositional Logic",
"Definition:Basic WFF of Predicate Logic",
"Definition:Language of Propositional Logic/Alphabet/Letter",
"Definition:Tautology/Formal Semantics/Boolean Interpretations",
"Definition:Basic WFF of Predicate Logic",
"De... | [
"Principle of Structural Induction",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Basic WFF of Predicate Logic",
"Definition:Boolean Interpretation",
"Definition:Basic WFF of Predicate Logic",
"Definition:Tautology/Formal Semantics/Boolean Interpretations",
"Definition:Tautology/... |
proofwiki-19448 | Integrated Linear Differential Mapping is Continuous | Let $C^1 \closedint a b := \map {C^1} {\closedint a b, \R}$ be the space of real functions of differentiability class $C^1$.
Let $S$ be the set of differentiable functions on closed real interval vanishing at their endpoints:
:$S := \set {\mathbf h \in C^1 \closedint a b : \map {\mathbf h} a = \map {\mathbf h} b = 0}$
... | We have that the Integrated Linear Differential Mapping is Linear.
For $\mathbf h \in S$ we have:
{{begin-eqn}}
{{eqn | l = \size {\map L {\mathbf h} }
| r = \size {\int_a^b \paren {\map {\mathbf A} t \map {\mathbf h} t + \map {\mathbf B} t \map {\mathbf h'} t }\rd t}
}}
{{eqn | o = \le
| r = \int_a^b \siz... | Let $C^1 \closedint a b := \map {C^1} {\closedint a b, \R}$ be the [[Definition:Space of Continuous Functions of Differentiability Class k|space of real functions of differentiability class $C^1$]].
Let $S$ be the [[Definition:Set|set]] of [[Definition:Differentiable Real Function|differentiable functions]] on [[Defin... | We have that the [[Integrated Linear Differential Mapping is Linear]].
For $\mathbf h \in S$ we have:
{{begin-eqn}}
{{eqn | l = \size {\map L {\mathbf h} }
| r = \size {\int_a^b \paren {\map {\mathbf A} t \map {\mathbf h} t + \map {\mathbf B} t \map {\mathbf h'} t }\rd t}
}}
{{eqn | o = \le
| r = \int_a^... | Integrated Linear Differential Mapping is Continuous | https://proofwiki.org/wiki/Integrated_Linear_Differential_Mapping_is_Continuous | https://proofwiki.org/wiki/Integrated_Linear_Differential_Mapping_is_Continuous | [
"Operator Theory",
"Continuous Mappings",
"Linear Transformations"
] | [
"Definition:Space of Continuous Functions of Differentiability Class k",
"Definition:Set",
"Definition:Differentiable Mapping/Real Function",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Endpoints",
"Definition:C^k Norm",
"Definition:Continuous Real Function/Subset",
"Definition:Integr... | [
"Integrated Linear Differential Mapping is Linear",
"Continuity of Linear Transformation/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)/Space"
] |
proofwiki-19449 | Partial Sum Congruent to P-adic Integer Modulo Power of p | Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $a \in \Z_p$.
Let $\ds a = \sum_{j=0}^\infty d_j p^j$ be the $p$-adic expansion of $a$
For all $n \in \N$, let $\ds a_n = \sum_{j=0}^n d_j p^j$ be the $n$-th partial sum of the $p$-adic expansion of $a$
Then:
:$\forall n \in \N : a_n \equiv a \pmod{p^{n+1}\Z_p... | We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = a - a_n
| r = \sum_{j = 0}^\infty d_j p^j - \sum_{j = 0}^n d_j p^j
| c = {{hypothesis}}
}}
{{eqn | r = \sum_{j = n+1}^\infty d_j p^j
| c = Removing first $n$ terms from the series
}}
{{eqn | r = p^{n+1} \sum_{j = 0}^\infty d_{j+n+1} p^j
... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
Let $a \in \Z_p$.
Let $\ds a = \sum_{j=0}^\infty d_j p^j$ be the [[Definition:P-adic Expansion|$p$-adic expansion]] of $a$
For all $n \in \N$, let $\ds a_n = \sum_{j=0}^n d_j p^j$ be the [[Definition:Par... | We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = a - a_n
| r = \sum_{j = 0}^\infty d_j p^j - \sum_{j = 0}^n d_j p^j
| c = {{hypothesis}}
}}
{{eqn | r = \sum_{j = n+1}^\infty d_j p^j
| c = Removing first $n$ terms from the [[Definition:Series|series]]
}}
{{eqn | r = p^{n+1} \sum_{j = 0}^... | Partial Sum Congruent to P-adic Integer Modulo Power of p | https://proofwiki.org/wiki/Partial_Sum_Congruent_to_P-adic_Integer_Modulo_Power_of_p | https://proofwiki.org/wiki/Partial_Sum_Congruent_to_P-adic_Integer_Modulo_Power_of_p | [
"P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:P-adic Expansion",
"Definition:Series/Sequence of Partial Sums",
"Definition:P-adic Expansion"
] | [
"Definition:Series",
"Definition:Series",
"Category:P-adic Integers"
] |
proofwiki-19450 | Lp Norm is Well-Defined | Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \hointr 1 \infty$.
Let $\map {\LL^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space of $\struct {X, \Sigma, \mu}$.
Let $\sim$ be the $\mu$-almost-everywhere equality relation on $\map {\LL^p} {X, \Sigma, \mu}$.
Let $\map {L^p} {X, \Sigma, \mu}$ be the $L^... | Note that:
:$\ds \int \size f^p \rd \mu$
is well-defined from the definition of the Lebesgue $p$-space.
We show that for $E \in \map {L^p} {X, \Sigma, \mu}$, $\norm E_p$ is independent of the representative chosen for $E$.
Let:
:$E = \eqclass f \sim = \eqclass g \sim$
for $\eqclass f \sim, \eqclass g \sim \in \map {... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]], and let $p \in \hointr 1 \infty$.
Let $\map {\LL^p} {X, \Sigma, \mu}$ be the [[Definition:Lebesgue Space|Lebesgue $p$-space]] of $\struct {X, \Sigma, \mu}$.
Let $\sim$ be the [[Definition:Almost-Everywhere Equality Relation|$\mu$-almost... | Note that:
:$\ds \int \size f^p \rd \mu$
is well-defined from the definition of the [[Definition:Lebesgue Space|Lebesgue $p$-space]].
We show that for $E \in \map {L^p} {X, \Sigma, \mu}$, $\norm E_p$ is independent of the [[Definition:Representative of Equivalence Class|representative]] chosen for $E$.
Let:
:$E... | Lp Norm is Well-Defined | https://proofwiki.org/wiki/Lp_Norm_is_Well-Defined | https://proofwiki.org/wiki/Lp_Norm_is_Well-Defined | [
"Lp Norms"
] | [
"Definition:Measure Space",
"Definition:Lebesgue Space",
"Definition:Almost-Everywhere Equality Relation",
"Definition:Lp Space",
"Definition:Lp Norm"
] | [
"Definition:Lebesgue Space",
"Definition:Equivalence Class/Representative",
"Equivalence Class Equivalent Statements",
"Definition:Almost-Everywhere Equality Relation",
"Definition:Almost Everywhere",
"Definition:Null Set",
"Definition:Almost Everywhere",
"A.E. Equal Positive Measurable Functions have... |
proofwiki-19451 | Equivalence of Definitions of Left Quasi-Reflexive Relation | Let $\RR \subseteq S \times S$ be a relation in $S$.
{{TFAE|def = Left Quasi-Reflexive Relation}} | === $(1)$ implies $(2)$ ===
Let $\RR$ be a left quasi-reflexive relation by definition $1$.
Then by definition:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR$
Let $x \in \Dom \RR$ be arbitrary.
Then by definition of domain:
:$\exists y \in S: \tuple {x, y} \in \RR$
Hence {{hypothesis}}:
:$\... | Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation in $S$]].
{{TFAE|def = Left Quasi-Reflexive Relation}} | === $(1)$ implies $(2)$ ===
Let $\RR$ be a [[Definition:Left Quasi-Reflexive Relation/Definition 1|left quasi-reflexive relation by definition $1$]].
Then by definition:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR$
Let $x \in \Dom \RR$ be arbitrary.
Then by definition of [[Definition:... | Equivalence of Definitions of Left Quasi-Reflexive Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Left_Quasi-Reflexive_Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Left_Quasi-Reflexive_Relation | [
"Left Quasi-Reflexive Relations"
] | [
"Definition:Relation"
] | [
"Definition:Left Quasi-Reflexive Relation/Definition 1",
"Definition:Domain (Set Theory)/Relation",
"Definition:Left Quasi-Reflexive Relation/Definition 2",
"Definition:Left Quasi-Reflexive Relation/Definition 2",
"Definition:Left Quasi-Reflexive Relation/Definition 1"
] |
proofwiki-19452 | Equivalence of Definitions of Right Quasi-Reflexive Relation | Let $\RR \subseteq S \times S$ be a relation in $S$.
{{TFAE|def = Right Quasi-Reflexive Relation}} | === $(1)$ implies $(2)$ ===
Let $\RR$ be a right quasi-reflexive relation by definition $1$.
Then by definition:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {y, y} \in \RR$
Let $y \in \Img \RR$ be arbitrary.
Then by definition of image set:
:$\exists x \in S: \tuple {x, y} \in \RR$
Hence {{hypothesis}}:... | Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation in $S$]].
{{TFAE|def = Right Quasi-Reflexive Relation}} | === $(1)$ implies $(2)$ ===
Let $\RR$ be a [[Definition:Right Quasi-Reflexive Relation/Definition 1|right quasi-reflexive relation by definition $1$]].
Then by definition:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {y, y} \in \RR$
Let $y \in \Img \RR$ be arbitrary.
Then by definition of [[Definitio... | Equivalence of Definitions of Right Quasi-Reflexive Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Right_Quasi-Reflexive_Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Right_Quasi-Reflexive_Relation | [
"Right Quasi-Reflexive Relations"
] | [
"Definition:Relation"
] | [
"Definition:Right Quasi-Reflexive Relation/Definition 1",
"Definition:Image (Set Theory)/Relation/Relation",
"Definition:Right Quasi-Reflexive Relation/Definition 2",
"Definition:Right Quasi-Reflexive Relation/Definition 2",
"Definition:Image (Set Theory)/Relation/Relation",
"Definition:Right Quasi-Reflex... |
proofwiki-19453 | Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\map {\mathcal M} {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.
Let $\sim$ be the almost-everywhere equality relation on $\map {\mathcal M} {X, \Sigma, \R}$.
Let $\map {\mathcal M} {X, \Sigma, \R}/\sim$ be the space of real-valu... | Let $E_1, E_2 \in \map {\mathcal M} {X, \Sigma, \R}/\sim$.
First, we show that if $E_1 = \eqclass f \sim$ and $E_2 = \eqclass g \sim$, that $\eqclass {f + g} \sim$ is well-understood.
This follows from Pointwise Sum of Measurable Functions is Measurable.
We now need to show that $E_1 + E_2$ is independent of the choic... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\map {\mathcal M} {X, \Sigma, \R}$ be the set of [[Definition:Measurable Real-Valued Function|real-valued $\Sigma$-measurable functions]] on $X$.
Let $\sim$ be the [[Definition:Almost-Everywhere Equality Relation|almost-everywhere e... | Let $E_1, E_2 \in \map {\mathcal M} {X, \Sigma, \R}/\sim$.
First, we show that if $E_1 = \eqclass f \sim$ and $E_2 = \eqclass g \sim$, that $\eqclass {f + g} \sim$ is well-understood.
This follows from [[Pointwise Sum of Measurable Functions is Measurable]].
We now need to show that $E_1 + E_2$ is independent of th... | Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined | https://proofwiki.org/wiki/Pointwise_Addition_on_Space_of_Real-Valued_Measurable_Functions_Identified_by_A.E._Equality_is_Well-Defined | https://proofwiki.org/wiki/Pointwise_Addition_on_Space_of_Real-Valued_Measurable_Functions_Identified_by_A.E._Equality_is_Well-Defined | [
"Space of Measurable Functions Identified by A.E. Equality",
"Space of Real-Valued Measurable Functions Identified by A.E. Equality",
"Space of Real-Valued Measurable Functions Identified by A.E. Equality"
] | [
"Definition:Measure Space",
"Definition:Measurable Function/Real-Valued Function",
"Definition:Almost-Everywhere Equality Relation",
"Definition:Space of Measurable Functions Identified by A.E. Equality/Real-Valued Function",
"Definition:Pointwise Addition on Space of Real-Valued Measurable Functions Identi... | [
"Pointwise Sum of Measurable Functions is Measurable",
"Definition:Equivalence Class/Representative",
"Equivalence Class Equivalent Statements",
"Pointwise Addition preserves A.E. Equality",
"Equivalence Class Equivalent Statements",
"Category:Space of Real-Valued Measurable Functions Identified by A.E. E... |
proofwiki-19454 | Pointwise Scalar Multiplication preserves A.E. Equality | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f, g : X \to \overline \R$ be extended real-valued functions such that:
:$f = g$ $\mu$-almost everywhere.
Let $\lambda \in \overline \R$.
Then:
:$\lambda \cdot f = \lambda \cdot g$ $\mu$-almost everywhere
where $\lambda \cdot f$ denotes pointwise scalar multipli... | Since:
:$f = g$ $\mu$-almost everywhere
there exists a $\mu$-null set $N \subseteq X$ such that:
:if $\map f x \ne \map g x$ then $x \in N$.
Note that if $\map f x = \map g x$ then $\lambda \map f x = \lambda \map g x$.
So, from the Rule of Transposition we have:
:if $\lambda \map f x \ne \lambda \map g x$ then $\ma... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f, g : X \to \overline \R$ be [[Definition:Extended Real-Valued Function|extended real-valued functions]] such that:
:$f = g$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]].
Let $\lambda \in \overline \R$.
Then:
:$\la... | Since:
:$f = g$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]]
there exists a [[Definition:Null Set|$\mu$-null set]] $N \subseteq X$ such that:
:if $\map f x \ne \map g x$ then $x \in N$.
Note that if $\map f x = \map g x$ then $\lambda \map f x = \lambda \map g x$.
So, from the [[Rule of Transpositio... | Pointwise Scalar Multiplication preserves A.E. Equality | https://proofwiki.org/wiki/Pointwise_Scalar_Multiplication_preserves_A.E._Equality | https://proofwiki.org/wiki/Pointwise_Scalar_Multiplication_preserves_A.E._Equality | [
"Measure Theory",
"Almost-Everywhere Equality Relation",
"Almost-Everywhere Equality Relation"
] | [
"Definition:Measure Space",
"Definition:Extended Real-Valued Function",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Pointwise Scalar Multiplication of Extended Real-Valued Functions"
] | [
"Definition:Almost Everywhere",
"Definition:Null Set",
"Rule of Transposition",
"Definition:Almost Everywhere",
"Category:Almost-Everywhere Equality Relation"
] |
proofwiki-19455 | Equivalence of Definitions of Quasi-Reflexive Relation | Let $\RR \subseteq S \times S$ be a relation in $S$.
{{TFAE|def = Quasi-Reflexive Relation}} | === $(1)$ implies $(3)$ ===
Let $\RR$ be a quasi-reflexive relation by definition $1$.
Then by definition:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR \land \tuple {y, y} \in \RR$
That is:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR$
and:
:$\forall x, y \in ... | Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation in $S$]].
{{TFAE|def = Quasi-Reflexive Relation}} | === $(1)$ implies $(3)$ ===
Let $\RR$ be a [[Definition:Quasi-Reflexive Relation/Definition 1|quasi-reflexive relation by definition $1$]].
Then by definition:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR \land \tuple {y, y} \in \RR$
That is:
:$\forall x, y \in S: \tuple {x, y} \in \RR ... | Equivalence of Definitions of Quasi-Reflexive Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Quasi-Reflexive_Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Quasi-Reflexive_Relation | [
"Quasi-Reflexive Relations"
] | [
"Definition:Relation"
] | [
"Definition:Quasi-Reflexive Relation/Definition 1",
"Definition:Left Quasi-Reflexive Relation/Definition 2",
"Definition:Right Quasi-Reflexive Relation/Definition 2",
"Definition:Quasi-Reflexive Relation/Definition 3",
"Definition:Quasi-Reflexive Relation/Definition 3",
"Definition:Left Quasi-Reflexive Re... |
proofwiki-19456 | Reflexive Relation is Quasi-Reflexive | Let $\RR$ be a reflexive relation on a set $S$.
Then $\RR$ is a quasi-reflexive relation on $S$. | By definition of reflexive relation:
:$\forall x \in S: \tuple {x, x} \in \RR$
Hence by definition of domain:
:$x \in \Dom \RR$
and hence by definition of field and Set is Subset of Union:
:$x \in \Field \RR$
That is:
:$\forall x \in \Field \RR: \tuple {x, x} \in \RR$
Hence the result by definition of quasi-reflexive r... | Let $\RR$ be a [[Definition:Reflexive Relation|reflexive relation]] on a [[Definition:Set|set]] $S$.
Then $\RR$ is a [[Definition:Quasi-Reflexive Relation|quasi-reflexive relation]] on $S$. | By definition of [[Definition:Reflexive Relation|reflexive relation]]:
:$\forall x \in S: \tuple {x, x} \in \RR$
Hence by definition of [[Definition:Domain of Relation|domain]]:
:$x \in \Dom \RR$
and hence by definition of [[Definition:Field of Relation|field]] and [[Set is Subset of Union]]:
:$x \in \Field \RR$
That... | Reflexive Relation is Quasi-Reflexive | https://proofwiki.org/wiki/Reflexive_Relation_is_Quasi-Reflexive | https://proofwiki.org/wiki/Reflexive_Relation_is_Quasi-Reflexive | [
"Quasi-Reflexive Relations",
"Reflexive Relations"
] | [
"Definition:Reflexive Relation",
"Definition:Set",
"Definition:Quasi-Reflexive Relation"
] | [
"Definition:Reflexive Relation",
"Definition:Domain (Set Theory)/Relation",
"Definition:Field of Relation",
"Set is Subset of Union",
"Definition:Quasi-Reflexive Relation",
"Category:Quasi-Reflexive Relations",
"Category:Reflexive Relations"
] |
proofwiki-19457 | Diagonal Relation is Reflexive (Class Theory) | Let $V$ be a basic universe.
Let $\Delta_V$ denote the diagonal relation on $V$:
:$\Delta_V = \set {\tuple {x, x}: x \in V}$
$\Delta_V$ is a reflexive relation. | {{begin-eqn}}
{{eqn | q = \forall x \in V
| l = x
| r = x
| c = {{Defof|Equals}}
}}
{{eqn | ll= \leadsto
| l = \tuple {x, x}
| o = \in
| r = \Delta_V
| c = {{Defof|Diagonal Relation}}
}}
{{end-eqn}}
So $\Delta_V$ is reflexive.
{{qed}} | Let $V$ be a [[Definition:Basic Universe|basic universe]].
Let $\Delta_V$ denote the [[Definition:Diagonal Relation (Class Theory)|diagonal relation]] on $V$:
:$\Delta_V = \set {\tuple {x, x}: x \in V}$
$\Delta_V$ is a [[Definition:Reflexive Relation (Class Theory)|reflexive relation]]. | {{begin-eqn}}
{{eqn | q = \forall x \in V
| l = x
| r = x
| c = {{Defof|Equals}}
}}
{{eqn | ll= \leadsto
| l = \tuple {x, x}
| o = \in
| r = \Delta_V
| c = {{Defof|Diagonal Relation}}
}}
{{end-eqn}}
So $\Delta_V$ is [[Definition:Reflexive Relation|reflexive]].
{{qed}} | Diagonal Relation is Reflexive (Class Theory) | https://proofwiki.org/wiki/Diagonal_Relation_is_Reflexive_(Class_Theory) | https://proofwiki.org/wiki/Diagonal_Relation_is_Reflexive_(Class_Theory) | [
"Diagonal Relation",
"Examples of Reflexive Relations"
] | [
"Definition:Basic Universe",
"Definition:Diagonal Relation/Class Theory",
"Definition:Reflexive Relation/Class Theory"
] | [
"Definition:Reflexive Relation"
] |
proofwiki-19458 | Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\map {\mathcal M} {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.
Let $\sim$ be the $\mu$-almost-everywhere equality relation on $\map {\mathcal M} {X, \Sigma, \R}$.
Let $\map {\mathcal M} {X, \Sigma, \R}/\sim$ be the space of rea... | Let $\lambda \in \R$.
Let $E \in \map {\mathcal M} {X, \Sigma, \R}/\sim$.
First, we show that if $E = \eqclass f \sim$, then $\eqclass {\lambda f} \sim$ is well-understood.
This follows from Pointwise Scalar Multiple of Measurable Function is Measurable.
We need to show that $\lambda \cdot E$ is independent of the choi... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\map {\mathcal M} {X, \Sigma, \R}$ be the set of [[Definition:Measurable Real-Valued Function|real-valued $\Sigma$-measurable functions]] on $X$.
Let $\sim$ be the [[Definition:Almost-Everywhere Equality Relation|$\mu$-almost-everyw... | Let $\lambda \in \R$.
Let $E \in \map {\mathcal M} {X, \Sigma, \R}/\sim$.
First, we show that if $E = \eqclass f \sim$, then $\eqclass {\lambda f} \sim$ is well-understood.
This follows from [[Pointwise Scalar Multiple of Measurable Function is Measurable]].
We need to show that $\lambda \cdot E$ is independent of ... | Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined | https://proofwiki.org/wiki/Pointwise_Scalar_Multiplication_on_Space_of_Real-Valued_Measurable_Functions_Identified_by_A.E._Equality_is_Well-Defined | https://proofwiki.org/wiki/Pointwise_Scalar_Multiplication_on_Space_of_Real-Valued_Measurable_Functions_Identified_by_A.E._Equality_is_Well-Defined | [
"Space of Real-Valued Measurable Functions Identified by A.E. Equality"
] | [
"Definition:Measure Space",
"Definition:Measurable Function/Real-Valued Function",
"Definition:Almost-Everywhere Equality Relation",
"Definition:Space of Measurable Functions Identified by A.E. Equality/Real-Valued Function",
"Definition:Pointwise Scalar Multiplication on Space of Real-Valued Measurable Fun... | [
"Pointwise Scalar Multiple of Measurable Function is Measurable",
"Definition:Equivalence Class/Representative",
"Equivalence Class Equivalent Statements",
"Pointwise Scalar Multiplication preserves A.E. Equality",
"Equivalence Class Equivalent Statements",
"Category:Space of Real-Valued Measurable Functi... |
proofwiki-19459 | Subset Relation is Ordering/Class Theory | Let $C$ be a class.
Then the subset relation $\subseteq$ is an ordering on $C$. | To establish that $\subseteq$ is an ordering, we need to show that it is reflexive, antisymmetric and transitive.
So, checking in turn each of the criteria for an ordering: | Let $C$ be a [[Definition:Class (Class Theory)|class]].
Then the [[Definition:Subset Relation|subset relation]] $\subseteq$ is an [[Definition:Ordering (Class Theory)|ordering]] on $C$. | To establish that $\subseteq$ is an [[Definition:Ordering (Class Theory)|ordering]], we need to show that it is [[Definition:Reflexive Relation (Class Theory)|reflexive]], [[Definition:Antisymmetric Relation (Class Theory)|antisymmetric]] and [[Definition:Transitive Relation (Class Theory)|transitive]].
So, checking i... | Subset Relation is Ordering/Class Theory | https://proofwiki.org/wiki/Subset_Relation_is_Ordering/Class_Theory | https://proofwiki.org/wiki/Subset_Relation_is_Ordering/Class_Theory | [
"Subset Relation is Ordering"
] | [
"Definition:Class (Class Theory)",
"Definition:Subset Relation",
"Definition:Ordering/Class Theory"
] | [
"Definition:Ordering/Class Theory",
"Definition:Reflexive Relation/Class Theory",
"Definition:Antisymmetric Relation/Class Theory",
"Definition:Transitive Relation/Class Theory",
"Definition:Ordering/Class Theory",
"Definition:Reflexive Relation/Class Theory",
"Definition:Antisymmetric Relation/Class Th... |
proofwiki-19460 | Characterization of Polynomial has Root in P-adic Integers | Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a polynomial over $\Z_p$.
Let $a \in \Z_p$.
Then:
:$\map F a = 0$
{{iff}}:
:there exists a sequence $\sequence {a_n}$ of integers:
::$(1): \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$
::$(2): \quad \map F {a_n} \equiv 0 \pm... | === Necessary Condition ===
{{:Characterization of Polynomial has Root in P-adic Integers/Necessary Condition}}{{qed|lemma}} | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]] over $\Z_p$.
Let $a \in \Z_p$.
Then:
:$\map F a = 0$
{{iff}}:
:there exists a [[Definition:Sequence|sequenc... | === [[Characterization of Polynomial has Root in P-adic Integers/Necessary Condition|Necessary Condition]] ===
{{:Characterization of Polynomial has Root in P-adic Integers/Necessary Condition}}{{qed|lemma}} | Characterization of Polynomial has Root in P-adic Integers | https://proofwiki.org/wiki/Characterization_of_Polynomial_has_Root_in_P-adic_Integers | https://proofwiki.org/wiki/Characterization_of_Polynomial_has_Root_in_P-adic_Integers | [
"P-adic Integers",
"Characterization of Polynomial has Root in P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Polynomial over Ring",
"Definition:Sequence",
"Definition:Integer"
] | [
"Characterization of Polynomial has Root in P-adic Integers/Necessary Condition"
] |
proofwiki-19461 | Smallest Element is Unique/Class Theory | Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be an ordering.
Let $A$ be a subclass of the field of $\RR$.
Suppose $A$ has a smallest element $s$ {{WRT}} $\RR$.
Then $s$ is unique. | Let $s$ and $t$ both be smallest elements of $A$.
Then by definition:
:$\forall y \in A: s \mathrel \RR y$
:$\forall y \in A: t \mathrel \RR y$
Thus it follows that:
:$s \preceq t$
:$t \preceq s$
But as $\preceq$ is an ordering, it is antisymmetric.
Hence by definition of antisymmetric, $a = b$.
{{qed}} | Let $V$ be a [[Definition:Basic Universe|basic universe]].
Let $\RR \subseteq V \times V$ be an [[Definition:Ordering (Class Theory)|ordering]].
Let $A$ be a [[Definition:Subclass|subclass]] of the [[Definition:Field of Relation (Class Theory)|field]] of $\RR$.
Suppose $A$ has a [[Definition:Smallest Element (Class... | Let $s$ and $t$ both be [[Definition:Smallest Element (Class Theory)|smallest elements]] of $A$.
Then by definition:
:$\forall y \in A: s \mathrel \RR y$
:$\forall y \in A: t \mathrel \RR y$
Thus it follows that:
:$s \preceq t$
:$t \preceq s$
But as $\preceq$ is an [[Definition:Ordering (Class Theory)|ordering]], it... | Smallest Element is Unique/Class Theory | https://proofwiki.org/wiki/Smallest_Element_is_Unique/Class_Theory | https://proofwiki.org/wiki/Smallest_Element_is_Unique/Class_Theory | [
"Smallest Element is Unique"
] | [
"Definition:Basic Universe",
"Definition:Ordering/Class Theory",
"Definition:Subclass",
"Definition:Field of Relation/Class Theory",
"Definition:Smallest Element/Class Theory",
"Definition:Unique"
] | [
"Definition:Smallest Element/Class Theory",
"Definition:Ordering/Class Theory",
"Definition:Antisymmetric Relation/Class Theory",
"Definition:Antisymmetric Relation/Class Theory"
] |
proofwiki-19462 | Greatest Element is Unique/Class Theory | Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be an ordering.
Let $A$ be a subclass of the field of $\RR$.
Suppose $A$ has a greatest element $g$ {{WRT}} $\RR$.
Then $g$ is unique. | Let $g$ and $h$ both be smallest elements of $A$.
Then by definition:
:$\forall y \in A: y \mathrel \RR g$
:$\forall y \in A: y \mathrel \RR h$
Thus it follows that:
:$g \preceq h$
:$h \preceq g$
But as $\preceq$ is an ordering, it is antisymmetric.
Hence by definition of antisymmetric, $g = h$.
{{qed}} | Let $V$ be a [[Definition:Basic Universe|basic universe]].
Let $\RR \subseteq V \times V$ be an [[Definition:Ordering (Class Theory)|ordering]].
Let $A$ be a [[Definition:Subclass|subclass]] of the [[Definition:Field of Relation (Class Theory)|field]] of $\RR$.
Suppose $A$ has a [[Definition:Greatest Element (Class... | Let $g$ and $h$ both be [[Definition:Smallest Element (Class Theory)|smallest elements]] of $A$.
Then by definition:
:$\forall y \in A: y \mathrel \RR g$
:$\forall y \in A: y \mathrel \RR h$
Thus it follows that:
:$g \preceq h$
:$h \preceq g$
But as $\preceq$ is an [[Definition:Ordering (Class Theory)|ordering]], it... | Greatest Element is Unique/Class Theory | https://proofwiki.org/wiki/Greatest_Element_is_Unique/Class_Theory | https://proofwiki.org/wiki/Greatest_Element_is_Unique/Class_Theory | [
"Greatest Element is Unique"
] | [
"Definition:Basic Universe",
"Definition:Ordering/Class Theory",
"Definition:Subclass",
"Definition:Field of Relation/Class Theory",
"Definition:Greatest Element/Class Theory",
"Definition:Unique"
] | [
"Definition:Smallest Element/Class Theory",
"Definition:Ordering/Class Theory",
"Definition:Antisymmetric Relation/Class Theory",
"Definition:Antisymmetric Relation/Class Theory"
] |
proofwiki-19463 | Characterization of Polynomial has Root in P-adic Integers/Necessary Condition | Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a polynomial over $\Z_p$.
Let $a \in \Z_p$.
Let $\map F a = 0$.
Then:
:there exists a sequence $\sequence {a_n}$ of integers:
::$\ds (1): \quad \lim_{n \mathop \to \infty} {a_n} = a$
::$(2): \quad \map F {a_n} \equiv 0 \pmod {... | Let $\map F a = 0$.
Let $\ds a = \sum_{j \mathop = 0}^\infty d_j p^j$ be the $p$-adic expansion of $a$.
For all $n \in \N_{>0}$, let:
:$\ds a_n = \sum_{j \mathop = 0}^{n - 1} d_j p^j$
By definition of $p$-adic expansion:
:$\ds (1): \quad \lim_{n \mathop \to \infty} {a_n} = a$
By definition of $p$-adic expansion of a $p... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]] over $\Z_p$.
Let $a \in \Z_p$.
Let $\map F a = 0$.
Then:
:there exists a [[Definition:Sequence|sequence]] ... | Let $\map F a = 0$.
Let $\ds a = \sum_{j \mathop = 0}^\infty d_j p^j$ be the [[Definition:P-adic Expansion|$p$-adic expansion]] of $a$.
For all $n \in \N_{>0}$, let:
:$\ds a_n = \sum_{j \mathop = 0}^{n - 1} d_j p^j$
By definition of [[Definition:P-adic Expansion|$p$-adic expansion]]:
:$\ds (1): \quad \lim_{n \matho... | Characterization of Polynomial has Root in P-adic Integers/Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Polynomial_has_Root_in_P-adic_Integers/Necessary_Condition | https://proofwiki.org/wiki/Characterization_of_Polynomial_has_Root_in_P-adic_Integers/Necessary_Condition | [
"Characterization of Polynomial has Root in P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Polynomial over Ring",
"Definition:Sequence",
"Definition:Integer",
"Definition:Congruence Modulo Ideal",
"Definition:Ideal of Ring"
] | [
"Definition:P-adic Expansion",
"Definition:P-adic Expansion",
"Definition:P-adic Expansion",
"Definition:P-adic Integer",
"Partial Sum Congruent to P-adic Integer Modulo Power of p",
"Polynomials of Congruent Ring Elements are Congruent"
] |
proofwiki-19464 | Characterization of Polynomial has Root in P-adic Integers/Sufficient Condition | Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a polynomial over $\Z_p$.
Let $a \in \Z_p$.
Let there exist a sequence $\sequence{a_n}$ of integers:
:$\ds (1): \quad \lim_{n \mathop \to \infty} {a_n} = a$
:$(2): \quad \map F {a_n} \equiv 0 \pmod {p^{n + 1} \Z_p}$
where $\map... | Let there exist a sequence $\sequence{a_n}$ of integers:
:$\ds (1): \quad \lim_{n \mathop \to \infty} {a_n} = a$
:$(2): \quad \map F {a_n} \equiv 0 \pmod {p^{n + 1} \Z_p}$
We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = a
| o = \equiv
| r = a_n
| rr= \pmod {p^{n + 1} \Z_p}
| c =... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]] over $\Z_p$.
Let $a \in \Z_p$.
Let there exist a [[Definition:Sequence|sequence]] $\sequence{a_n}$ of [[Defin... | Let there exist a [[Definition:Sequence|sequence]] $\sequence{a_n}$ of [[Definition:Integer|integers]]:
:$\ds (1): \quad \lim_{n \mathop \to \infty} {a_n} = a$
:$(2): \quad \map F {a_n} \equiv 0 \pmod {p^{n + 1} \Z_p}$
We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = a
| o = \equiv
| r = a... | Characterization of Polynomial has Root in P-adic Integers/Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Polynomial_has_Root_in_P-adic_Integers/Sufficient_Condition | https://proofwiki.org/wiki/Characterization_of_Polynomial_has_Root_in_P-adic_Integers/Sufficient_Condition | [
"Characterization of Polynomial has Root in P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Polynomial over Ring",
"Definition:Sequence",
"Definition:Integer",
"Definition:Congruence Modulo Ideal",
"Definition:Ideal of Ring"
] | [
"Definition:Sequence",
"Definition:Integer",
"Partial Sum Congruent to P-adic Integer Modulo Power of p",
"Polynomials of Congruent Ring Elements are Congruent",
"Characterization of Closed Ball in P-adic Numbers",
"Squeeze Theorem/Sequences/Real Numbers",
"Sequence of Powers of Number less than One",
... |
proofwiki-19465 | Subclass of Subclass is Subclass | Let $A$, $B$ and $C$ be classes.
Let $A$ be a subclass of $B$.
Let $B$ be a subclass of $C$.
Then $A$ is a subclass of $C$. | Let $x \in A$ be arbitrary.
It follows by definition of subclass that $x \in B$.
It further follows by definition of subclass that $x \in C$.
So we have that $x \in A$ implies that $x \in C$.
As $x$ is arbitrary, the result follows.
{{qed}}
Category:Subclasses
9i86a2uac5awae3004l5cgl1wu13l7g | Let $A$, $B$ and $C$ be [[Definition:Class (Class Theory)|classes]].
Let $A$ be a [[Definition:Subclass|subclass]] of $B$.
Let $B$ be a [[Definition:Subclass|subclass]] of $C$.
Then $A$ is a [[Definition:Subclass|subclass]] of $C$. | Let $x \in A$ be arbitrary.
It follows by definition of [[Definition:Subclass|subclass]] that $x \in B$.
It further follows by definition of [[Definition:Subclass|subclass]] that $x \in C$.
So we have that $x \in A$ implies that $x \in C$.
As $x$ is arbitrary, the result follows.
{{qed}}
[[Category:Subclasses]]
9i... | Subclass of Subclass is Subclass | https://proofwiki.org/wiki/Subclass_of_Subclass_is_Subclass | https://proofwiki.org/wiki/Subclass_of_Subclass_is_Subclass | [
"Subclasses"
] | [
"Definition:Class (Class Theory)",
"Definition:Subclass",
"Definition:Subclass",
"Definition:Subclass"
] | [
"Definition:Subclass",
"Definition:Subclass",
"Category:Subclasses"
] |
proofwiki-19466 | Subclass of Well-Ordered Class is Well-Ordered | Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation.
Let $A$ be a subclass of $V$ which is well-ordered under $\RR$.
Let $B$ be a subclass of $A$.
Then $B$ is also well-ordered under $\RR$. | First suppose $B$ is the empty class.
From Empty Class is Subclass of All Classes, $B$ is a subclass of $A$.
Then by Empty Class is Well-Ordered, $\O$ is well-ordered under $\RR$.
Otherwise, let $X$ be an arbitrary non-empty class subclass of $B$.
By Subclass of Subclass is Subclass, $X$ is a subclass of $A$.
Hence by ... | Let $V$ be a [[Definition:Basic Universe|basic universe]].
Let $\RR \subseteq V \times V$ be a [[Definition:Relation (Class Theory)|relation]].
Let $A$ be a [[Definition:Subclass|subclass]] of $V$ which is [[Definition:Well-Ordered Class|well-ordered under $\RR$]].
Let $B$ be a [[Definition:Subclass|subclass]] of $... | First suppose $B$ is the [[Definition:Empty Class|empty class]].
From [[Empty Class is Subclass of All Classes]], $B$ is a [[Definition:Subclass|subclass]] of $A$.
Then by [[Empty Class is Well-Ordered]], $\O$ is [[Definition:Well-Ordered Class|well-ordered under $\RR$]].
Otherwise, let $X$ be an arbitrary [[Defini... | Subclass of Well-Ordered Class is Well-Ordered | https://proofwiki.org/wiki/Subclass_of_Well-Ordered_Class_is_Well-Ordered | https://proofwiki.org/wiki/Subclass_of_Well-Ordered_Class_is_Well-Ordered | [
"Well-Orderings",
"Subclasses"
] | [
"Definition:Basic Universe",
"Definition:Relation/Class Theory",
"Definition:Subclass",
"Definition:Well-Ordered Class",
"Definition:Subclass",
"Definition:Well-Ordered Class"
] | [
"Definition:Empty Class",
"Empty Class is Subclass of All Classes",
"Definition:Subclass",
"Empty Class is Well-Ordered",
"Definition:Well-Ordered Class",
"Definition:Non-Empty Set/Class Theory",
"Definition:Subclass",
"Subclass of Subclass is Subclass",
"Definition:Subclass",
"Definition:Well-Ord... |
proofwiki-19467 | Image of Convex Set under Linear Transformation is Convex | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ and $Y$ be vector spaces over $\Bbb F$.
Let $C \subseteq X$ be convex.
Let $T : X \to Y$ be a linear transformation.
Then $\map T C \subseteq Y$ is convex. | Let $y_1, y_2 \in \map T C$ and $\lambda \in \closedint 0 1$.
Then, there exists $x_1, x_2 \in C$ such that:
:$y_1 = T x_1$
and:
:$y_2 = T x_2$
Then, we have:
:$\lambda y_1 + \paren {1 - \lambda} y_2 = \lambda T x_1 + \paren {1 - \lambda} T x_2 = \map T {\lambda x_1 + \paren {1 - \lambda} x_2}$
Since $C$ is convex,... | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Vector Space|vector spaces]] over $\Bbb F$.
Let $C \subseteq X$ be [[Definition:Convex Set (Vector Space)|convex]].
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Then $\map T C \subseteq Y$ is [[Definition:Convex S... | Let $y_1, y_2 \in \map T C$ and $\lambda \in \closedint 0 1$.
Then, there exists $x_1, x_2 \in C$ such that:
:$y_1 = T x_1$
and:
:$y_2 = T x_2$
Then, we have:
:$\lambda y_1 + \paren {1 - \lambda} y_2 = \lambda T x_1 + \paren {1 - \lambda} T x_2 = \map T {\lambda x_1 + \paren {1 - \lambda} x_2}$
Since $C$ is ... | Image of Convex Set under Linear Transformation is Convex | https://proofwiki.org/wiki/Image_of_Convex_Set_under_Linear_Transformation_is_Convex | https://proofwiki.org/wiki/Image_of_Convex_Set_under_Linear_Transformation_is_Convex | [
"Convex Sets (Vector Spaces)",
"Linear Transformations"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Linear Transformation",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Category:Convex Sets (Vector Spaces)",
"Category:Linear Transformations"
] |
proofwiki-19468 | Open Ball Centred at Origin in Normed Vector Space is Symmetric | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $\map B {0, r}$ be the open ball in $X$ centered at $0$ with radius $r$.
Then $\map B {0, r}$ is symmetric. | Let $x \in \map B {0, r}$.
Then:
:$\norm x < r$
We then have:
:$\norm {-x} = \cmod {-1} \norm x = \norm x < r$
So $-x \in \map B {0, r}$.
So $\map B {0, r}$ is symmetric.
{{qed}}
Category:Symmetric Subsets of Vector Spaces
Category:Open Balls
bguvmmwlmpaxhz26mj9z9nl405ex5iu | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map B {0, r}$ be the [[Definition:Open Ball in Normed Vector Space|open ball]] in $X$ [[Definition:Center of Open Ball|centered]] at $0$ with [[Definition:Radius of Open Ball|radius]] $r$.
Then $\map B {0, r}$ is [[Defi... | Let $x \in \map B {0, r}$.
Then:
:$\norm x < r$
We then have:
:$\norm {-x} = \cmod {-1} \norm x = \norm x < r$
So $-x \in \map B {0, r}$.
So $\map B {0, r}$ is [[Definition:Symmetric Subset of Vector Space|symmetric]].
{{qed}}
[[Category:Symmetric Subsets of Vector Spaces]]
[[Category:Open Balls]]
bguvmmwlmpax... | Open Ball Centred at Origin in Normed Vector Space is Symmetric | https://proofwiki.org/wiki/Open_Ball_Centred_at_Origin_in_Normed_Vector_Space_is_Symmetric | https://proofwiki.org/wiki/Open_Ball_Centred_at_Origin_in_Normed_Vector_Space_is_Symmetric | [
"Symmetric Subsets of Vector Spaces",
"Open Balls"
] | [
"Definition:Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Definition:Symmetric Set/Vector Space"
] | [
"Definition:Symmetric Set/Vector Space",
"Category:Symmetric Subsets of Vector Spaces",
"Category:Open Balls"
] |
proofwiki-19469 | Image of Symmetric Set under Linear Transformation is Symmetric | Let $X$ and $Y$ be vector spaces over a subfield of $\C$.
Let $C$ be a symmetric subset of $X$.
Let $T : X \to Y$ be a linear transformation.
Then $\map T C$ is a symmetric subset of $Y$. | Let $y \in \map T C$.
Then there exists $x \in C$ such that $y = T x$.
Then from linearity we have $-y = \map T {-x}$.
Since $C$ is symmetric, we have $-x \in C$.
So $-y \in \map T C$.
So $\map T C$ is symmetric.
{{qed}}
Category:Symmetric Subsets of Vector Spaces
bakajtjrqucx8uln7bmkjbm01ekrc4r | Let $X$ and $Y$ be [[Definition:Vector Space|vector spaces]] over a [[Definition:Subfield|subfield]] of $\C$.
Let $C$ be a [[Definition:Symmetric Subset of Vector Space|symmetric subset]] of $X$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Then $\map T C$ is a [[Definition:Sym... | Let $y \in \map T C$.
Then there exists $x \in C$ such that $y = T x$.
Then from [[Definition:Linear Transformation|linearity]] we have $-y = \map T {-x}$.
Since $C$ is [[Definition:Symmetric Subset of Vector Space|symmetric]], we have $-x \in C$.
So $-y \in \map T C$.
So $\map T C$ is [[Definition:Symmetric Subs... | Image of Symmetric Set under Linear Transformation is Symmetric | https://proofwiki.org/wiki/Image_of_Symmetric_Set_under_Linear_Transformation_is_Symmetric | https://proofwiki.org/wiki/Image_of_Symmetric_Set_under_Linear_Transformation_is_Symmetric | [
"Symmetric Subsets of Vector Spaces"
] | [
"Definition:Vector Space",
"Definition:Subfield",
"Definition:Symmetric Set/Vector Space",
"Definition:Linear Transformation",
"Definition:Symmetric Set/Vector Space"
] | [
"Definition:Linear Transformation",
"Definition:Symmetric Set/Vector Space",
"Definition:Symmetric Set/Vector Space",
"Category:Symmetric Subsets of Vector Spaces"
] |
proofwiki-19470 | Closure of Symmetric Subset of Normed Vector Space is Symmetric | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $A \subseteq X$ be symmetric.
Then the topological closure of $A$ is symmetric. | Let $A^-$ be the topological closure of $A$.
Let $a \in A^-$.
Then from Point in Closure of Subset of Metric Space iff Limit of Sequence, we have:
:there exists a sequence $\sequence {a_n}_{n \mathop \in \N}$ in $A$ such that $a_n \to a$.
Since $A$ is symmetric, we have:
:$-a_n \in A$ for each $n \in \N$.
We also ha... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $A \subseteq X$ be [[Definition:Symmetric Subset of Vector Space|symmetric]].
Then the [[Definition:Topological Closure|topological closure]] of $A$ is [[Definition:Symmetric Subset of Vector Space|symmetric]]. | Let $A^-$ be the [[Definition:Topological Closure|topological closure]] of $A$.
Let $a \in A^-$.
Then from [[Point in Closure of Subset of Metric Space iff Limit of Sequence]], we have:
:there exists a [[Definition:Sequence|sequence]] $\sequence {a_n}_{n \mathop \in \N}$ in $A$ such that $a_n \to a$.
Since $A$ is... | Closure of Symmetric Subset of Normed Vector Space is Symmetric | https://proofwiki.org/wiki/Closure_of_Symmetric_Subset_of_Normed_Vector_Space_is_Symmetric | https://proofwiki.org/wiki/Closure_of_Symmetric_Subset_of_Normed_Vector_Space_is_Symmetric | [
"Set Closures",
"Symmetric Subsets of Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Symmetric Set/Vector Space",
"Definition:Closure (Topology)",
"Definition:Symmetric Set/Vector Space"
] | [
"Definition:Closure (Topology)",
"Point in Closure of Subset of Metric Space iff Limit of Sequence",
"Definition:Sequence",
"Definition:Symmetric Set/Vector Space",
"Multiple Rule for Sequences in Normed Vector Space",
"Category:Set Closures",
"Category:Symmetric Subsets of Vector Spaces"
] |
proofwiki-19471 | Empty Set is Well-Ordered | Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a relation on $S$.
Let $\O$ denote the empty set.
Let $\RR_\O$ denote the restriction of $\RR$ to $\O$.
Then $\struct {\O, \RR_\O}$ is a well-ordered set. | Let $V$ be a basic universe.
By definition of basic universe, $\O$ is an element of $V$.
By the Axiom of Transitivity, $\O$ is a class.
The result follows from Empty Class is Well-Ordered.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] on $S$.
Let $\O$ denote the [[Definition:Empty Set|empty set]].
Let $\RR_\O$ denote the [[Definition:Restriction of Relation|restriction]] of $\RR$ to $\O$.
Then $\struct {\O, \RR_\O}$ is a [[Definition:Well-O... | Let $V$ be a [[Definition:Basic Universe|basic universe]].
By definition of [[Definition:Basic Universe|basic universe]], $\O$ is an [[Definition:Element|element]] of $V$.
By [[Axiom:Axiom of Transitivity|the Axiom of Transitivity]], $\O$ is a [[Definition:Class (Class Theory)|class]].
The result follows from [[Empt... | Empty Set is Well-Ordered/Proof 2 | https://proofwiki.org/wiki/Empty_Set_is_Well-Ordered | https://proofwiki.org/wiki/Empty_Set_is_Well-Ordered/Proof_2 | [
"Empty Set",
"Well-Orderings",
"Empty Set is Well-Ordered"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Empty Set",
"Definition:Restriction/Relation",
"Definition:Well-Ordered Set"
] | [
"Definition:Basic Universe",
"Definition:Basic Universe",
"Definition:Element",
"Axiom:Axiom of Transitivity",
"Definition:Class (Class Theory)",
"Empty Class is Well-Ordered"
] |
proofwiki-19472 | Empty Class is Well-Ordered | Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation.
Let $\O$ denote the empty class.
Then $\O$ is well-ordered under $\RR$. | We have that $\O$ is well-ordered under $\RR$ {{iff}} every non-empty subclass of $\O$ has a smallest element under $\RR$.
But $\O$ has no non-empty subclass.
Hence this condition is satisfied vacuously.
The result follows.
{{qed}}
Category:Empty Class
Category:Well-Orderings
k0efs1sd9vqr2i3dw69066dtmt5mbfx | Let $V$ be a [[Definition:Basic Universe|basic universe]].
Let $\RR \subseteq V \times V$ be a [[Definition:Relation (Class Theory)|relation]].
Let $\O$ denote the [[Definition:Empty Class|empty class]].
Then $\O$ is [[Definition:Well-Ordered Class|well-ordered]] under $\RR$. | We have that $\O$ is [[Definition:Well-Ordered Class|well-ordered]] under $\RR$ {{iff}} every [[Definition:Non-Empty Class|non-empty]] [[Definition:Subclass|subclass]] of $\O$ has a [[Definition:Smallest Element|smallest element]] under $\RR$.
But $\O$ has no [[Definition:Non-Empty Class|non-empty]] [[Definition:Subcl... | Empty Class is Well-Ordered | https://proofwiki.org/wiki/Empty_Class_is_Well-Ordered | https://proofwiki.org/wiki/Empty_Class_is_Well-Ordered | [
"Empty Class",
"Well-Orderings"
] | [
"Definition:Basic Universe",
"Definition:Relation/Class Theory",
"Definition:Empty Class",
"Definition:Well-Ordered Class"
] | [
"Definition:Well-Ordered Class",
"Definition:Non-Empty Set/Class Theory",
"Definition:Subclass",
"Definition:Smallest Element",
"Definition:Non-Empty Set/Class Theory",
"Definition:Subclass",
"Definition:Vacuous Truth",
"Category:Empty Class",
"Category:Well-Orderings"
] |
proofwiki-19473 | Infinite Series preserves Strict Inequality | Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be real sequences such that:
:$x_n \le y_n$ for all $n \in \N$
and:
:$x_m < y_m$ for at least one $m \in \N$.
Suppose that:
:$\ds \sum_{n \mathop = 1}^\infty x_n$ and $\ds \sum_{n \mathop = 1}^\infty y_n$ both converge.
Then:
:$\ds \sum... | We have:
:$\ds \sum_{n \mathop = 1}^N x_n \le \sum_{n \mathop = 1}^N y_n$ for each $N \in \N$.
Then:
:$\ds \sum_{n \mathop = 1}^N \paren {y_n - x_n} \ge 0$ for each $N \in \N$.
From the Well-Ordering Principle, there exists a least $m \in \N$ such that $x_m < y_m$.
Then for this $m$ we have $y_m - x_m > 0$.
So for ... | Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be [[Definition:Real Sequence|real sequences]] such that:
:$x_n \le y_n$ for all $n \in \N$
and:
:$x_m < y_m$ for at least one $m \in \N$.
Suppose that:
:$\ds \sum_{n \mathop = 1}^\infty x_n$ and $\ds \sum_{n \mathop = 1}^\infty y_... | We have:
:$\ds \sum_{n \mathop = 1}^N x_n \le \sum_{n \mathop = 1}^N y_n$ for each $N \in \N$.
Then:
:$\ds \sum_{n \mathop = 1}^N \paren {y_n - x_n} \ge 0$ for each $N \in \N$.
From the [[Well-Ordering Principle]], there exists a least $m \in \N$ such that $x_m < y_m$.
Then for this $m$ we have $y_m - x_m > 0$... | Infinite Series preserves Strict Inequality | https://proofwiki.org/wiki/Infinite_Series_preserves_Strict_Inequality | https://proofwiki.org/wiki/Infinite_Series_preserves_Strict_Inequality | [
"Series"
] | [
"Definition:Real Sequence",
"Definition:Convergent Series"
] | [
"Well-Ordering Principle",
"Linear Combination of Convergent Series",
"Definition:Convergent Series",
"Inequality Rule for Real Sequences",
"Category:Series"
] |
proofwiki-19474 | Class which has Injection to Subclass of Well-Orderable Class is Well-Orderable | Let $B$ be a well-orderable class.
Let $A$ be a class such that there exists an injection $f: A \to C$, where $C$ is a subclass of $B$.
Then $A$ is a well-orderable class. | Let $\RR$ be a well-ordering that can be established on $B$.
This can always be done, as $B$ is a well-orderable class.
Let $F$ be an injection that maps each element $x$ of $A$ to an element $\map F x$ of $B$.
Let $\preccurlyeq$ be the class of all ordered pairs $\tuple {x, y}$ of elements of $A$ such that $\tuple {\m... | Let $B$ be a [[Definition:Well-Orderable Class|well-orderable class]].
Let $A$ be a [[Definition:Class (Class Theory)|class]] such that there exists an [[Definition:Injection|injection]] $f: A \to C$, where $C$ is a [[Definition:Subclass|subclass]] of $B$.
Then $A$ is a [[Definition:Well-Orderable Class|well-orderab... | Let $\RR$ be a [[Definition:Well-Ordering (Class Theory)|well-ordering]] that can be established on $B$.
This can always be done, as $B$ is a [[Definition:Well-Orderable Class|well-orderable class]].
Let $F$ be an [[Definition:Injection|injection]] that maps each [[Definition:Element of Class|element]] $x$ of $A$ to ... | Class which has Injection to Subclass of Well-Orderable Class is Well-Orderable | https://proofwiki.org/wiki/Class_which_has_Injection_to_Subclass_of_Well-Orderable_Class_is_Well-Orderable | https://proofwiki.org/wiki/Class_which_has_Injection_to_Subclass_of_Well-Orderable_Class_is_Well-Orderable | [
"Well-Orderings"
] | [
"Definition:Well-Orderable Set/Class Theory",
"Definition:Class (Class Theory)",
"Definition:Injection",
"Definition:Subclass",
"Definition:Well-Orderable Set/Class Theory"
] | [
"Definition:Well-Ordering/Class Theory",
"Definition:Well-Orderable Set/Class Theory",
"Definition:Injection",
"Definition:Element/Class",
"Definition:Element/Class",
"Definition:Class (Class Theory)",
"Definition:Ordered Pair",
"Definition:Element/Class",
"Definition:Well-Ordered Class",
"Definit... |
proofwiki-19475 | Norm of Summation | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
Then:
:$\ds \norm {\sum_{k \mathop = 1}^n x_k} \le \sum_{k \mathop = 1}^n \norm {x_k}$ for all $n \in \N$. | The proof proceeds by induction.
For each $n \in \N$, let $\map P n$ be the proposition:
:$\ds \norm {\sum_{k \mathop = 1}^n x_k} \le \sum_{k \mathop = 1}^n \norm {x_k}$ for all $n \in \N$. | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$.
Then:
:$\ds \norm {\sum_{k \mathop = 1}^n x_k} \le \sum_{k \mathop = 1}^n \norm {x_k}$ for all $n \in \N$. | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For each $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \norm {\sum_{k \mathop = 1}^n x_k} \le \sum_{k \mathop = 1}^n \norm {x_k}$ for all $n \in \N$. | Norm of Summation | https://proofwiki.org/wiki/Norm_of_Summation | https://proofwiki.org/wiki/Norm_of_Summation | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-19476 | Multiple Rule for Sequence in Normed Vector Space | Let $\Bbb F$ be a subfield of $\C$.
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ converging to $x$.
Let $\lambda \in \Bbb F$.
Then:
:$\lambda x_n \to \lambda x$ | The case $\lambda = 0$ follows from Constant Sequence in Normed Vector Space Converges.
Now take $\lambda \ne 0$.
Since $\sequence {x_n}_{n \mathop \in \N}$ converges to $x$, for each $\epsilon > 0$ there exists $N \in \N$ such that:
:$\ds \norm {x_n - x} < \frac \epsilon {\cmod \lambda}$ for all $n \ge N$.
Noting th... | Let $\Bbb F$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ [[Definition:Convergent Sequence in Normed Vector Space|convergi... | The case $\lambda = 0$ follows from [[Constant Sequence in Normed Vector Space Converges]].
Now take $\lambda \ne 0$.
Since $\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence in Normed Vector Space|converges]] to $x$, for each $\epsilon > 0$ there exists $N \in \N$ such that:
:$\ds \norm {x_n - ... | Multiple Rule for Sequence in Normed Vector Space | https://proofwiki.org/wiki/Multiple_Rule_for_Sequence_in_Normed_Vector_Space | https://proofwiki.org/wiki/Multiple_Rule_for_Sequence_in_Normed_Vector_Space | [
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Subfield",
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Constant Sequence in Normed Vector Space Converges",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Norm/Vector Space",
"Category:Convergent Sequences (Normed Vector Spaces)"
] |
proofwiki-19477 | Constant Sequence in Normed Vector Space Converges | Let $\Bbb F$ be a subfield of $\C$.
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence with $x_n = x$ for all $n \in \N$.
Then:
:$x_n \to x$ | We have:
:$\norm {x_n - x} = 0$
for all $n \in \N$.
So, for all $\epsilon > 0$, we have:
:$\norm {x_n - x} < \epsilon$ for all $n \in \N$.
So:
:$x_n \to x$
{{qed}}
Category:Convergent Sequences (Normed Vector Spaces)
fzpigffj0s6m8ayucmia3ww46youo3z | Let $\Bbb F$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] with $x_n = x$ for all $n \in \N$.
Then:
:$x_n \to... | We have:
:$\norm {x_n - x} = 0$
for all $n \in \N$.
So, for all $\epsilon > 0$, we have:
:$\norm {x_n - x} < \epsilon$ for all $n \in \N$.
So:
:$x_n \to x$
{{qed}}
[[Category:Convergent Sequences (Normed Vector Spaces)]]
fzpigffj0s6m8ayucmia3ww46youo3z | Constant Sequence in Normed Vector Space Converges | https://proofwiki.org/wiki/Constant_Sequence_in_Normed_Vector_Space_Converges | https://proofwiki.org/wiki/Constant_Sequence_in_Normed_Vector_Space_Converges | [
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Subfield",
"Definition:Normed Vector Space",
"Definition:Sequence"
] | [
"Category:Convergent Sequences (Normed Vector Spaces)"
] |
proofwiki-19478 | Banach Isomorphism Theorem | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be Banach spaces.
Let $T : X \to Y$ be a bijective bounded linear transformation.
Then the inverse of $T$ is a bounded linear transformation.
That is, $T$ is a normed vector space isomorphism. | Let $B_X^-$ be the closed unit ball of $X$.
Let $B_Y^-$ be the closed unit ball of $Y$.
Let $T^{-1} : Y \to X$ be the inverse of $T$.
From Inverse of Linear Transformation is Linear Transformation, $T^{-1} : Y \to X$ is a linear transformation.
It remains to show that $T^{-1}$ is bounded.
Since $T$ is bijective, it is ... | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be [[Definition:Banach Space|Banach spaces]].
Let $T : X \to Y$ be a [[Definition:Bijection|bijective]] [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Then the [[Definition:Inverse Mapping|inverse]] of $T$ is a [[Definitio... | Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $X$.
Let $B_Y^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $Y$.
Let $T^{-1} : Y \to X$ be the [[Definition:Inverse Mapping|inverse]] of $T$.
From [[Inverse of Linear Transformation is Linear Transformation]], $T^{-1} : Y \to X$ i... | Banach Isomorphism Theorem | https://proofwiki.org/wiki/Banach_Isomorphism_Theorem | https://proofwiki.org/wiki/Banach_Isomorphism_Theorem | [
"Linear Transformations on Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Bijection",
"Definition:Bounded Linear Transformation",
"Definition:Inverse Mapping",
"Definition:Bounded Linear Transformation",
"Definition:Normed Vector Space Isomorphism"
] | [
"Definition:Closed Unit Ball",
"Definition:Closed Unit Ball",
"Definition:Inverse Mapping",
"Inverse of Linear Transformation is Linear Transformation",
"Definition:Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Bijection",
"Definition:Surjection",
"Banach-Schauder T... |
proofwiki-19479 | Countable Set is Well-Orderable | Let $S$ be a countable set.
Then $S$ is well-orderable. | By the Well-Ordering Principle, the set of natural numbers $\N$ under the usual ordering $\le$ forms a well-ordered set.
By definition of countable set, there exists an injection:
:$f: S \to \N$
Let $V$ be a basic universe.
By definition of basic universe:
:$S \in V$
and:
:$\N \in V$
By the {{axiom-link|Transitivity}},... | Let $S$ be a [[Definition:Countable Set|countable set]].
Then $S$ is [[Definition:Well-Orderable Set|well-orderable]]. | By the [[Well-Ordering Principle]], the [[Definition:Natural Numbers|set of natural numbers]] $\N$ under the [[Definition:Usual Ordering|usual ordering]] $\le$ forms a [[Definition:Well-Ordered Set|well-ordered set]].
By definition of [[Definition:Countable Set|countable set]], there exists an [[Definition:Injection|i... | Countable Set is Well-Orderable | https://proofwiki.org/wiki/Countable_Set_is_Well-Orderable | https://proofwiki.org/wiki/Countable_Set_is_Well-Orderable | [
"Countable Sets",
"Well-Orderings"
] | [
"Definition:Countable Set",
"Definition:Well-Orderable Set"
] | [
"Well-Ordering Principle",
"Definition:Natural Numbers",
"Definition:Usual Ordering",
"Definition:Well-Ordered Set",
"Definition:Countable Set",
"Definition:Injection",
"Definition:Basic Universe",
"Definition:Basic Universe",
"Definition:Class (Class Theory)",
"Class which has Injection to Subcla... |
proofwiki-19480 | Rational Numbers are Well-Orderable | The set $\Q$ of rational numbers is well-orderable. | From Rational Numbers are Countably Infinite, $\Q$ is a countable set.
The result follows from Countable Set is Well-Orderable.
{{qed}} | The [[Definition:Set|set]] $\Q$ of [[Definition:Rational Number|rational numbers]] is [[Definition:Well-Orderable Set|well-orderable]]. | From [[Rational Numbers are Countably Infinite]], $\Q$ is a [[Definition:Countable Set|countable set]].
The result follows from [[Countable Set is Well-Orderable]].
{{qed}} | Rational Numbers are Well-Orderable/Proof 1 | https://proofwiki.org/wiki/Rational_Numbers_are_Well-Orderable | https://proofwiki.org/wiki/Rational_Numbers_are_Well-Orderable/Proof_1 | [
"Rational Numbers",
"Well-Orderings",
"Rational Numbers are Well-Orderable"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Well-Orderable Set"
] | [
"Rational Numbers are Countably Infinite",
"Definition:Countable Set",
"Countable Set is Well-Orderable"
] |
proofwiki-19481 | Rational Numbers are Well-Orderable | The set $\Q$ of rational numbers is well-orderable. | The rational numbers are arranged thus:
:$\dfrac 0 1, \dfrac 1 1, \dfrac {-1} 1, \dfrac 1 2, \dfrac {-1} 2, \dfrac 2 1, \dfrac {-2} 1, \dfrac 1 3, \dfrac 2 3, \dfrac {-1} 3, \dfrac {-2} 3, \dfrac 3 1, \dfrac 3 2, \dfrac {-3} 1, \dfrac {-3} 2, \dfrac 1 4, \dfrac 3 4, \dfrac {-1} 4, \dfrac {-3} 4, \dfrac 4 1, \dfrac 4 3... | The [[Definition:Set|set]] $\Q$ of [[Definition:Rational Number|rational numbers]] is [[Definition:Well-Orderable Set|well-orderable]]. | The [[Definition:Rational Number|rational numbers]] are arranged thus:
:$\dfrac 0 1, \dfrac 1 1, \dfrac {-1} 1, \dfrac 1 2, \dfrac {-1} 2, \dfrac 2 1, \dfrac {-2} 1, \dfrac 1 3, \dfrac 2 3, \dfrac {-1} 3, \dfrac {-2} 3, \dfrac 3 1, \dfrac 3 2, \dfrac {-3} 1, \dfrac {-3} 2, \dfrac 1 4, \dfrac 3 4, \dfrac {-1} 4, \dfra... | Rational Numbers are Well-Orderable/Proof 2 | https://proofwiki.org/wiki/Rational_Numbers_are_Well-Orderable | https://proofwiki.org/wiki/Rational_Numbers_are_Well-Orderable/Proof_2 | [
"Rational Numbers",
"Well-Orderings",
"Rational Numbers are Well-Orderable"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Well-Orderable Set"
] | [
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Bijection",
"Definition:Rational Number",
"Definition:Ordering",
"Definition:Usual Ordering",
"Definition:Well-Ordering"
] |
proofwiki-19482 | Well-Ordering is not necessarily Usual Ordering | Let $S$ be a set of numbers.
According to Zermelo's Well-Ordering Theorem, $S$ can be well-ordered.
However, the usual ordering on $S$ may not necessarily be a well-ordering. | From Rational Numbers are Well-Orderable, it is possible to apply a well-ordering to the set of rational numbers $\Q$.
However, the usual ordering on $\Q$ is not a well-ordering.
Indeed:
:$\set {x \in \Q: x \le 0}$
has no smallest element.
{{qed}} | Let $S$ be a [[Definition:Set|set]] of [[Definition:Number|numbers]].
According to [[Zermelo's Well-Ordering Theorem]], $S$ can be [[Definition:Well-Ordered Set|well-ordered]].
However, the [[Definition:Usual Ordering|usual ordering]] on $S$ may not necessarily be a [[Definition:Well-Ordering|well-ordering]]. | From [[Rational Numbers are Well-Orderable]], it is possible to apply a [[Definition:Well-Ordering|well-ordering]] to the [[Definition:Rational Number|set of rational numbers]] $\Q$.
However, the [[Definition:Usual Ordering|usual ordering]] on $\Q$ is not a [[Definition:Well-Ordering|well-ordering]].
Indeed:
:$\set {... | Well-Ordering is not necessarily Usual Ordering | https://proofwiki.org/wiki/Well-Ordering_is_not_necessarily_Usual_Ordering | https://proofwiki.org/wiki/Well-Ordering_is_not_necessarily_Usual_Ordering | [
"Well-Orderings"
] | [
"Definition:Set",
"Definition:Number",
"Zermelo's Well-Ordering Theorem",
"Definition:Well-Ordered Set",
"Definition:Usual Ordering",
"Definition:Well-Ordering"
] | [
"Rational Numbers are Well-Orderable",
"Definition:Well-Ordering",
"Definition:Rational Number",
"Definition:Usual Ordering",
"Definition:Well-Ordering",
"Definition:Smallest Element"
] |
proofwiki-19483 | Poincaré Recurrence Theorem | Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.
Then for each $A \in \BB$:
:$\ds \map \mu {A \setminus \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A} = 0$
That is, for $\mu$-almost all $x\in A$ there are integers $0 < n_1 < n_2 < \cdots$ such that $\map {T^{n_i} ... | Let:
:$\ds A_\infty := \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A$
For $N \in \N$, let:
:$\ds A_N := \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A$
so that:
:$\ds A_\infty = \bigcap _{N \mathop = 1} ^\infty A_N$
Now, we need to show:
:$\ds \map \mu {A \setminus A_\infty} = 0$
First, ... | Let $\struct {X, \BB, \mu, T}$ be a [[Definition:Measure-Preserving Dynamical System|measure-preserving dynamical system]].
Then for each $A \in \BB$:
:$\ds \map \mu {A \setminus \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A} = 0$
That is, for $\mu$-[[Definition:Almost All|almost all... | Let:
:$\ds A_\infty := \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A$
For $N \in \N$, let:
:$\ds A_N := \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A$
so that:
:$\ds A_\infty = \bigcap _{N \mathop = 1} ^\infty A_N$
Now, we need to show:
:$\ds \map \mu {A \setminus A_\infty} = 0$
Fir... | Poincaré Recurrence Theorem | https://proofwiki.org/wiki/Poincaré_Recurrence_Theorem | https://proofwiki.org/wiki/Poincaré_Recurrence_Theorem | [
"Measures",
"Ergodic Theory"
] | [
"Definition:Measure-Preserving Dynamical System",
"Definition:Almost All",
"Definition:Integer"
] | [
"Preimage of Union under Mapping/Family of Sets",
"Definition:Measure-Preserving Transformation",
"Measure is Monotone",
"De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection/Corollary",
"Measure is Countably Subadditive",
"Measure of Set Difference with Subset"
] |
proofwiki-19484 | Characterization of Integer has Square Root in P-adic Integers/Necessary Condition | Let $\Z_p$ be the $p$-adic integers for some prime $p \ne 2$.
Let $a \in Z$ be an integer such that $p \nmid a$.
Let the exist $x \in \Z_p$ such that $x^2 = a$.
Then:
:$a$ is a quadratic residue of $p$.
That is:
:an integer $a$ not divisible by $p$ has a square root in $\Z_p$ ($p \ne 2$)
{{iff}}:
:$a$ is a quadratic re... | Let there exist $x$ such that $x^2 = a$.
By definition of root of polynomial:
:$\map F X$ has a root in $\Z_p$.
From Characterization of Integer Polynomial has Root in $p$-adic Integers:
:there exists an integer sequence $\sequence {a_n}$ such that:
::$(1): \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$
::$(2): \quad... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p \ne 2$.
Let $a \in Z$ be an [[Definition:Integer|integer]] such that $p \nmid a$.
Let the exist $x \in \Z_p$ such that $x^2 = a$.
Then:
:$a$ is a [[Definition:Quadratic Residue|quadratic residue]] of $p$.... | Let there exist $x$ such that $x^2 = a$.
By definition of [[Definition:Root of Polynomial|root of polynomial]]:
:$\map F X$ has a [[Definition:Root of Polynomial|root]] in $\Z_p$.
From [[Characterization of Integer Polynomial has Root in P-adic Integers|Characterization of Integer Polynomial has Root in $p$-adic Int... | Characterization of Integer has Square Root in P-adic Integers/Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Integer_has_Square_Root_in_P-adic_Integers/Necessary_Condition | https://proofwiki.org/wiki/Characterization_of_Integer_has_Square_Root_in_P-adic_Integers/Necessary_Condition | [
"Characterization of Integer has Square Root in P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Integer",
"Definition:Quadratic Residue",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Square Root",
"Definition:Quadratic Residue"
] | [
"Definition:Root of Polynomial",
"Definition:Root of Polynomial",
"Characterization of Integer Polynomial has Root in P-adic Integers",
"Definition:Integer Sequence",
"Definition:Quadratic Residue"
] |
proofwiki-19485 | Characterization of Integer has Square Root in P-adic Integers/Sufficient Condition | Let $\Z_p$ be the $p$-adic integers for some prime $p \ne 2$.
Let $a \in Z$ be an integer such that $p \nmid a$.
Let $a$ be a quadratic residue of $p$.
Then:
:$\exists x \in \Z_p : x^2 = a$ | Let $a$ be a quadratic residue of $p$.
By definition of quadratic residue of $p$:
:$\exists b \in \Z : a \equiv b^2 \pmod p$
Then:
:$\map F b = b^2 - a \equiv 0 \pmod p$
and
:$\map {F'} b = 2b$
By hypothesis:
:$p \nmid 2$
and
:$p \nmid b^2$
From the contrapositive statement of Divisor Divides Multiple:
:$p \nmid b$
Fro... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p \ne 2$.
Let $a \in Z$ be an [[Definition:Integer|integer]] such that $p \nmid a$.
Let $a$ be a [[Definition:Quadratic Residue|quadratic residue]] of $p$.
Then:
:$\exists x \in \Z_p : x^2 = a$ | Let $a$ be a [[Definition:Quadratic Residue|quadratic residue]] of $p$.
By definition of [[Definition:Quadratic Residue|quadratic residue]] of $p$:
:$\exists b \in \Z : a \equiv b^2 \pmod p$
Then:
:$\map F b = b^2 - a \equiv 0 \pmod p$
and
:$\map {F'} b = 2b$
By hypothesis:
:$p \nmid 2$
and
:$p \nmid b^2$
From th... | Characterization of Integer has Square Root in P-adic Integers/Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Integer_has_Square_Root_in_P-adic_Integers/Sufficient_Condition | https://proofwiki.org/wiki/Characterization_of_Integer_has_Square_Root_in_P-adic_Integers/Sufficient_Condition | [
"Characterization of Integer has Square Root in P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Integer",
"Definition:Quadratic Residue"
] | [
"Definition:Quadratic Residue",
"Definition:Quadratic Residue",
"Definition:Contrapositive Statement",
"Divisor Divides Multiple",
"Definition:Contrapositive Statement",
"Euclid's Lemma for Prime Divisors",
"Congruence Modulo Equivalence for Integers in P-adic Integers",
"Hensel's Lemma/P-adic Integer... |
proofwiki-19486 | Categories of Elements under Well-Ordering | Let $A$ be a class.
Let $\preccurlyeq$ be a well-ordering on $A$.
Let $x \in A$ be an element of $A$.
Then $x$ falls into one of the following $3$ categories:
:$(1): \quad x$ is the smallest element of $A$ {{WRT}} $\preccurlyeq$
:$(2): \quad x$ is the immediate successor of another element $y \in A$ {{WRT}} $\preccurly... | Let $x \in A$.
;$(1): \quad x$ is the smallest element of $A$ {{WRT}} $\preccurlyeq$:
We note that $\preccurlyeq$ is a well-ordering on $A$.
Hence as $A \subseteq A$ we have that $A$ has a smallest element.
Hence there exists one element of $A$ which is that smallest element of $A$.
Let $x$ be that smallest element of ... | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $\preccurlyeq$ be a [[Definition:Well-Ordering (Class Theory)|well-ordering]] on $A$.
Let $x \in A$ be an [[Definition:Element of Class|element]] of $A$.
Then $x$ falls into one of the following $3$ categories:
:$(1): \quad x$ is the [[Definition:Smallest ... | Let $x \in A$.
;$(1): \quad x$ is the [[Definition:Smallest Element (Class Theory)|smallest element]] of $A$ {{WRT}} $\preccurlyeq$:
We note that $\preccurlyeq$ is a [[Definition:Well-Ordering (Class Theory)|well-ordering]] on $A$.
Hence as $A \subseteq A$ we have that $A$ has a [[Definition:Smallest Element (Class... | Categories of Elements under Well-Ordering | https://proofwiki.org/wiki/Categories_of_Elements_under_Well-Ordering | https://proofwiki.org/wiki/Categories_of_Elements_under_Well-Ordering | [
"Well-Orderings"
] | [
"Definition:Class (Class Theory)",
"Definition:Well-Ordering/Class Theory",
"Definition:Element/Class",
"Definition:Smallest Element/Class Theory",
"Definition:Immediate Successor Element/Class Theory",
"Definition:Element/Class",
"Definition:Limit Element under Well-Ordering"
] | [
"Definition:Smallest Element/Class Theory",
"Definition:Well-Ordering/Class Theory",
"Definition:Smallest Element/Class Theory",
"Definition:Element/Class",
"Definition:Smallest Element/Class Theory",
"Definition:Smallest Element/Class Theory",
"Definition:Smallest Element/Class Theory",
"Definition:I... |
proofwiki-19487 | Characterization of P-adic Unit has Square Root in P-adic Units/Condition 1 implies Condition 2 | Let $\Z_p$ be the $p$-adic integers for some prime $p \ne 2$.
Let $Z_p^\times$ be the set of $p$-adic units.
Let $u \in Z_p^\times$ be a $p$-adic unit.
Let $u = c_0 + c_1p + c_2p^2 + \ldots$ be the $p$-adic expansion of $u$.
Let there exist $x \in \Z_p^\times$ such that $x^2 = u$.
Then:
:$c_0$ is a quadratic residue o... | Let there exist $x \in \Z_p^\times$ such that $x^2 = u$
Then:
$x^2 \equiv c_0 \pmod {p\Z_p}$
Let $x = x_0 + x_1p + x_2p^2 + x_3p^3 + \ldots$ be the $p$-adic expansion of $x$.
From Partial Sum Congruent to P-adic Integer Modulo Power of p:
:$x \equiv x_0 \pmod {p\Z_p}$
Then:
:$x_0^2 \equiv x^2 \equiv c_0 \pmod {p\Z_p}$
... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p \ne 2$.
Let $Z_p^\times$ be the [[Definition:Set|set]] of [[Definition:P-adic Unit|$p$-adic units]].
Let $u \in Z_p^\times$ be a [[Definition:P-adic Unit|$p$-adic unit]].
Let $u = c_0 + c_1p + c_2p^2 + \l... | Let there exist $x \in \Z_p^\times$ such that $x^2 = u$
Then:
$x^2 \equiv c_0 \pmod {p\Z_p}$
Let $x = x_0 + x_1p + x_2p^2 + x_3p^3 + \ldots$ be the [[Definition:P-adic Expansion|$p$-adic expansion]] of $x$.
From [[Partial Sum Congruent to P-adic Integer Modulo Power of p]]:
:$x \equiv x_0 \pmod {p\Z_p}$
Then:
:$x_... | Characterization of P-adic Unit has Square Root in P-adic Units/Condition 1 implies Condition 2 | https://proofwiki.org/wiki/Characterization_of_P-adic_Unit_has_Square_Root_in_P-adic_Units/Condition_1_implies_Condition_2 | https://proofwiki.org/wiki/Characterization_of_P-adic_Unit_has_Square_Root_in_P-adic_Units/Condition_1_implies_Condition_2 | [
"Characterization of P-adic Unit has Square Root in P-adic Units"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Set",
"Definition:P-adic Unit",
"Definition:P-adic Unit",
"Definition:P-adic Expansion",
"Definition:Quadratic Residue"
] | [
"Definition:P-adic Expansion",
"Partial Sum Congruent to P-adic Integer Modulo Power of p",
"Definition:Quadratic Residue",
"Definition:Quadratic Residue"
] |
proofwiki-19488 | Characterization of P-adic Unit has Square Root in P-adic Units/Condition 2 implies Condition 3 | Let $\Z_p$ be the $p$-adic integers for some prime $p \ne 2$.
Let $Z_p^\times$ be the set of $p$-adic units.
Let $u \in Z_p^\times$ be a $p$-adic unit.
Let $u = c_0 + c_1p + c_2p^2 + \ldots$ be the $p$-adic expansion of $u$.
Let $c_0$ be a quadratic residue of $p$.
Then:
:$\exists y \in \Z_p : y^2 \equiv u \pmod{p\Z_p... | Let $c_0$ be a quadratic residue of $p$.
By definition of quadratic residue:
:$\exists x_0 \in \Z : x_0^2 \equiv c_0 \pmod {p\Z_p}$
Then:
:$x_0^2 \equiv u \pmod {p\Z_p}$ | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p \ne 2$.
Let $Z_p^\times$ be the [[Definition:Set|set]] of [[Definition:P-adic Unit|$p$-adic units]].
Let $u \in Z_p^\times$ be a [[Definition:P-adic Unit|$p$-adic unit]].
Let $u = c_0 + c_1p + c_2p^2 + \l... | Let $c_0$ be a [[Definition:Quadratic Residue|quadratic residue]] of $p$.
By definition of [[Definition:Quadratic Residue|quadratic residue]]:
:$\exists x_0 \in \Z : x_0^2 \equiv c_0 \pmod {p\Z_p}$
Then:
:$x_0^2 \equiv u \pmod {p\Z_p}$ | Characterization of P-adic Unit has Square Root in P-adic Units/Condition 2 implies Condition 3 | https://proofwiki.org/wiki/Characterization_of_P-adic_Unit_has_Square_Root_in_P-adic_Units/Condition_2_implies_Condition_3 | https://proofwiki.org/wiki/Characterization_of_P-adic_Unit_has_Square_Root_in_P-adic_Units/Condition_2_implies_Condition_3 | [
"Characterization of P-adic Unit has Square Root in P-adic Units"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Set",
"Definition:P-adic Unit",
"Definition:P-adic Unit",
"Definition:P-adic Expansion",
"Definition:Quadratic Residue"
] | [
"Definition:Quadratic Residue",
"Definition:Quadratic Residue"
] |
proofwiki-19489 | Characterization of P-adic Unit has Square Root in P-adic Units/Condition 3 implies Condition 1 | Let $\Z_p$ be the $p$-adic integers for some prime $p \ne 2$.
Let $Z_p^\times$ be the set of $p$-adic units.
Let $u \in Z_p^\times$ be a $p$-adic unit.
Let there exist $y \in \Z_p$ such that $y^2 \equiv u \pmod {p \Z_p}$.
Then:
:$\exists x \in \Z_p^\times : x^2 = u$ | Let there exist $y \in \Z_p$ such that $y^2 \equiv u \pmod {p\Z_p}$
Let $y = y_0 + y_1 p + y_2 p^2 + y_3 p^3 + \ldots$ be the $p$-adic expansion of $y$.
By definition of $p$-adic expansion:
:$y_0 \in {0, 1, \ldots, p - 1}$
From $p$-adic Expansion of $p$-adic Unit:
:$y_0 \ne 0$
Hence:
:$y_0 \in {1, 2, \ldots, p - 1}$
It... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p \ne 2$.
Let $Z_p^\times$ be the [[Definition:Set|set]] of [[Definition:P-adic Unit|$p$-adic units]].
Let $u \in Z_p^\times$ be a [[Definition:P-adic Unit|$p$-adic unit]].
Let there exist $y \in \Z_p$ such... | Let there exist $y \in \Z_p$ such that $y^2 \equiv u \pmod {p\Z_p}$
Let $y = y_0 + y_1 p + y_2 p^2 + y_3 p^3 + \ldots$ be the [[Definition:P-adic Expansion|$p$-adic expansion]] of $y$.
By definition of [[Definition:P-adic Expansion|$p$-adic expansion]]:
:$y_0 \in {0, 1, \ldots, p - 1}$
From [[P-adic Expansion of P-... | Characterization of P-adic Unit has Square Root in P-adic Units/Condition 3 implies Condition 1 | https://proofwiki.org/wiki/Characterization_of_P-adic_Unit_has_Square_Root_in_P-adic_Units/Condition_3_implies_Condition_1 | https://proofwiki.org/wiki/Characterization_of_P-adic_Unit_has_Square_Root_in_P-adic_Units/Condition_3_implies_Condition_1 | [
"Characterization of P-adic Unit has Square Root in P-adic Units"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Set",
"Definition:P-adic Unit",
"Definition:P-adic Unit"
] | [
"Definition:P-adic Expansion",
"Definition:P-adic Expansion",
"P-adic Expansion of P-adic Unit",
"Partial Sum Congruent to P-adic Integer Modulo Power of p",
"Definition:Polynomial over Ring",
"Definition:Formal Derivative of Polynomial",
"Definition:Contrapositive Statement",
"Euclid's Lemma for Prim... |
proofwiki-19490 | Lower Closure is Strict Lower Closure of Immediate Successor | Let $\struct {S, \preccurlyeq}$ be a totally ordered set.
Let $b$ be the immediate successor element of $a$:
Then:
:$a^\preccurlyeq = b^\prec$
where:
:$a^\preccurlyeq$ is the lower closure of $a$
:$b^\prec$ is the strict lower closure of $b$. | Let:
:$x \in b^\prec$
By the definition of strict upper closure:
:$x \prec b$
By the definition of total ordering:
:$a \prec x$ or $x \preccurlyeq a$
If $a \prec x$ then $a \prec x \prec b$, contradicting the premise.
Thus:
:$x \preccurlyeq a$
and so:
:$x \in a^\preccurlyeq$
By definition of subset:
:$b^\prec \subseteq... | Let $\struct {S, \preccurlyeq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $b$ be the [[Definition:Immediate Successor Element|immediate successor element]] of $a$:
Then:
:$a^\preccurlyeq = b^\prec$
where:
:$a^\preccurlyeq$ is the [[Definition:Lower Closure of Element|lower closure]] of $a$
:$... | Let:
:$x \in b^\prec$
By the definition of [[Definition:Strict Upper Closure of Element|strict upper closure]]:
:$x \prec b$
By the definition of [[Definition:Total Ordering|total ordering]]:
:$a \prec x$ or $x \preccurlyeq a$
If $a \prec x$ then $a \prec x \prec b$, [[Definition:Contradiction|contradicting]] the [[... | Lower Closure is Strict Lower Closure of Immediate Successor | https://proofwiki.org/wiki/Lower_Closure_is_Strict_Lower_Closure_of_Immediate_Successor | https://proofwiki.org/wiki/Lower_Closure_is_Strict_Lower_Closure_of_Immediate_Successor | [
"Total Orderings",
"Lower Closures"
] | [
"Definition:Totally Ordered Set",
"Definition:Immediate Successor Element",
"Definition:Lower Closure/Element",
"Definition:Strict Lower Closure/Element"
] | [
"Definition:Strict Upper Closure/Element",
"Definition:Total Ordering",
"Definition:Contradiction",
"Definition:Premise",
"Definition:Subset",
"Definition:Upper Closure/Element",
"Extended Transitivity",
"Definition:Subset",
"Definition:Set Equality",
"Category:Total Orderings",
"Category:Lower ... |
proofwiki-19491 | Characterisation of Limit Element under Well-Ordering | Let $A$ be a class.
Let $\preccurlyeq$ be a well-ordering on $A$.
Let $x \in A$ be an element of $A$ such that $x$ is not the smallest element of $A$ under $\preccurlyeq$.
Then:
:$x$ is a limit element of $A$ under $\preccurlyeq$
{{iff}}:
:$x^\prec$ has no greatest element {{WRT}} $\preccurlyeq$
where $x^\prec$ denotes... | Suppose $x^\prec$ has a greatest element $y$.
Then $x$ is the immediate successor of $y$, that is:
:$\nexists z \in A : y < z < x$
Hence $x$ is not a limit element.
Therefore, if $x$ is a limit element, then $x^\prec$ cannot have a greatest element.
{{qed|lemma}}
Suppose $x$ is an immediate successor.
Then the immediat... | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $\preccurlyeq$ be a [[Definition:Well-Ordering (Class Theory)|well-ordering]] on $A$.
Let $x \in A$ be an [[Definition:Element of Class|element]] of $A$ such that $x$ is not the [[Definition:Smallest Element (Class Theory)|smallest element]] of $A$ under $\p... | Suppose $x^\prec$ has a [[Definition:Greatest Element (Class Theory)|greatest element]] $y$.
Then $x$ is the [[Definition:Immediate Successor Element (Class Theory)|immediate successor]] of $y$, that is:
:$\nexists z \in A : y < z < x$
Hence $x$ is not a [[Definition:Limit Element under Well-Ordering|limit element]].... | Characterisation of Limit Element under Well-Ordering | https://proofwiki.org/wiki/Characterisation_of_Limit_Element_under_Well-Ordering | https://proofwiki.org/wiki/Characterisation_of_Limit_Element_under_Well-Ordering | [
"Well-Orderings",
"Limit Elements"
] | [
"Definition:Class (Class Theory)",
"Definition:Well-Ordering/Class Theory",
"Definition:Element/Class",
"Definition:Smallest Element/Class Theory",
"Definition:Limit Element under Well-Ordering",
"Definition:Greatest Element/Class Theory",
"Definition:Strict Lower Closure/Element/Class Theory"
] | [
"Definition:Greatest Element/Class Theory",
"Definition:Immediate Successor Element/Class Theory",
"Definition:Limit Element under Well-Ordering",
"Definition:Limit Element under Well-Ordering",
"Definition:Greatest Element/Class Theory",
"Definition:Immediate Successor Element/Class Theory",
"Definitio... |
proofwiki-19492 | Lp Norm is Norm | Let $\struct {X, \Sigma, \mu}$ be a measure space and let $p \in \hointr 1 \infty$.
Let $\map {L^p} {X, \Sigma, \mu}$ be the $L^p$ vector space on $\struct {X, \Sigma, \mu}$.
Let $\norm \cdot_p$ be the $L^p$ norm.
Then $\norm \cdot_p$ is a norm on $\map {L^p} {X, \Sigma, \mu}$. | Let $\map {\mathcal L^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space.
Let $\sim$ be the $\mu$-almost everywhere equality relation on $\map {\mathcal L^p} {X, \Sigma, \mu}$.
Let $\eqclass f \sim \in \map {L^p} {X, \Sigma, \mu}$.
Then, we have by the definition of the $L^p$ norm we have:
:$\norm {\eqclass f \sim}_p = ... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]] and let $p \in \hointr 1 \infty$.
Let $\map {L^p} {X, \Sigma, \mu}$ be the [[Definition:Lp Vector Space|$L^p$ vector space]] on $\struct {X, \Sigma, \mu}$.
Let $\norm \cdot_p$ be the [[Definition:Lp Norm|$L^p$ norm]].
Then $\norm \cdot_p... | Let $\map {\mathcal L^p} {X, \Sigma, \mu}$ be the [[Definition:Lebesgue Space|Lebesgue $p$-space]].
Let $\sim$ be the [[Definition:Almost-Everywhere Equality Relation on Lebesgue Space|$\mu$-almost everywhere equality relation on $\map {\mathcal L^p} {X, \Sigma, \mu}$]].
Let $\eqclass f \sim \in \map {L^p} {X, \Sigm... | Lp Norm is Norm | https://proofwiki.org/wiki/Lp_Norm_is_Norm | https://proofwiki.org/wiki/Lp_Norm_is_Norm | [
"Lp Norms",
"Normed Vector Spaces"
] | [
"Definition:Measure Space",
"Definition:Lp Space/Vector Space",
"Definition:Lp Norm",
"Definition:Norm/Vector Space"
] | [
"Definition:Lebesgue Space",
"Definition:Almost-Everywhere Equality Relation/Lebesgue Space",
"Definition:Lp Norm",
"P-Seminorm is Seminorm",
"Definition:Mapping",
"Definition:Positive/Real Number",
"Axiom:Vector Space Norm Axioms",
"Definition:Lp Norm",
"P-Seminorm is Seminorm"
] |
proofwiki-19493 | Kernel of Linear Transformation between Finite-Dimensional Normed Vector Spaces is Closed | Let $m, n \in \N_{> 0}$ be natural numbers.
Let $A \in \R^{m \times n}$ be a matrix.
Let $\ker A = \set {\mathbf x \in \R^n : A \mathbf x = 0}$ be the kernel of $A$.
Then $\ker A$ is a closed subspace of $\R^n$. | Let $T_A : \R^n \to \R^m$ be the linear transformation such that:
:$\forall \mathbf x \in \R^n : T_A \mathbf x := A \mathbf x$
By Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous, $T_A : \R^n \to \R^m$ is continuous.
We have that Singleton in Normed Vector Space is Closed.
Hence, ... | Let $m, n \in \N_{> 0}$ be [[Definition:Natural Number|natural numbers]].
Let $A \in \R^{m \times n}$ be a [[Definition:Matrix|matrix]].
Let $\ker A = \set {\mathbf x \in \R^n : A \mathbf x = 0}$ be the [[Definition:Kernel of Linear Transformation on Vector Space|kernel]] of $A$.
Then $\ker A$ is a [[Definition:Clo... | Let $T_A : \R^n \to \R^m$ be the [[Definition:Linear Transformation on Vector Space|linear transformation]] such that:
:$\forall \mathbf x \in \R^n : T_A \mathbf x := A \mathbf x$
By [[Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous]], $T_A : \R^n \to \R^m$ is [[Definition:Contin... | Kernel of Linear Transformation between Finite-Dimensional Normed Vector Spaces is Closed | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_between_Finite-Dimensional_Normed_Vector_Spaces_is_Closed | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_between_Finite-Dimensional_Normed_Vector_Spaces_is_Closed | [
"Operator Theory",
"Continuous Mappings",
"Linear Transformations"
] | [
"Definition:Natural Numbers",
"Definition:Matrix",
"Definition:Kernel of Linear Transformation/Vector Space",
"Definition:Closed Set/Normed Vector Space",
"Definition:Vector Subspace"
] | [
"Definition:Linear Transformation/Vector Space",
"Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous",
"Definition:Continuous Linear Transformation Space",
"Singleton in Normed Vector Space is Closed",
"Definition:Closed Set/Normed Vector Space",
"Mapping is Continuous i... |
proofwiki-19494 | Equivalence of Definitions of Commutative Local Ring | Let $A$ be a commutative ring with unity.
{{TFAE|def = Commutative Local Ring}} | === Definition 1 implies Definition 2 ===
Let $\mathfrak m \subsetneq A$ be the unique maximal ideal.
First, $A$ is nontrivial, since $1 \notin \mathfrak m$.
Secondly, let $x, y \in A$ be non-units.
Let $\ideal x$ and $\ideal y$ be the principal ideals generated by $x$ and $y$, respectively.
In view of the unique maxim... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
{{TFAE|def = Commutative Local Ring}} | === Definition 1 implies Definition 2 ===
Let $\mathfrak m \subsetneq A$ be the [[Definition:Unique|unique]] [[Definition:Maximal Ideal of Ring|maximal ideal]].
First, $A$ is [[Definition:Non-Trivial Ring|nontrivial]], since $1 \notin \mathfrak m$.
Secondly, let $x, y \in A$ be non-[[Definition:Unit of Ring|units]].... | Equivalence of Definitions of Commutative Local Ring | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Commutative_Local_Ring | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Commutative_Local_Ring | [
"Definitions/Local Rings"
] | [
"Definition:Commutative and Unitary Ring"
] | [
"Definition:Unique",
"Definition:Maximal Ideal of Ring",
"Definition:Non-Trivial Ring",
"Definition:Unit of Ring",
"Definition:Principal Ideal of Ring",
"Definition:Unique",
"Definition:Maximal Ideal of Ring",
"Definition:Unit of Ring",
"Definition:Non-Trivial Ring",
"Definition:Unit of Ring",
"... |
proofwiki-19495 | Strict Lower Closure of Limit Element is Infinite | Let $A$ be a class.
Let $\preccurlyeq$ be a well-ordering on $A$.
Let $x \in A$ be a limit element of $A$ under $\preccurlyeq$.
Let $x^\prec$ denotes the strict lower closure of $x$ in $A$ under $\preccurlyeq$.
Then $x^\prec$ is an infinite set. | Let $x \in A$ be a limit element of $A$ under $\preccurlyeq$.
From Characterisation of Limit Element under Well-Ordering it follows that $x^\prec$ has no greatest element {{WRT}} $\preccurlyeq$.
The result follows.
{{qed}} | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $\preccurlyeq$ be a [[Definition:Well-Ordering (Class Theory)|well-ordering]] on $A$.
Let $x \in A$ be a [[Definition:Limit Element under Well-Ordering|limit element]] of $A$ under $\preccurlyeq$.
Let $x^\prec$ denotes the [[Definition:Strict Lower Closure ... | Let $x \in A$ be a [[Definition:Limit Element under Well-Ordering|limit element]] of $A$ under $\preccurlyeq$.
From [[Characterisation of Limit Element under Well-Ordering]] it follows that $x^\prec$ has no [[Definition:Greatest Element (Class Theory)|greatest element]] {{WRT}} $\preccurlyeq$.
The result follows.
{{q... | Strict Lower Closure of Limit Element is Infinite | https://proofwiki.org/wiki/Strict_Lower_Closure_of_Limit_Element_is_Infinite | https://proofwiki.org/wiki/Strict_Lower_Closure_of_Limit_Element_is_Infinite | [
"Well-Orderings",
"Lower Closures",
"Limit Elements"
] | [
"Definition:Class (Class Theory)",
"Definition:Well-Ordering/Class Theory",
"Definition:Limit Element under Well-Ordering",
"Definition:Strict Lower Closure/Element/Class Theory",
"Definition:Infinite Set"
] | [
"Definition:Limit Element under Well-Ordering",
"Characterisation of Limit Element under Well-Ordering",
"Definition:Greatest Element/Class Theory"
] |
proofwiki-19496 | First Principle of Transfinite Induction | Let $A$ be a class.
Let $\preccurlyeq$ be a well-ordering on $A$.
Let $P$ be a property that satisfies the following condition:
:For all $x \in A$, if $P$ holds for every $y \prec x$, then $P$ holds for $x$.
Then $P$ holds for all $x \in A$. | Let $s$ be the smallest element of $A$.
Then vacuously $P$ holds for every element $y \in A$ such that $y \prec s$.
Thus $P$ holds for $s$.
{{AimForCont}} $P$ fails to hold for some $z \in A$.
The class of elements of $A$ for which $P$ does not hold is therefore non-empty.
We have that $A$ is a well-ordered class.
Henc... | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $\preccurlyeq$ be a [[Definition:Well-Ordering (Class Theory)|well-ordering]] on $A$.
Let $P$ be a [[Definition:Property|property]] that satisfies the following condition:
:For all $x \in A$, if $P$ holds for every $y \prec x$, then $P$ holds for $x$.
Then... | Let $s$ be the [[Definition:Smallest Element (Class Theory)|smallest element]] of $A$.
Then [[Definition:Vacuous Truth|vacuously]] $P$ holds for every [[Definition:Element of Class|element]] $y \in A$ such that $y \prec s$.
Thus $P$ holds for $s$.
{{AimForCont}} $P$ fails to hold for some $z \in A$.
The [[Definiti... | First Principle of Transfinite Induction | https://proofwiki.org/wiki/First_Principle_of_Transfinite_Induction | https://proofwiki.org/wiki/First_Principle_of_Transfinite_Induction | [
"Well-Orderings",
"Mathematical Induction"
] | [
"Definition:Class (Class Theory)",
"Definition:Well-Ordering/Class Theory",
"Definition:Property"
] | [
"Definition:Smallest Element/Class Theory",
"Definition:Vacuous Truth",
"Definition:Element/Class",
"Definition:Class (Class Theory)",
"Definition:Element/Class",
"Definition:Non-Empty Set/Class Theory",
"Definition:Well-Ordered Class",
"Definition:Smallest Element/Class Theory",
"Definition:Contrad... |
proofwiki-19497 | Second Principle of Transfinite Induction | Let $A$ be a class.
Let $\preccurlyeq$ be a well-ordering on $A$.
Let $P$ be a property that satisfies the following $3$ conditions:
:$(1): \quad P$ holds for the smallest element of $A$.
:$(2): \quad$ For all $x \in A$ which have an immediate successor $\map S x$, if $P$ holds for $x$, then $P$ holds for $\map S x$.
:... | {{AimForCont}} $P$ fails to hold for some $z \in A$.
The class of elements of $A$ for which $P$ does not hold is therefore non-empty.
We have that $A$ is a well-ordered class.
Hence there must be some smallest element $x \in A$ for which $P$ fails to hold.
By $(1)$, this cannot be the smallest element of $A$.
By $(2)$,... | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $\preccurlyeq$ be a [[Definition:Well-Ordering (Class Theory)|well-ordering]] on $A$.
Let $P$ be a [[Definition:Property|property]] that satisfies the following $3$ conditions:
:$(1): \quad P$ holds for the [[Definition:Smallest Element (Class Theory)|small... | {{AimForCont}} $P$ fails to hold for some $z \in A$.
The [[Definition:Class (Class Theory)|class]] of [[Definition:Element of Class|elements]] of $A$ for which $P$ does not hold is therefore [[Definition:Non-Empty Class|non-empty]].
We have that $A$ is a [[Definition:Well-Ordered Class|well-ordered class]].
Hence th... | Second Principle of Transfinite Induction | https://proofwiki.org/wiki/Second_Principle_of_Transfinite_Induction | https://proofwiki.org/wiki/Second_Principle_of_Transfinite_Induction | [
"Well-Orderings",
"Mathematical Induction"
] | [
"Definition:Class (Class Theory)",
"Definition:Well-Ordering/Class Theory",
"Definition:Property",
"Definition:Smallest Element/Class Theory",
"Definition:Immediate Successor Element/Class Theory",
"Definition:Limit Element under Well-Ordering"
] | [
"Definition:Class (Class Theory)",
"Definition:Element/Class",
"Definition:Non-Empty Set/Class Theory",
"Definition:Well-Ordered Class",
"Definition:Smallest Element/Class Theory",
"Definition:Smallest Element/Class Theory",
"Definition:Immediate Successor Element/Class Theory",
"Definition:Limit Elem... |
proofwiki-19498 | Pointwise Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\map {\mathcal M} {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.
Let $\sim$ be the almost-everywhere equality equivalence relation on $\map {\mathcal M} {X, \Sigma, \R}$.
Let $\map {\mathcal M} {X, \Sigma, \R}/\sim$ be the space ... | Let $E_1, E_2 \in \map {\mathcal M} {X, \Sigma, \R}/\sim$.
We need to show that $E_1 \cdot E_2$ is independent of the choice of representative for $E_1$ and $E_2$.
Suppose that:
:$\eqclass f \sim = \eqclass F \sim = E_1$
and:
:$\eqclass g \sim = \eqclass G \sim = E_2$
From Equivalence Class Equivalent Statements, we h... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\map {\mathcal M} {X, \Sigma, \R}$ be the set of [[Definition:Measurable Real-Valued Function|real-valued $\Sigma$-measurable functions]] on $X$.
Let $\sim$ be the [[Definition:Almost-Everywhere Equality Relation|almost-everywhere e... | Let $E_1, E_2 \in \map {\mathcal M} {X, \Sigma, \R}/\sim$.
We need to show that $E_1 \cdot E_2$ is independent of the choice of [[Definition:Representative of Equivalence Class|representative]] for $E_1$ and $E_2$.
Suppose that:
:$\eqclass f \sim = \eqclass F \sim = E_1$
and:
:$\eqclass g \sim = \eqclass G \sim =... | Pointwise Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Space_of_Real-Valued_Measurable_Functions_Identified_by_A.E._Equality_is_Well-Defined | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Space_of_Real-Valued_Measurable_Functions_Identified_by_A.E._Equality_is_Well-Defined | [
"Space of Real-Valued Measurable Functions Identified by A.E. Equality"
] | [
"Definition:Measure Space",
"Definition:Measurable Function/Real-Valued Function",
"Definition:Almost-Everywhere Equality Relation",
"Definition:Space of Measurable Functions Identified by A.E. Equality/Real-Valued Function",
"Definition:Pointwise Multiplication on Space of Real-Valued Measurable Functions ... | [
"Definition:Equivalence Class/Representative",
"Equivalence Class Equivalent Statements",
"Pointwise Multiplication preserves A.E. Equality",
"Equivalence Class Equivalent Statements",
"Category:Space of Real-Valued Measurable Functions Identified by A.E. Equality"
] |
proofwiki-19499 | Lp Space is Vector Space | Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.
Let $\map \MM {X, \Sigma, \R}/\sim_\mu$ be the space of real-valued measurable functions identified by $\mu$-A.E. equality.
Let $\map {L^p} {X, \Sigma, \mu}$ be the $L^p$ space of $\struct {X, \Sigma, \mu}$.
Let $+$ denote pointwise... | It is sufficient to show that $\struct {\map {L^p} {X, \Sigma, \mu}, +, \cdot}_\R$ is a vector subspace of $\struct {\map \MM {X, \Sigma, \R}/\sim_\mu, +, \cdot}_\R$.
From $L^p$ Space is Subset of Space of Measurable Functions Identified by A.E. Equality:
:$\map {L^p} {X, \Sigma, \mu} \subseteq \map \MM {X, \Sigma, \R}... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]], and let $p \in \closedint 1 \infty$.
Let $\map \MM {X, \Sigma, \R}/\sim_\mu$ be the [[Definition:Space of Measurable Functions Identified by A.E. Equality|space of real-valued measurable functions identified by $\mu$-A.E. equality]].
Let ... | It is sufficient to show that $\struct {\map {L^p} {X, \Sigma, \mu}, +, \cdot}_\R$ is a [[Definition:Vector Subspace|vector subspace]] of $\struct {\map \MM {X, \Sigma, \R}/\sim_\mu, +, \cdot}_\R$.
From [[Lp Space is Subset of Space of Real-Valued Measurable Functions Identified by A.E. Equality|$L^p$ Space is Subset... | Lp Space is Vector Space | https://proofwiki.org/wiki/Lp_Space_is_Vector_Space | https://proofwiki.org/wiki/Lp_Space_is_Vector_Space | [
"Lp Spaces",
"Vector Spaces"
] | [
"Definition:Measure Space",
"Definition:Space of Measurable Functions Identified by A.E. Equality",
"Definition:Lp Space",
"Definition:Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality",
"Definition:Pointwise Scalar Multiplication on Space of Real-Valued Measurable ... | [
"Definition:Vector Subspace",
"Lp Space is Subset of Space of Real-Valued Measurable Functions Identified by A.E. Equality",
"Definition:Lebesgue Space",
"Definition:Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality",
"Lebesgue Space is Vector Space",
"Lp Space is... |
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