# User contributions for Caliburn

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#### 14 March 2024

- 14:0214:02, 14 March 2024 diff hist +59 Talk:Finite Dimensional Subspace of Normed Vector Space is Closed No edit summary
- 14:0114:01, 14 March 2024 diff hist +12 Talk:Finite Dimensional Subspace of Normed Vector Space is Closed No edit summary
- 14:0014:00, 14 March 2024
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**+816** Talk:Finite Dimensional Subspace of Normed Vector Space is Closed No edit summary

#### 13 March 2024

- 18:2618:26, 13 March 2024 diff hist +186 Talk:Finite Dimensional Subspace of Normed Vector Space is Closed No edit summary
- 18:2418:24, 13 March 2024 diff hist +8 Talk:Finite Dimensional Subspace of Normed Vector Space is Closed No edit summary
- 18:2418:24, 13 March 2024 diff hist +431 Talk:Finite Dimensional Subspace of Normed Vector Space is Closed No edit summary

#### 10 January 2024

- 15:0415:04, 10 January 2024 diff hist −19 Talk:Element of Spectrum of Self-Adjoint Densely-Defined Linear Operator is Approximate Eigenvalue No edit summary current
- 15:0115:01, 10 January 2024 diff hist +288 N Talk:Element of Spectrum of Self-Adjoint Densely-Defined Linear Operator is Approximate Eigenvalue Created page with "Writing up something else and just realised that this could easily be reframed as approximate eigenvalue $\implies$ in point spectrum or continuous spectrum. The self-adjoint bit is entirely extraneous. ~~~~"

#### 20 November 2023

- 19:0419:04, 20 November 2023 diff hist −6 Characterization of Norm-to-Weak and Weak-to-Weak Continuity of Linear Transformations No edit summary

#### 29 October 2023

- 16:3816:38, 29 October 2023 diff hist +19 Euler Phi Function of Product wrong one current
- 16:3716:37, 29 October 2023 diff hist +193 Euler Phi Function of Prime Power No edit summary
- 13:5113:51, 29 October 2023 diff hist +1 Euler Phi Function is Multiplicative/Proof 2 No edit summary
- 13:5113:51, 29 October 2023 diff hist +29 Euler Phi Function is Multiplicative/Proof 2 No edit summary
- 13:5113:51, 29 October 2023
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**−2,814** Euler Phi Function is Multiplicative No edit summary - 13:5013:50, 29 October 2023
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**+3,286** N Euler Phi Function is Multiplicative/Proof 1 Created page with "== Theorem == {{:Euler Phi Function is Multiplicative}} == Proof == <onlyinclude> Let $R = \set {r_1, r_2, \ldots, r_{\map \phi m} }$ and $S = \set {s_1, s_2, \ldots, s_{\map \phi n} }$ be the reduced residue systems for the respective moduli $m$ and $n$. We are to show that the set of $\map \phi m \map \phi n$ integers: :$T = \set {n r + m s: r \in..." current - 13:4913:49, 29 October 2023 diff hist +425 N Euler Phi Function is Multiplicative/Proof 2 Created page with "== Theorem == {{:Euler Phi Function is Multiplicative}} == Proof == From Euler Phi Function of Product, we have: :$\ds \map \phi {m n} = \map \phi m \map \phi n \paren {\frac {\map \gcd {m, n} } {\map \phi {\map \gcd {m, n} } } }$ From hypothesis, we have: :$\map \gcd {m, n} = 1$ From Euler Phi Function of 1, we have $\map \phi = 1$, giving the result. {{qed}} Category:Euler Phi Function is Multiplicative"
- 13:4613:46, 29 October 2023 diff hist +27 Euler Phi Function is Multiplicative No edit summary
- 13:4413:44, 29 October 2023
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**+2,676** N Euler Phi Function of Product not sure if this is already here, names are not enlightening, Euler Phi Function is Multiplicative is proved without it - 13:4213:42, 29 October 2023 diff hist −103 Euler Phi Function of Product with Prime No edit summary current
- 13:4113:41, 29 October 2023 diff hist 0 m Euler Phi Function preserves Divisibility Caliburn moved page Euler Phi Function of Product with Prime/Corollary to Euler Phi Function preserves Divisibility without leaving a redirect: unintuitive name current
- 12:3112:31, 29 October 2023 diff hist +205 Euler Phi Function of Integer No edit summary
- 12:3012:30, 29 October 2023 diff hist +212 Euler Phi Function in terms of Möbius Function No edit summary
- 12:2712:27, 29 October 2023 diff hist −1 Sum of Euler Phi Function over Divisors No edit summary
- 12:2712:27, 29 October 2023 diff hist +3 Sum of Euler Phi Function over Divisors No edit summary
- 12:2412:24, 29 October 2023 diff hist +3 Definition:Euler Phi Function No edit summary
- 12:2412:24, 29 October 2023 diff hist +211 Sum of Euler Phi Function over Divisors No edit summary
- 12:2312:23, 29 October 2023 diff hist +226 Definition:Euler Phi Function No edit summary
- 12:2112:21, 29 October 2023 diff hist −10 Sum of Möbius Function over Divisors No edit summary
- 12:1912:19, 29 October 2023 diff hist +49 Sum of Möbius Function over Divisors No edit summary
- 12:1912:19, 29 October 2023 diff hist +59 Euler Phi Function in terms of Möbius Function No edit summary
- 12:1512:15, 29 October 2023 diff hist +268 N Talk:Euler Phi Function in terms of Möbius Function Created page with "I would actually want to call Sum of Möbius Function over Divisors/Lemma, "sum of Mobius function over divisors" and call this "Euler Phi Function in terms of Möbius Function". ~~~~"
- 11:3211:32, 29 October 2023 diff hist +26 Definition talk:Identity Arithmetic Function No edit summary current
- 11:3111:31, 29 October 2023 diff hist +244 Definition talk:Identity Arithmetic Function No edit summary
- 11:2411:24, 29 October 2023 diff hist +218 Sum of Möbius Function over Divisors No edit summary
- 11:2311:23, 29 October 2023 diff hist +204 Definition:Möbius Function No edit summary
- 11:2011:20, 29 October 2023 diff hist +144 Definition:Arithmetic Function No edit summary current
- 11:1011:10, 29 October 2023 diff hist +81 Square of Expectation of Product is Less Than or Equal to Product of Expectation of Squares I think this proof should remain since it doesn't need any measure theory current

#### 24 October 2023

- 10:2610:26, 24 October 2023 diff hist +130 Category talk:Chi-Squared Distribution No edit summary

#### 15 October 2023

- 11:5511:55, 15 October 2023 diff hist +430 N Definition:Normal Element of *-Algebra Created page with "== Definition == <onlyinclude> Let $\struct {A, \ast}$ be a $\ast$-algebra. Let $a \in A$. We say that $a$ is '''normal''' {{iff}}: :$a^\ast a = a a^\ast$ </onlyinclude> == Sources == * {{BookReference|C*-Algebras and Operator Theory|1990|Gerard J. Murphy|prev = Product of Element in *-Star Algebra with its Star is Hermitian|next = ?}}: $2.1$: $C^\ast$-Algebras Category:Definitions/*-Algebras" current
- 11:5211:52, 15 October 2023
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**+993** N Product of Element in *-Star Algebra with its Star is Hermitian not sure about title, will come back to do C* norm axiom stuff current - 11:5111:51, 15 October 2023 diff hist +3 Element of *-Algebra Uniquely Decomposes into Hermitian Elements No edit summary current
- 11:4511:45, 15 October 2023 diff hist +19 Element of *-Algebra Uniquely Decomposes into Hermitian Elements No edit summary
- 11:4411:44, 15 October 2023
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**+1,895** N Element of *-Algebra Uniquely Decomposes into Hermitian Elements Created page with "== Theorem == <onlyinclude> Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$. Let $a \in A$. Then there exists unique Hermitian elements $b, c \in A$ such that: :$a = b + i c$ </onlyinclude> == Proof == === Proof of Existence === {{finish|fill in $\text C^\ast x$ with template}} Let: :$b = \dfrac 1 2 \paren {a + a^\ast}$ and: :$c = \dfrac 1 {2 i} \paren {a - a^\ast}$ Then we have using $(..." - 11:2611:26, 15 October 2023 diff hist +436 N Definition:Hermitian Element of *-Algebra Created page with "== Definition == <onlyinclude> Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$. Let $a \in A$. We say that $a$ is '''Hermitian''' {{iff}} $a^\ast = a$. </onlyinclude> == Sources == * {{BookReference|C*-Algebras and Operator Theory|1990|Gerard J. Murphy|prev = ?|next = Element of *-Algebra Uniquely Decomposes into Hermitian Elements}}: $2.1$: $C^\ast$-Algebras Category:Definitions/*-Algebras" current
- 11:2311:23, 15 October 2023
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**+526** N Definition:*-Subalgebra Created page with "== Definition == <onlyinclude> Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$. Let $B \subseteq A$ be a self-adjoint subalgebra of $A$. We say that $B$ is a '''$\ast$-subalgebra''' of $A$. </onlyinclude> == Sources == * {{BookReference|C*-Algebras and Operator Theory|1990|Gerard J. Murphy|prev = Definition:Self-Adjoint Subset of *-Algebra|next = ?}}: $2.1$: $C^..." current - 11:2011:20, 15 October 2023
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**+513** N Definition:Self-Adjoint Subset of *-Algebra Created page with "== Definition == <onlyinclude> Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$. Let $S \subseteq A$ be a subset of $A$ such that: :for each $a \in S$ we have $a^\ast \in S$. We say that $S$ is a '''self-adjoint subset''' of $A$. </onlyinclude> == Sources == * {{BookReference|C*-Algebras and Operator Theory|1990|Gerard J. Murphy|prev = Definition:*-Algebra|next = Definition:*-Subalgebra}}: $2.1$: $C^\ast$-Algebra..." current - 11:1511:15, 15 October 2023 diff hist +485 N Definition:*-Algebra Created page with "== Definition == <onlyinclude> Let $A$ be an algebra over $\C$. Let $\ast$ be an involution on $A$. We call $\struct {A, \ast}$ a '''$\ast$-algebra'''. </onlyinclude> == Sources == * {{BookReference|C*-Algebras and Operator Theory|1990|Gerard J. Murphy|prev = Definition:Involution on Algebra|next = Definition:Self-Adjoint Subset of *-Algebra}}: $2.1$: $C^\ast$-Algebras Category:Definitions/*-..." current
- 11:1311:13, 15 October 2023 diff hist +1 Definition:Involution on Algebra No edit summary current
- 11:0911:09, 15 October 2023 diff hist −86 Definition:Involution on Algebra not sure how I managed that
- 11:0811:08, 15 October 2023
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**+1,057** N Definition:Involution on Algebra Created page with "== Definition == <onlyinclude> Let $A$ be an algebra over $\C$. Let $\ast : A \to A$ be a mapping satisfying: {{begin-axiom}} {{axiom | n = \text C^* 1 | q = \forall x \in A | ml= x^{**} | mo= = | mr= x }} {{axiom | n = \text C^* 2 | q = \forall x \in A | ml= x^* + y^* | mo= = | mr= \paren {x + y}^* }} {{axiom | n = \text C^* 3 | q = \forall..."