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Combined display of all available logs of ProofWiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).
- 15:01, 10 January 2024 Caliburn talk contribs created page 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. ~~~~")
- 13:50, 29 October 2023 Caliburn talk contribs created page 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: :$T = \set {n r + m s: r \in...")
- 13:49, 29 October 2023 Caliburn talk contribs created page Euler Phi Function is Multiplicative/Proof 2 (Created page with "== Theorem == {{:Euler Phi Function is Multiplicative}} == Proof == From Euler Phi Function of Product, we have: :$\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")
- 13:44, 29 October 2023 Caliburn talk contribs created page 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)
- 13:41, 29 October 2023 Caliburn talk contribs moved page Euler Phi Function of Product with Prime/Corollary to Euler Phi Function preserves Divisibility without leaving a redirect (unintuitive name)
- 12:15, 29 October 2023 Caliburn talk contribs created page Talk:Sum of Möbius Function over Divisors (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". ~~~~")
- 11:55, 15 October 2023 Caliburn talk contribs created page 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")
- 11:52, 15 October 2023 Caliburn talk contribs created page Product of Element in *-Star Algebra with its Star is Hermitian (not sure about title, will come back to do C* norm axiom stuff)
- 11:44, 15 October 2023 Caliburn talk contribs created page 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 $(...")
- 11:26, 15 October 2023 Caliburn talk contribs created page 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")
- 11:23, 15 October 2023 Caliburn talk contribs created page 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^...")
- 11:20, 15 October 2023 Caliburn talk contribs created page 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...")
- 11:15, 15 October 2023 Caliburn talk contribs created page 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/*-...")
- 11:08, 15 October 2023 Caliburn talk contribs created page 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...")
- 09:57, 15 October 2023 Caliburn talk contribs created page Complex Numbers form Preordered Vector Space (Created page with "== Theorem == <onlyinclude> Consider the complex numbers $\C$ as a vector space over itself. Define the relation $\ge$ by: :$z \ge w$ {{iff}}: :$z - w \in \hointr 0 \infty$ for each $z, w \in \C$. Then $\struct {\C, \ge}$ is a preordered vector space. </onlyinclude> == Proof == From Characterization of Preordered Vector Spaces, it is enough to...")
- 15:38, 12 October 2023 Caliburn talk contribs created page Definition:Positive Element of Preordered Vector Space (Created page with "== Definition == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {X, \succeq}$ be a preordered vector space over $\GF$. Let ${\mathbf 0}_X$ be the zero vector of $X$. Let $x \in X$. We say that $x$ is a '''positive element''' {{iff}} $x \succeq {\mathbf 0}_X$. </onlyinclude> == Sources == * {{BookReference|C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics|2023|Jean-Be...")
- 15:32, 12 October 2023 Caliburn talk contribs created page Characterization of Preordered Vector Spaces (from definitions)
- 11:46, 12 October 2023 Caliburn talk contribs created page Definition:Preordered Vector Space (need this to define "positive linear functional" in good conscience)
- 11:35, 12 October 2023 Caliburn talk contribs created page Definition:Preordering Induced by Convex Cone (Created page with "== Definition == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a vector space over $\GF$. Let $P \subseteq X$ be a convex cone in $X$. Define a relation $\succeq^P$ by: :$v \succeq^P v'$ {{iff}} $v - v' \in P$ for each $v, v' \in X$. We say that $\succeq^P$ is the '''preordering on $X$ induced by $P$'''. </onlyinclude> == Sources == * {{BookReference|C*-Algebras and Mathematica...")
- 11:29, 12 October 2023 Caliburn talk contribs created page Convex Cone is Convex Set (Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a vector space over $\GF$. Let $P \subseteq X$ be a convex cone in $X$. Then $P$ is convex. </onlyinclude> == Proof == Let $x, y \in P$. Let $t \in \closedint 0 1$ so that: :$t \ge 0$ and $1 - t \ge 0$. Since $P$ is a cone, we have: :$t x \in P$ and $\paren {1 - t} y...")
- 11:25, 12 October 2023 Caliburn talk contribs created page Definition:Convex Cone (Created page with "== Definition == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a vector space over $\GF$. Let $P \subseteq X$ be a cone in $X$. We say that $P$ is a '''convex cone''' {{iff}}: :for each $v, v' \in P$, we have $v + v' \in P$. </onlyinclude> == Also see == * Convex Cone is Convex Set == Sources == * {{BookReference|C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics|...")
- 11:23, 12 October 2023 Caliburn talk contribs created page Definition:Cone (Vector Space) (weird book to cite but this was the first one I thought of, has a lot of advanced FA in one place)
- 19:50, 11 October 2023 Caliburn talk contribs created page Talk:Complement of Bounded Set has Exactly One Unbounded Component (Created page with "I have a feeling that the spirit of your proof transfers to infinite-dimensional NVSs as well. Thanks for doing this anyway, I will need it. ~~~~")
- 13:42, 1 October 2023 Caliburn talk contribs created page Talk:Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function (Created page with "This was extracted out from Approximation to Stirling's Formula for Gamma Function for reference. I would guess the domain is the same as that theorem but I haven't seriously looked at special functions for a while. Binmore was cited for this but apparently this does not appear in the text. ~~~~")
- 15:33, 29 September 2023 Caliburn talk contribs created page Composition of Linear Isometries is Linear Isometry (Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {X, \norm \cdot_X}$, $\struct {Y, \norm \cdot_Y}$ and $\struct {Z, \norm \cdot_Z}$ be normed vector spaces over $\GF$. Let $T : X \to Y$ and $S : Y \to Z$ be linear isometries. Then $S T$ is a linear isometry. </onlyinclude> == Proof == From Composition of Linear Transformations is Linear Transform...")
- 15:01, 29 September 2023 Caliburn talk contribs created page Completion Theorem/Normed Vector Space (Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Then there exists a Banach space $\struct {\widetilde X, \widetilde {\norm {\, \cdot \,} } }$ and a linear isometry $\phi : X \to \widetilde X$ such that $\phi \sqbrk X$ is dense in $\widetilde X$. Further, the Definiti...")
- 14:44, 29 September 2023 Caliburn talk contribs created page Talk:Completion Theorem (Metric Space)/Lemma 4 (Created page with "I will check this later, currently doing completion for NVSs. But in general there will be no inclusions between $A_1$ and $A_2$, and even if there was $\phi_1^{-1} \circ \phi_2$ would have domain $A$ not $\phi_1 \sqbrk A$. ~~~~")
- 13:01, 29 September 2023 Caliburn talk contribs created page Seminorm on Vector Space induces Norm on Quotient (Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a vector space over $\GF$. Let $p$ be a seminorm on $X$. Let: :$N = \set {x \in X : \map p x = 0}$ From Set of Points for which Seminorm is Zero is Vector Subspace, $N$ is a vector subspace. Let $X/N$ be the quotient vector space of $X$ modulo $N$. Let $\pi : X \to X/N$ be the...")
- 15:25, 28 September 2023 Caliburn talk contribs created page Condition for Closure of Open Ball to be Closed Ball of Same Radius (useful)
- 10:30, 26 August 2023 Caliburn talk contribs created page Category:Gelfand's Spectral Radius Formula (Created page with "{{SubjectCategory|result = Gelfand's Spectral Radius Formula}} Category:Spectral Theory")
- 10:29, 26 August 2023 Caliburn talk contribs created page Gelfand's Spectral Radius Formula (Created page with "== Theorem == <onlyinclude> === Bounded Linear Operator === {{:Gelfand's Spectral Radius Formula/Bounded Linear Operator}} === Banach Algebra === {{:Gelfand's Spectral Radius Formula/Banach Algebra}} </onlyinclude> Category:Gelfand's Spectral Radius Formula")
- 10:27, 26 August 2023 Caliburn talk contribs created page Gelfand's Spectral Radius Formula/Banach Algebra (Created page with "== Theorem == <onlyinclude> Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$. Let $x \in A$. Let $\map {r_A} x$ be the spectral radius of $x$ in $A$. Then, we have: :$\ds \map {r_A} x = \inf_{n \mathop \in \N} \norm {x^n}^{1/n} = \lim_{n \mathop \to \infty} \norm {x^n}^{1/n}$ </onlyinclude> == Proof == First suppose that $A$ is Definition:Unital Banach Algebra|unital...")
- 21:33, 25 August 2023 Caliburn talk contribs moved page Kernel of Character on Commutative Banach Algebra is Maximal Ideal to Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal without leaving a redirect
- 09:53, 25 August 2023 Caliburn talk contribs created page Spectrum of Element in Maximal Commutative Subalgebra of Unital Banach Algebra (Created page with "== Theorem == <onlyinclude> Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra. Let $C$ be a commutative subalgebra of $A$ that is maximal with respect to set inclusion. Let $x \in C$. Let $\map {\sigma_A} x$ and $\map {\sigma_C} x$ be the Definition:Spectrum (Spectral Theory)...")
- 19:01, 24 August 2023 Caliburn talk contribs created page Maximal Commutative Subalgebra of Unital Algebra is Unital (Created page with "== Theorem == <onlyinclude> Let $K$ be a field. Let $A$ be a unital algebra over $K$. Let $B$ be a commutative subalgebra of $A$ that is maximal with respect to set inclusion. Then $B$ is unital. </onlyinclude> == Proof == We show that $B + K {\mat...")
- 18:48, 24 August 2023 Caliburn talk contribs created page Maximal Subalgebra in Normed Algebra is Closed (Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$. Let $B$ be a subalgebra of $A$ that is maximal with respect to set inclusion. Then $B$ is closed. </onlyinclude> == Proof == From Closure of Subalgebra in Normed Algebra is Subalgebra, the Def...")
- 18:45, 24 August 2023 Caliburn talk contribs created page Closure of Subalgebra in Normed Algebra is Subalgebra (Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$. Let $B$ be a subalgebra of $A$. Then $B^-$ is a subalgebra of $A$. </onlyinclude> == Proof == From Closure of Subspace of Normed Vector Space is Subspace, $B^-$ is a vector subspace of $A$. Now let $x, y \in B^-$. From t...")
- 18:36, 24 August 2023 Caliburn talk contribs created page Category:Spectrum of Element in Unital Subalgebra (Created page with "{{SubjectCategory|result = Spectrum of Element in Unital Subalgebra}} Category:Spectra (Spectral Theory) Category:Unital Subalgebras")
- 18:35, 24 August 2023 Caliburn talk contribs created page Spectrum of Element in Unital Subalgebra/Corollary (Created page with "== Theorem == <onlyinclude> Let $A$ be a non-unital algebra over $\C$. Let $B$ be a subalgebra of $A$. Let $x \in B$. Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the spectra of $x$ in $A$ and $B$ respectively. Then: :$\map {\sigma_A} x \subseteq \map {\sigma_B} x$ </onlyinclude> == Proof == Let $A_+$ and $B_+$ be the Defini...")
- 18:17, 24 August 2023 Caliburn talk contribs created page Unitization of Algebra over Field preserves Subalgebra Relation (Created page with "== Theorem == <onlyinclude> Let $K$ be a field. Let $A$ be a commutative algebra over $K$. Let $B$ be a subalgebra of $A$. Let $A_+$ and $B_+$ be the unitizations of $A$ and $B$ respectively. Then $B_+$ is a unital subalgebra of $A_+$. </onlyinclude> == Proof == Let $\struct {x, s},...")
- 22:49, 23 August 2023 Caliburn talk contribs moved page Unitization of Commutative Algebra is Commutative to Unitization of Commutative Algebra over Field is Commutative without leaving a redirect
- 22:49, 23 August 2023 Caliburn talk contribs created page Unitization of Commutative Algebra is Commutative (Created page with "== Theorem == <onlyinclude> Let $K$ be a field. Let $A$ be a commutative algebra over $K$. Let $A_+$ be the unitization of $A$. Then $A_+$ is commutative. </onlyinclude> == Proof == Let $\tuple {x, \lambda}, \tuple {y, \mu} \in A_+$. Then, we have: {{begin-eqn}} {{eqn | l = \tuple {x, \la...")
- 19:41, 23 August 2023 Caliburn talk contribs created page Talk:Spectrum of Element in Unital Subalgebra (Created page with "sorry I've been making quite a few mistakes last week or so, blame the heat idk. thanks for pointing them all out. ~~~~")
- 10:01, 23 August 2023 Caliburn talk contribs created page Definition talk:Generator of Algebra (Created page with "Feels like the title is talking about $S$ while the page seems to be talking about the subalgebra generated by $S$. I would want a separate page for "generated subalgebra" ~~~~")
- 09:58, 23 August 2023 Caliburn talk contribs created page Intersection of Subalgebras is Subalgebra (Created page with "== Theorem == <onlyinclude> Let $K$ be a field. Let $\struct {A, +, \circ, \cdot}_K$ be an algebra over $K$. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of subalgebras of $A$. Let: :$\ds B = \bigcap_{\alpha \mathop \in I} A_\alpha$ Then $B$ is a subalgebra of $A$. </onlyinclude> == Proof == From Set...")
- 09:34, 23 August 2023 Caliburn talk contribs created page Spectrum of Element in Unital Subalgebra (Created page with "== Theorem == <onlyinclude> Let $A$ be an algebra over $\C$. Let $B$ be a unital subalgebra of $A$. Let $x \in B$. Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the spectra of $x$ in $A$ and $B$ respectively. Then: :$\map {\sigma_A} x \subseteq \map {\sigma_B} x$ </onlyinclude> == Proof == Let $\map G A$ and $\map G B$ be the Definition:Group of Units|...")
- 09:26, 23 August 2023 Caliburn talk contribs created page Group of Units of Submonoid is Subgroup (Created page with "== Theorem == <onlyinclude> Let $T$ be a monoid. Let $S$ be a submonoid of $T$. Let $\map G T$ and $\map G S$ be the groups of units of $T$ and $S$ respectively. Then $\map G S \subseteq \map G T$ and $\map G S$ is a subgroup of $\map G T$. </onlyinclude> == Proof == Let $x \in \map G S$. Then $x \in S$ and there exists $y \in S$ such that $x y = y x = e$. Since $...")
- 17:08, 21 August 2023 Caliburn talk contribs created page Talk:Resolvent Mapping is Analytic/Banach Algebra (Created page with "I don't think it's a mistake, just an omission on the derivative page I linked. Will take a look later. ~~~~")
- 15:35, 21 August 2023 Caliburn talk contribs created page Spectral Mapping Theorem for Polynomials (Created page with "== Theorem == <onlyinclude> Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$. Let $p : \C \to \C$ be a polynomial with: :$\ds \map p z = \sum_{j \mathop = 0}^n a_j z^j$ for all $z \in \C$, for some $a_0, \ldots, a_n \in \C$. Define: :$\ds \map p y = a_0 {\mathbf 1}_A + \sum_{j \mathop = 1}^n a_j y^j$ for each $y \in A$. Let $x \in A$. Let $\map {\sigma_A} x$ and $\map {\sigma_A} {\map p x...")
- 11:10, 21 August 2023 Caliburn talk contribs created page Spectrum of Element of Banach Algebra is Non-Empty (Created page with "== Theorem == <onlyinclude> Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$. Let $x \in A$. Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$. Then $\map {\sigma_A} x \ne \O$. </onlyinclude> == Proof == Suppose first that $\struct {A, \norm {\, \cdot \,} }$ is a unital banach algebra. Let $\map {\rho_A} x$ be the Definition:Reso...")