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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).
- 21:10, 16 October 2023 Julius talk contribs created page Set of Invertible Continuous Transformations is Open Subset of Continuous Linear Transformations in Supremum Operator Norm Topology (Created page with "== Theorem == Let $X$ be a Banach space. Let $\map {CL} X$ be the continuous linear operator space on $X$. Let $\map {GL} X$ denote the set of all invertible continuous linear operators on $X$. Then $\map {GL} X \subseteq \map {CL} X$ in the supremum operator norm topology....")
- 11:33, 17 July 2023 Julius talk contribs created page Properties of Product of Identity plus Operator Raised to Powers of 2 (Created page with "{{WIP|links, extra steps, notation etc.}} == Theorem == Let $X$ be a Banach space. Let $\map {CL} X$ be a continuous linear transformation sapce. Let $\norm {\, \cdot \,}$ be the supremum operator norm. Let $A \in \map {CL} X$ be such that $\norm A < 1$. Let $I$ be the identity mapping. Let $\circ$ be the Definiti...")
- 23:50, 14 July 2023 Julius talk contribs created page Definition:Differential of Mapping/Manifolds (Created page with "== Definition == {{WIP|Polishing}} Let $M$ and $N$ be smooth manifolds with or without boundary. Let $F : M \to N$ be a smooth map. Let $T_p M$ be the tangent space at $p \in M$. Then the '''differential of $F$''', denoted by $d F$, is the mapping $d F_p : T_p M \to T_{\map F p} N$ such that: :$\forall f \in \map {C^\infty} N : \forall p \in...")
- 12:35, 6 July 2023 Julius talk contribs created page System of Linear Equations as Continuous Linear Transformation (Created page with "== Theorem == Let $x_1, x_2 \in \R$ be real numbers. Consider the following system of simultaneous linear equations $(S)$: {{begin-eqn}} {{eqn | l = x_1 | r = \frac 1 2 x_1 + \frac 1 3 x_2 + 1 }} {{eqn | l = x_2 | r = \frac 1 3 x_1 + \frac 1 4 x_2 + 2 }} {{end-eqn}} Let $\norm {\, \cdot \,}_2$ be the $2$-norm. Let $\norm {\, \cdot \,}$ be the Definition:Suprem...")
- 11:52, 6 July 2023 Julius talk contribs created page Category:Definitions/Neumann Series (Created page with "{{DefinitionCategory|Neumann Series}}")
- 11:50, 6 July 2023 Julius talk contribs created page Category:Neumann Series (Created page with "{{SubjectCategory}}")
- 11:47, 6 July 2023 Julius talk contribs deleted page Conditions for Identity Minus Operator to be Bijective (content was: "== Theorem == Let $X$ be a Banach space. Let $\map {CL} X$ be the continous linear transformation space. Let $\norm {\, \cdot \,}$ be the supremum operator norm. Let $A \in \map {CL} X$ be such that $\norm A < 1$. Then the mapping $I...", and the only contributor was "Julius" (talk))
- 11:44, 6 July 2023 Julius talk contribs created page Neumann Series Theorem/Corollary 2 (Created page with "== Theorem == Let $X$ be a Banach space. Let $\map {CL} X$ be the continous linear transformation space. Let $\norm {\, \cdot \,}$ be the supremum operator norm. Let $A \in \map {CL} X$ be such that $\norm A < 1$. <onlyinclude> The mapping $\paren {I - A}^{-1} : X \to X$ is continuous. </only...")
- 11:36, 6 July 2023 Julius talk contribs created page Neumann Series Theorem/Corollary 1 (Created page with "== Theorem == Let $X$ be a Banach space. Let $\map {CL} X$ be the continous linear transformation space. Let $\norm {\, \cdot \,}$ be the supremum operator norm. Let $A \in \map {CL} X$ be such that $\norm A < 1$. <onlyinclude> Then the mapping $I - A : X \to X$ is bijective. </onlyinclude> == Proof...")
- 11:29, 6 July 2023 Julius talk contribs created page Conditions for Identity Minus Operator to be Bijective (Created page with "== Theorem == Let $X$ be a Banach space. Let $\map {CL} X$ be the continous linear transformation space. Let $\norm {\, \cdot \,}$ be the supremum operator norm. Let $A \in \map {CL} X$ be such that $\norm A < 1$. Then the mapping $I - A : X \to X$ is bijective. == Proof == {{ProofWanted}} == So...")
- 12:15, 23 June 2023 Julius talk contribs created page Definition:Neumann Series (Created page with "== Definition == Let $T$ be an operator. Let $\circ$ denote the composition. Let $I$ be the identity mapping. For any $k \in \N$ let $T^k = \underbrace{T \circ T \circ \ldots \circ T \circ T}_{k \text{ times} }$ and $T^0 = I$ Then the series $\ds \sum_{k \mathop = 0}^\infty T^k$ are known as '''Neumann series'''. == Sources == * {{BookReference|A...")
- 12:01, 23 June 2023 Julius talk contribs created page Neumann Series Theorem (Created page with "== Theorem == Let $X$ be a Banach space. Let $\map {CL} X$ be the continous linear transformation space. Let $\norm {\, \cdot \,}$ be the supremum operator norm. Let $A \in \map {CL} X$ be such that $\norm A < 1$. Let $\circ$ be the composition of mappings. Then: :$I - A$ is Definition:Invertible Continuo...")
- 14:21, 5 June 2023 Julius talk contribs deleted page Cauchy-Schwarz Inequality/Continuous Linear Transformation Space with Supremum Operator Norm (Saved under a different name)
- 14:16, 5 June 2023 Julius talk contribs created page Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative (Created page with "== Theorem == Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$, $\struct {Z, \norm {\, \cdot \,}_Z}$ be normed vector spaces. Let $A : Y \to Z$ and $B : X \to Y$ be continuous linear transformations. Let $\norm {\, \cdot \,}$ be the supremum operator norm. Let $\circ$ denote the Definition:Composition of Mappings|...")
- 13:03, 2 June 2023 Julius talk contribs created page Cauchy-Schwarz Inequality/Continuous Linear Transformation Space with Supremum Operator Norm (Created page with "== Theorem == <onlyinclude> Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$, $\struct {Z, \norm {\, \cdot \,}_Z}$ be normed vector spaces. Let $A : Y \to Z$ and $B : X \to Y$ be continuous linear transformations. Let $\norm {\, \cdot \,}$ be the supremum operator norm. Let $\circ$ denote the Definition:Composition...")
- 12:10, 2 June 2023 Julius talk contribs created page Talk:Conformality is Equivalence Relation on Set of Riemannian Metrics (Created page with "== Notation == The set $\set {g}$ is supposed to denote a collection of all Riemannian metrics which are admissible to $M$. The precise condtions come from the topology and differentiable structure of $M$. Then we are supposed to pick a subset of these metrics which in addition are related by conformal transformations. Then the conformal transformation as a relation on this subset of metrics is an equivalence relation. Maybe what we need is a set of all Riemannian metri...")
- 21:44, 26 May 2023 Julius talk contribs created page Definition talk:Metric Compatible Connection (Created page with "==Mistake== What happened here is that I left out the notion of general covariant derivative (for personal reasons, I only had time for most basic notions). The current definition at hand is defined only for vector fields. However, this covariant derivative induces a covariant derivative of arbitrary tensor fields including scalars. And for scalars it reduces to exactly what you wrote. So what we really have to do here is to created a theorem page for existence of such...")
- 10:34, 25 May 2023 Julius talk contribs created page Definition talk:Minkowski Spacetime (Created page with "== Equivalence == Clearly equivalence here is problematic, because we have geometer's Minkowski spacetime (any dimension) and physicist's Minkowski spacetime (3+1 dimensions). As always, mathematicians stole the physical notation and generalized to higher dimensions and stripped its material aspects. Also, for physicists, the meaning of time and space are not just geometrical, i.e. direction of time imposes a causal structure of events. For geometers, Minkowski spacetim...")
- 10:14, 25 May 2023 Julius talk contribs created page Talk:P-Norm of Real Sequence is Strictly Decreasing Function of P (Created page with "== Inequality == The inequality simply compares a sum of non-negative numbers with any single number of that sum. Should we add something else?--~~~~")
- 13:11, 10 May 2023 Julius talk contribs created page Definition talk:Dot Product (Created page with "== Equivalence == Please bring me up to the topic. Which way is it? Does Equivalence of Definitions of Dot Product show the equivalence or not?--~~~~")
- 09:40, 9 May 2023 Julius talk contribs created page Talk:Invertibility of Identity Minus Operator (Created page with "== Various questions == I was about to type out the proof for Neumann series theorem when I noticed this page. I have not taken it apart yet, but I expect it essentially to be the proof I found elsewhere. Ideally, I would like to adjust it minimally by adding a few extra steps here and there. I have not read about bounded linear operators (maybe you could provide a reference for this), but I wonder if they are equivalent to continuous linear transformations (the bounded...")
- 12:16, 8 May 2023 Julius talk contribs created page Existence of Transformations whose Commutator equals Identity (Created page with "== Theorem == Let $\map {C^\infty} \R$ be the space of smooth real functions. Let $\circ$ denote the composition of mappings. Let $A : \map {C^\infty} \R \to \map {C^\infty} \R$ be the mapping such that: :$\forall \phi \in \map {C^\infty} \R : \forall x \in \R : \map {\paren{ A \circ \phi } } x := \map {\dfrac {d \phi} {d x} } x$ Let $B : \map {C^\infty} \R \t...")
- 11:22, 8 May 2023 Julius talk contribs deleted page Nonexistence of Continuous Linear Transformations whose Commutator equals Identity (content was: "#REDIRECT Nonexistence of Continuous Linear Transformations over Finite Dimensional Vector Space whose Commutator equals Identity", and the only contributor was "Julius" (talk))
- 11:21, 8 May 2023 Julius talk contribs moved page Nonexistence of Continuous Linear Transformations whose Commutator equals Identity to Nonexistence of Continuous Linear Transformations over Finite Dimensional Vector Space whose Commutator equals Identity
- 16:28, 4 May 2023 Julius talk contribs created page Nonexistence of Continuous Linear Transformations whose Commutator equals Identity (Created page with "== Theorem == Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be the space of continuous linear transformations. Let $A, B \in \map {CL} {X, X}$ be continuous linear transformations. Let $I$ be the Definition:Identit...")
- 22:57, 27 March 2023 Julius talk contribs created page Nonexistence of Complex Matrices whose Commutator equals Identity (Created page with "== Theorem == Let $d \in \N_{> 0}$ be a positive natural number. Let $\mathbf A, \mathbf B \in \C^{d \times d}$ be complex matrices. Let $\mathbf I$ be the $d \times d$ identity matrix. Then there is no $\mathbf A$, $\mathbf B$ such that $\mathbf A \mathbf B - \mathbf B \mathbf A = \mathbf I$. == Proof == {{AimForCont}} there are $\mathbf A$, $\mat...")
- 23:42, 22 March 2023 Julius talk contribs deleted page Invertibility of Unit Transformation Plus Product of Two Continuous Linear Transformations (content was: "#REDIRECT Invertibility of Identity Transformation Plus Product of Two Continuous Linear Transformations", and the only contributor was "Julius" (talk))
- 23:41, 22 March 2023 Julius talk contribs moved page Invertibility of Unit Transformation Plus Product of Two Continuous Linear Transformations to Invertibility of Identity Transformation Plus Product of Two Continuous Linear Transformations
- 23:41, 22 March 2023 Julius talk contribs created page Invertibility of Unit Transformation Plus Product of Two Continuous Linear Transformations (Created page with "== Theorem == Let $\struct {X, \norm{, \cdot ,} }$ be the normed vector space. Let $I : X \to X$ be the identity mapping. Let $\map {CL} X := \map {CL} {X, X}$ be the continuous linear transformation space on $X$. Suppose $I + A \circ B$ is invertible, where $\circ$ denotes the Definition:Compo...")
- 18:13, 15 March 2023 Julius talk contribs created page Talk:UFD is GCD Domain (Created page with "== Call for improvement == I just found the following comment on stack exchange: [https://math.stackexchange.com/questions/4655615/ufd-implies-gcd]. Should we do anthing? Is anyone here capable of improving this page?--~~~~")
- 09:04, 15 March 2023 Julius talk contribs created page Definition talk:Analytic Function/Banach Space Valued Function (Created page with "== Something missing? == The Banach space $\struct {X, \norm{\, \cdot \,}}$ is only introduced, but never referred to. What's it's purpose here? $E$ is technically also undefined, but from the context it should belong to $\C$, right?--~~~~")
- 01:33, 15 March 2023 Julius talk contribs deleted page Condition for Diagonal Operator to be Invertible (content was: "#REDIRECT Necessary and Sufficient Condition for Diagonal Operator to be Invertible", and the only contributor was "Julius" (talk))
- 01:33, 15 March 2023 Julius talk contribs moved page Condition for Diagonal Operator to be Invertible to Necessary and Sufficient Condition for Diagonal Operator to be Invertible
- 01:32, 15 March 2023 Julius talk contribs created page Condition for Diagonal Operator to be Invertible (Created page with "== Theorem == Let $\mathbb F \in \set {\R, \C}$. Let $\sequence {\lambda_n}_{n \mathop \in \N_{> 0} }$ be a bounded sequence in $\mathbb F$. Let $\ell^2$ be the $2$-sequence space. Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space on $\ell^2$. Let $\Lambda \in \map {CL} {\ell^2}$ be the Def...")
- 20:57, 12 March 2023 Julius talk contribs created page Talk:Normed Dual Space is Banach Space (Created page with "== Theorem formulation == It feels to me that not all "let" constructions have the same meaning. Is it your choice to present both inputs and outputs of the theorem together with auxiliary structures in the theorem statment? I usually leave the theorem only for inputs and outputs, and introduce auxiliary structures in the proof. I wonder if this is what I see here?--~~~~")
- 18:52, 5 March 2023 Julius talk contribs moved page Explicit Form of Supremum Operator Norm of Elementwise Multiplication of 2-Sequence by Bounded Sequence to Supremum Operator Norm of Diagonal Operator over 2-Sequence Space (More standard and precise terminology)
- 18:25, 5 March 2023 Julius talk contribs moved page Elementwise Multiplication of 2-Sequence by Bounded Sequence is Continuous Linear Transformation to Diagonal Operator over 2-Sequence Space is Continuous Linear Transformation (More standard and precise terminology)
- 00:40, 3 March 2023 Julius talk contribs created page Definition:Diagonal Operator (Created page with "== Definition == Let $\struct {X, \norm {\, \cdot \,}_X}$ be the normed vector space. Let $\dim X$ be the dimension of $X$. Let $\set {e_i}_{1 \mathop \le i \mathop \le \dim X}$ be the basis of $X$. Let $\Lambda : X \to X$ be a mapping such that: :$\forall i \in \N_{> 0} : i \le \dim X : \exists \lambda_i \in \C : \map...")
- 00:18, 1 March 2023 Julius talk contribs created page Bijective Continuous Linear Operator is not necessarily Invertible (Created page with "{{WIP|Cleanup in process}} == Theorem == Let $\struct {X, \norm {\, \cdot \,} }$ be the normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $I \in \map {CL} X$ be the identity element. Suppose $A \in \map {CL} X$ is bijective. Then $A$ is not Def...")
- 00:25, 28 February 2023 Julius talk contribs created page Invertible Continuous Linear Operator is Bijective (Created page with "== Theorem == Let $\struct {X, \norm {\, \cdot \,} }$ be the normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $I \in \map {CL} X$ be the identity element. Suppose $A \in \map {CL} X$ is invertible. Then $A$ is Definition:Bijection|bij...")
- 00:01, 28 February 2023 Julius talk contribs created page Definition:Inverse of Continuous Linear Operator (Created page with "== Definition == Let $\struct {X, \norm {\, \cdot \,} }$ be the normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $I \in \map {CL} X$ be the identity element. Suppose $A \in \map {CL} X$ is invertible. Then the Definition:Unique|unique...")
- 15:05, 25 February 2023 Julius talk contribs created page Invertible Continuous Linear Operator has Unique Inverse (Created page with "== Theorem == Let $\struct {X, \norm {\, \cdot \,} }$ be the normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $I \in \map {CL} X$ be the identity element. Suppose $A \in \map {CL} X$ is invertible. Then: :$\exists ! B \in \map {CL} X :...")
- 14:51, 25 February 2023 Julius talk contribs created page Continuous Linear Operator over Infinite Dimensional Vector Space is not necessarily Invertible (Created page with "== Theorem == Let $\struct {X, \norm {\, \cdot\,}_X}$ be a normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $I \in \map {CL} X$ be the identity element. Let $S, T \in \map {CL} X$. Suppose the dimension of $X$ is [[Definition:Finite Cardinal|finite]...")
- 23:55, 22 February 2023 Julius talk contribs created page Continuous Linear Operator over Finite Dimensional Vector Space is Invertible (Created page with "== Theorem == Let $\struct {X, \norm {\, \cdot\,}_X}$ be a normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $I \in \map {CL} X$ be the identity element. Let $S, T \in \map {CL} X$. Suppose the dimension of $X$ is [[Definition:Finite Cardinal|finite]...")
- 00:09, 22 February 2023 Julius talk contribs created page Definition:Invertible Continuous Linear Operator (Created page with "== Definition == Let $\struct {X, \norm {\, \cdot\,}_X}$ be a normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $I \in \map {CL} X$ be the identity element. Let $A \in \map {CL} X$. Suppose: :$\exists B \in \map {CL} X : A \circ B = B \circ A = I$ where $\circ$ denotes the Definition:Co...")
- 23:53, 21 February 2023 Julius talk contribs created page Continuous Linear Transformation Space as Banach Algebra (Created page with "== Theorem == Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $*_X : X \times X \to X$ and $* : \map {CL} X \times \map {CL} X \to \map {CL} X$ be bilinear mappings. Suppose $\struct {\struct {X, \norm {\, \cdot \,}_X}, *_X}$ is a Definition:...")
- 11:05, 21 February 2023 Julius talk contribs created page Definition talk:Equation of Wave Motion (Created page with "== Comment on nomenclature == I am a physicist by training, and from my experience the "wave equation" is usually reserved for what here is called "equation of wave motion". On the other hand, "Schrödinger Wave Equation" is almost always called simply as the "Schrödinger Equation". One actually has to work a bit with "Schrödinger Equation" to make it look like a "Wave equation", because a superficial inspection suggests that it is actually a diffusion equation. Only...")
- 00:57, 21 February 2023 Julius talk contribs created page Definition talk:Algebra over Ring (Created page with "== Algebra vs Algebraic Structure == If I understand correctly, eventually all such uses of "algebra" will have to be renamed as "algebraic structure". Even though there is not much action here at the moment, this is the ultimate goal, right?--~~~~")
- 00:52, 21 February 2023 Julius talk contribs created page Continuous Linear Transformation Algebra with Supremum Operator Norm is Normed Algebra (Created page with "== Theorem == Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $\struct {\map {CL} X, *}$ be an associative algebra. Let $\norm {\, \cdot \,}$ be the supremum operator norm. Then $\struct {\struct {\map...")
- 00:45, 21 February 2023 Julius talk contribs created page Continuous Linear Transformation Algebra has Two-Sided Identity (Created page with "== Theorem == Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space. Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space. Let $\struct {\map {CL} X, *}$ be an associative algebra. Then there exists an identity element $I \in \map {CL} X$ such that: :$\forall x \in X : \m...")