# Tagged Questions

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### How do they call the topological tensor product that classifies operators from Hilbert space?

Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the ...
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### operator onto-theorem

I have this theorem: Let $V$ a Banach space, reflexive,separable, and let $A$ an operator monotonic, bounded, semi-continuos, coercive. Then, $A$ is onto. Where we can find the proof of this ...
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### Reference request for nonlinear functional analysis notes.

I'm currently trying to read a paper on Fixed Points of Asymptotic Contractions" by W.A. Kirk. A small excerpt can be seen here. Those with accounts on the Elsevier page can see the whole content. ...
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### Summability of Fourier series from Banach space point of view

I am under the impression the following is true (any pointer to a reference would be appreciated ): Theorem (Katznelson?) For any $f \in C[0,1]$ with Fourier coefficients $\{ \hat{f}(n)\}$, there ...
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### Patching up basis for $L^2(\mathbb{R}^n)$

Given an orthogonal basis for $L^2(I)$ where $I\subset\mathbb{R}^n$ is the unit cube, can we construct an orthogonal basis for $L^2(\mathbb{R}^n)$ by translations/dilations etc.? Any reference to ...
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### Reference on $\mathcal{L}^p(I;X)$

I am doing some reading on evolution equations, and $\mathcal{L}^p$ spaces with functions with values in a Banach space $X$ appears rather often. However I have not found a comprehensive reference ...
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### Reproducing kernel Hilbert sapce

I encountered the following claim (verbatim): Theorem Let $V$ be a subspace of $L^2(\mathbb{R})$ and $\{e_n\}$ be a orthonormal basis of $V$. The $V$ is a reproducing kernel Hilbert space with kernel ...
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### $H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required

Let $\Omega$ be a bounded domain and define $V=L^2(\Omega)$ and $H=H^{-1}(\Omega)$. Endow $H$ with the inner product $$(f,g)_{H} = \langle f, (-\Delta)^{-1}g \rangle_{H^{-1}, H^1}$$ where ...
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### Popular Topics in mathematical analysis(Functional analysis)

I am writing a text(as a duty by my mentor) dealing with the recently popular topics(including open problems) in mathematical analysis. At first part, I briefly introduced the mathematical ...
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### operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
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### Normal operators spectral theory

Can anyone guide me to a good resource for proving the spectral theory for normal operators and proving they admit invariant subspaces. When I google it, either it is just the finite dimensional case ...
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### Weak periodic solution of parabolic PDE

Take $$u_t(t) + A(t)u(t) = f(t),$$ $$u(0) = u(T),$$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
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### $L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?
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### Can the “inducing” vector norm be deduced or “recovered” from an induced norm?

Can the "inducing" vector norm be deduced or "recovered" from an induced (operator) norm? This question occurred to me after seeing this question. I'm hoping that perhaps there exists something like ...
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### Functional analysis problem involving maximal ideal space

For the past week, I've been trying to solve, as a practice homework exercise, Problem 6 of Chapter 11 in Rudin's Functional Analysis, but have not gotten very far it seems. The problem is as follows: ...
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### $K$ and $L$ homeomorphic, then $C(K)$ is isomorphic to $C(L)$

Can someone sketch the proof (or give me some reference) of the following fact : If $K$ and $L$, compact and Hausdorff spaces, are homeomorphic then the lattices $C(K)$ and $C(L)$ are isomorphic. (I ...
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### Textbook for functional analysis in the style of Amann/Escher

most textbooks I've seen so far are not concise enough for my taste and try to give way too much motivation. Or they're written with a too large focus on applications... Rudin wasn't bad contentwise, ...
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### What is Newton's theorem?

I'm reading a paper about mathematical physics at the moment and am wondering about the following: Let $w\colon\mathbb{R}^2\to\mathbb{R}$ be defined by $w(x)=-\log|x|$ and ...
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### Tempered fundamental solutions

According to the Malgrangeâ€“Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a ...
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### Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...
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### Projection on intersection as strong limit

For orthogonal projections $P,Q$ on some Hilbert space $H$ let $P\wedge Q$ denote the orthogonal projection on $\text{Im }P\cap\text{Im }Q$. Clearly if $P,Q$ commute $P\wedge Q$ is given by $PQ$. I ...
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### Spectral theorem for $n$-tuples of selfadjoint operators

I need a 'good' reference to the following version of the Spectral Theorem: Given $n$ commuting selfadjoint operators on an infinite-dimensional Hilbert space, there exist a Borel measure $\mu$ on ...
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### Does distributional convergence imply weak convergence

let $g_k,g\in H^1(\Omega)$ (bounded domain) be given, with $g_k\to g$ in $L^2(\Omega)$. Unfortunately, I don't know whether the $g_k$ are uniformly bounded in $H^1$. I want to show that ...
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### Any suggestions about good Analysis Textbooks that cover the following topics?

I am an undergraduate math major student. I took two courses in Advanced Calculus (Real Analysis): one in Single variable Analysis, and the second in Multivariable Analysis. We basically used Rudin's ...
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### References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
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### Reference request: Graduate Algebra book for self study

I have had little exposure to algebra during my undergraduate degree, covering essentially only the basics of group theory with an emphasis on the symmetries of Euclidean space, and a course on Galois ...
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### Existence of solutions to linear evolution equation with a noncoercive operator

Consider the Gelfand triple $V\hookrightarrow H \hookrightarrow V'$ and, for given $T>0$, the Sobolev-Bochner space $$\mathcal W(0,T) := \{ v \in L^2(0,T;V): \dot v \in L^2(0,T;V')\}.$$ Consider ...
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### Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
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### Frechet derivative of compact operator is compact

... this seems to be a well known fact as mentioned in this and in this manuscript. However, I was not able to find a proof or to prove it by myself. So my question is: How to prove this? Any hint ...
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### Comprehensive references on partial differential equations

How do the three volumes by Taylor's "Partial differential equations" compare with the two volumes with the same title by Friedrich Sauvigny's as a reference for study? What are the good and bad ...
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### Question about SOT and compact operators

I need some help with functional analysis / Hilbert space theory. If you have a favorite text to recommend, please let me know~ Here is my question: Given $v_t$ be the "squeeze operator" on ...
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### Do monotone operators have positive Frechet derivatives?

If a scalar function $f\colon \mathbb R \to \mathbb R$ is monotone and differentiable, then $f'\geq 0$. Monotonicity is generalized for an operator $A\colon V \to V^*$, where $V$ is a Banach spaces ...
### $C^1$-extension of function on a normal doamin
Let $f(x,t)$ be defined on the set $N:=\{(x,t): x\in(a,b), 0\leq t \leq g(x)\}$ where $g(x)\in C^1([a,b])$, $g>0$ and $f(\in C^1(\bar N))$. Is it possible to extend $f$ smoothly on the set ...