For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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Compact operator

If $H$ and $K$ are Hilbert spaces,show that if $T:H\longrightarrow K$ is a compact operator and $\{e_{n}\}$ is any orthonormal sequence in $H$ then $\|Te_{n}\|\to0$.Is the converse true? thanks.
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1answer
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Orthogonal projection on the Hilbert space .

I want to prove the following: If $X$ is a Hilbert space and $Y$ is a closed subspace of $X$, then every $x\in X$ can be written as $x=y+z $ where $y\in Y$, $z \in Y^\perp$. The ...
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1answer
586 views

Weak convergence, together with convergence of norms, implies strong convergence in a Hilbert space.

Let $(x_n)$ be a weakly convergent sequence in a Hilbert space $H$. If $\| x_n \| \to \| x \|$, show that $x_n$ converges strongly to $x$. Context This problem comes from a question in my exam ...
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1answer
64 views

Spectral Measures: Domain Criterion

Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Then the criterion holds: ...
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1answer
207 views

Normal operators in Hilbert spaces

Let $H$ be a separable Hilbert space and let $T:H\to H$ be a continues linear map such that there exists an orthonormal basis of $H$ that consists of the eigenvectors of $T$. Show that $T$ is normal. ...
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Summary: Spectrum vs. Numerical Range

Reference A proof of the statement below is split into: Normal Operators: Spectrum vs. Numerical Range Spectral Measures: Spectrum vs. Numerical Range Problem Given a Hilbert space ...
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1answer
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Showing a certain subspace of Hilbert space is dense

Let H be the Hilbert space of square-summable sequences of reals. A few years ago I thought I had proved that the subspace Z of real sequences with only finitely many nonzero terms, such that they ...
5
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1answer
428 views

Bounded operator and Compactness problem

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a ...
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1answer
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Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
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1answer
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Question about adjoint map and strong operator topology (SOT)

I am wondering if there is any condition one can apply (e.g. uniform boundedness?) that ensure the adjoint of a net of SOT-continuous elements is again SOT-continuous? My major question is ...
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1answer
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proof for a basis in $L^2$

I know, correct me if I am wrong, that the functions $H_n(x)\exp(-x^2/2)$ form a complete basis in $L^2(\mathbb{R},dx)$, where $H_n(x)$ is the $n$th Hermite polynomial. This must be true also for ...
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2answers
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Stone's Theorem Integral: Avanced Integral

Reference This problem grew out from: Stone's Theorem Integral: Basic Integral Problem Given the real line as measure space $\mathbb{R}$ and a Hilbert space $\mathcal{H}$. Consider a strongly ...
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1answer
158 views

Prove or disprove that the given expression is “always” positive

I have previously asked a question and I tried to solve it by my own and it led to the question below: Prove or disprove that ...
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2answers
241 views

What is the norm of the operator $L((x_n)) \equiv \sum_{n=1}^\infty \frac{x_n}{\sqrt{n(n+1)}}$ on $\ell_2$?

Let $(x_n) \subset \ell_2$ and let operator $L:\ell_2\to \mathbb R$ be defined by: $\displaystyle L((x_n)) := \sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n(n+1)}}$. Find the norm of L.
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2answers
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I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed.

Let $x_n$ be a sequence in a Hilbert space such that $\left\Vert x_n \right\Vert=1$ and $ \langle x_n,\ x_m \rangle =0 $, for all $n \neq m$. Let $ K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\} $. ...
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1answer
25 views

Spectral Measures: Multi Version (III)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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0answers
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Lemma 3.3-7 and Theorem 3.6-2 in Kreyszig's “Introductory Functional Analysis With Applications”: What if completeness is lost? [duplicate]

Let $X$ be an inner product space, and let $M$ be a non-empty subset of $X$. Then we have the following: (a) If the space of $M$ is dense in $X$, then $M^\perp = \{0 \}$, that is, $x \in X$, $x ...
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1answer
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Spectral Measures: Embedding

This thread is just a note! Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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2answers
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about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
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1answer
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Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence

Let $\{e_n\}$ be a complete orthonormal sequence in an Hilbert space $H$ and let $\{f_n\}$ be an arbitrary sequence of elements in $H$ s.t $$\sum_{n=1}^\infty\|f_n-e_n\|^2<1$$Show that ...
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2answers
147 views

Characterisation of norm convergence

Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$): We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and we have $x_n ...
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1answer
195 views

weak convergence condition

Let $l^{2}=\left\{x=(x^{(1)},x^{(2)},...):\sum_{i=1}^{\infty }\left\vert x ^{(i)}\right\vert ^{2}<\infty \right\} $. Would you help me to prove that $({\vert|x_n |\vert})$ is bounded sequence and ...
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2answers
379 views

Question about limits of weakly convergent sequence in $H^1_0(\Omega)$

Let $H = H_{0}^{1}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ whose boundary $\partial\Omega$ is a smooth manifold. We know that the embedding $$H\hookrightarrow L^s(\Omega)$$ is compact for ...
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2answers
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Stone's Theorem Integral: Basic Integral

Disclaimer This thread has been renewed: Stone's Theorem Integral: Advanced Integral Problem Given a finite Borel measure $\mu$ and a Hilbert space $\mathcal{H}$. Consider a strongly continuous ...
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1answer
140 views

Weak limits and subsequences

Let $S:X \to X$ be a (nonlinear) map between a Hilbert space $X$. I want to show that $S$ is weakly continuous, so if $x_n \rightharpoonup x$, then $S(x_n) \rightharpoonup S(x)$. To do this, I have ...
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1answer
117 views

If a sequence is not a frame

A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0<A\leq B<\infty$ such that $$ A\|f\|^{2} \leq \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq ...
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2answers
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Isometric <=> Left Inverse Adjoint

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
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1answer
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Range of $Df(a)$ contained in the subspace $\{f(a)\}^{\perp}$ with $f$ differentiable

Let $A$ a open set in a Hilbert Space $H$, suppose that $f:A\to F$ is differentiable at $a\in A$ and that $||f(x)||=c$ forall $x\in A$. Show that range of $Df(a)$ is contained in the subspace ...
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Given two basis sets for a finite Hilbert space, does an unbiased vector exist?

Let $\{A_n\}$ and $\{B_n\}$ be two bases for an $N$-dimensional Hilbert space. Does there exist a unit vector $V$ such that: $$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ ...
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1answer
460 views

Over ZF, does “every Hilbert space have a basis” imply AC?

I know there is a similar result due to Blass [1] that over ZF, "every vector space has a (Hamel) basis" implies AC. Looking around, however, I can't find any results on the question for Hilbert ...
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1answer
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Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
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An idempotent operator is compact if and only if it is of finite rank

Would you help me to solve this problem. Show that an idempotent operator on hilbert space is compact if and only if it has finite rank.
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1answer
841 views

A few questions about the Hilbert triple/Gelfand triple

I am attempting to fully understand Hilbert triples by reading Brezis' Function Analysis book. Consider $V \subset H \subset V^*$, where $V$ is Banach and $H$ is Hilbert. $V$ is dense in $H$. ...
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1answer
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orthonormal system in a Hilbert space

Let $\{e_n\}$ be an orthonormal basis for a Hilbert space $H$. Let $\{f_n\}$ be an orthonormal set in $H$ such that $\sum_{n=1}^{\infty}{\|f_n-e_n\|}<1$. How do I show that $\{f_n\}$ is also an ...
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1answer
276 views

Criteria of compactness of an operator

Suppose $K$ is a linear operator in a separable Hilbert space $H$ such that for any Hilbert basis $\{e_i\}$ of $H$ we have $\lim_{i,j \to \infty} (Ke_i,e_j) = 0$. Is it true that $K$ is compact? ...
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572 views

Isometric to Dual implies Hilbertable?

Let $X$ be a Banach space and suppose that $X$ is isometric to its continuous dual space $X^*$. Must $X$ be hilbertable in the sense that there exists an inner product which induces the norm on $X$? ...
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981 views

Is a closed set with the “unique nearest point” property convex?

A friend of mind had a question that I couldn't answer. It is well-known that if $K$ is a closed, convex subset of a Hilbert space $H$ (say over the reals) then, for any point $p \in H$, there exists ...
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1answer
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How to Prove the Semi-parametric Representer Theorem

This question concerns the generalized Representer Theorem, due to Schölkopf, Herbrich, and Smola. In this magnificent work, the authors provide two versions of the Representer Theorem, a ...
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2answers
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Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
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1answer
312 views

Two different definitions of ellipticity

This is a question originating in another mathematics forum, matematicamente.it (in Italian). In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
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$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
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Bounded operator that does not attain its norm

What is a bounded operator on a Hilbert space that does not attain its norm? An example in $L^2$ or $l^2$ would be preferred. All of the simple examples I have looked at (the identity operator, the ...
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1answer
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Bounded sequence in Hilbert space contains weak convergent subsequence

In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence. Is there any short proof? Thanks a lot.
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2answers
280 views

Counterexample for the stability of orthogonal projections

Let $V$ be a seperable Banach space, which is dense and continuously embedded in a Hilbert Space $H$. Let $(V_m)$ be a Galerkin scheme (See definition below) for $V$. Using the embedding we can ...
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0answers
429 views

Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
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1answer
472 views

Reproducing kernel Hilbert spaces and the isomorphism theorem

A reproducing kernel Hilbert space is a Hilbert space in which the evaluation functional $L_x : f \rightarrow f(x)$ is continuous. By continuity, the Riesz representation theorem says that this ...
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1answer
686 views

Proof Complex positive definite => self-adjoint

I am looking for a proof of the theorem that says: A is a complex positive definite endomorphism and therefore is A self-adjoint. Does anybody of you know how to do this?
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Tensor Product: Hilbert Spaces

This question has been modified... Problem Given Hilbert spaces. In general, their algebraic tensor product isn't complete: ...
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384 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
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0answers
141 views

Is this projection operator onto a subspace of a Hilbert space bounded?

(I copy and paste and edit from Is this operator bounded? Hilbert space projection, my question is almost the same) Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense and ...