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

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2
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1answer
62 views

compute the norm of a compact operator on $l^2$

Let $a_j\to 0$ and let $T:l^2 \to l^2$ be the operator defined by $ T(s_1,s_2,s_3,...)=(0,a_1s_1,a_2s_2,...)$. Compute the operator norm $||T||$. The hint of the problem is prove that $T$ is a ...
2
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1answer
117 views

If E is a Hilbert space and $T \in B(E)$ is compact, show $T(E)$ does not contain a closed infinite dimensional subspace

It's the problem from "Essential Results of Functional Analysis," R.J. Zimmer, Chapter 3, problem 3.1. I try to prove this problem and I am confused with the condition "closed infinite dimensional." ...
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0answers
65 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
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0answers
181 views

Spectral theorem and projection

This should be simple, but I'm stuck. Let $A$ be an unbounded self-adjoint operator on a Hilbert space $H$. The spectral theorem says that there is a decomposition of $H$ into a direct sum for which ...
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2answers
215 views

prove that this operator is not compact

Let $g\in C[0,1]$ be a continuous function and $g\ne 0$. Let $G:C[0,1]\to C[0,1]$ the operator defined by: $G(f)(x)=f(x)g(x)$. I proved that the operator is linear and continuous. I want to prove that ...
3
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1answer
92 views

Stone's theorem for 1-parameter groups of unitary multipliers?

Let $A$ be a nonunital C*-algebra and let $M(A)$ denote its multiplier algebra. Let $(u_t)_{t \in \mathbb{R}}$ be a strictly continuous 1-parameter group of unitary multipliers. That is, $u_t x \to x$ ...
2
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1answer
45 views

About norm and duality in $\mathcal S(\mathbb R^n)$

Reading the book "Classical and multilinear harmonic analysis, Vol. 1" by Muscalu, Schlag, 2013; I have a problem understanding the first step of the proof of Lemma 11.3. The relevant parts are: ...
2
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1answer
71 views

$\lambda_k \to 0$ implies $T$ is compact?

I am doing an exercise which asks to show that if $\{\varphi_k\}$ is an orthonormal basis in a Hilbert space with $T$ a bounded operator satisfying $T\varphi_k = \lambda_k \varphi_k$, then $\lambda_k ...
3
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2answers
164 views

A dense subset of a Hilbert space

I am curious about the following problem: Consider the Hilbert space (a weighted $L^2(\mathbb{R})$ space): $$\mathscr{H}=\bigg\{f: \mathbb{R}\to\mathbb{R}\text{ Lebesgue ...
4
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1answer
266 views

Is $L^2(\Omega)$ dense in $H^{-1}(\Omega)$?

Is it true that $L^2(\Omega)$, identified with its own dual, is dense in $H^{-1}(\Omega)$? $H^{-1}(\Omega)$ is the dual of $H^1_0(\Omega)$ and $H^1_0(\Omega)$ is the $H^1$-closure of smooth functions ...
3
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0answers
119 views

An orthonormal basis for a Hilbert space

Can anyone give me some hint on the following problem without using any knowledge about complex analysis or Fourier analysis? Thanks a lot! Consider the Hilbert space $$\mathscr{H}:=\bigg\{f\text{ ...
4
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1answer
233 views

Are two hilbert spaces with the same algebraic dimension (their hamel bases have the same cardinality) isomorphic?

We know that two hilbert spaces tat have orthonormal bases of the same cardinality are isomorphic (as an inner product spaces). my question is what can we say when we know that their hamel bases ...
4
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1answer
156 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...
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0answers
23 views

inequality related to transformations and inner products

Let $T$ be a bounded transformation from a hilbert space to itself. Suppose that if $||f||\leq 1$ and $||g||<1$ then $|\text{Re}(Tf,g)|\leq M$ where we are taking the real part of the inner ...
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0answers
49 views

Square-integrable representations of noncompact groups

In Marc Rieffel's paper "Square-integrable Representations of Hilbert Algebras," he establishes (Corollary 5.12) that a nonfinite discrete group has no square-integrable, irreducible representations. ...
0
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1answer
57 views

From a Hilbert space to another Hilber space, is a norm preserving invertible transformation also unitary?

Given two Hilber spaces - $H_1, H_2$ and a transformation $T:H_1 \to H_2$ that is norm preserving and invertable, does this imply that $T$ is also unitary transformation, namely that it preserves the ...
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3answers
256 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
5
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1answer
481 views

The sup norm on $C[0,1]$ is not equivalent to another one, induced by some inner product

Let $\mathrm{C}[0,1]$ be the space of continuous functions $[0,1]\rightarrow \mathbb{R}$ endowed with the norm $||x||_{\infty}=\mathrm{max}_{t\in [0,1]}|x(t)|$. It is easy to verify that this norm is ...
0
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1answer
49 views

How to show that the limit of sequence of eigenvectors (same eigenvalue) is also an eigenvector?

Let $H$ be a continuous Hermitian operator on an infinite dimensional Hilbert space. Also, let $f_n$ be a sequence approaching $f$ as $n\to\infty$, where each $f_n$ is an eigenvector of the same ...
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1answer
70 views

inf sup duality in Hilbert spaces

Let $Y$ be a Hilbert space, for all $y \in Y$ and $X$ a closed subspace of $Y$, I want to prove the following duality result: $$\inf_{g \in X} || y -g|| = \sup_{(f,X)=0} \frac{(y,f)}{||f||},$$ where ...
3
votes
1answer
74 views

Orthonormal bases for Hilbert spaces

In Reed and Simon (Functional Analysis) Theorem II.6 states that, given an orthonormal basis $\{ x_\alpha \}_{\alpha \in A}$ (not necessarily countable)for a Hilbert space $H$, every $y \in H$ can be ...
0
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2answers
218 views

Hilbert Spaces are Reflexive

I want to show that all Hilbert spaces are reflexive. I have found the following proof on StackExchange: Hilbert Space is reflexive However, I do not understand it. Essentially, we want to show ...
3
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1answer
1k 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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4answers
115 views

Why is the inner product not an element of the Hilbert space?

What I know about Hilbert space is that, elements in that space can be complex numbers. But I was confused to read this statement from a book: The inner product, being a complex number, is not an ...
2
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1answer
282 views

Adjoint of resolvent of self-adjoint, densely-defined operator on a Hilbert space

Let $H$ be a Hilbert space, $T=T^*$ a densely-defined linear operator on $H$. Denote the resolvent set of $T$ as $\rho(T)=\{\lambda\in\mathbb{C}~|~T-\lambda$ has bounded, everywhere-defined inverse}, ...
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1answer
465 views

Non-commuting projection operators on a Hilbert space

Let $H$ be a separable Hilbert space. Can you provide an example of 3 orthogonal projection operators which are mutually non-commuting?
5
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1answer
697 views

The Hahn-Banach theorem for Hilbert spaces follows from Riesz's theorem

How does the Hahn-Banach theorem for Hilbert spaces follow from Riesz's representation theorem?
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0answers
58 views

Is this operator closed?

Consider the linear operator $H$ with domain $D(H) = S(\mathbb R)\subset L^2(\mathbb R)$, where $S(\mathbb R)$ is Schwartz space, defined by \begin{align} H\psi(x) = -ix^3\frac{d\psi}{dx}(x) -i ...
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2answers
93 views

Orthogonal completion in nonhilbert spaces [duplicate]

Let $X$ be some Hilbert space. There is the widely known theorem in functional analysis which states that for each closed subspace $H\subset X$ we have $H\bigoplus H^{\perp}=X$. Now we do not suppose ...
0
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1answer
74 views

Is the separable infinite-dimensional Hilbert space over $\Bbb C$ a Lie group?

Does the infinite-dimensional separable Hilbert space over $\Bbb C$ form a Lie group? It is a Banach space, that is, a complete normed space. Could you please guide me to answer this question?
4
votes
1answer
142 views

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 ...
2
votes
1answer
139 views

How to show that time-dependent norm is continuous (please verify my proof)

For each $t \in [0,T]$, let $H_t$ be a Hilbert space. Suppose for each $t$, the operator $T_t:H_0 \to H_t$ is a linear homeomorphism with inverse $T_{-t}:H_t \to H_0$ also linear homeomorphism. ...
2
votes
1answer
91 views

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 ...
0
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1answer
388 views

Example of non-orthogonal projection on Hilbert space

Can anybody cook up an example of a projection operator $P$ on a Hilbert space $H$ that is non-orthogonal? I.e., one where $PH$ and $(1-P)H$ are not orthogonal subspaces of $H$. I'm completely ...
0
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2answers
116 views

Subspace of $C^1 [0,1]$

Consider the inner product space of continuously differentiable functions, $C^1 [0,1]$ with inner product:$$\left<f,g\right> =\int_{0}^1f(x)\overline{g(x)}\,dx + ...
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1answer
121 views

What is a concrete example of a non-compact Hermitian operator on an infinite-dimensional Hilbert space whose eigenvectors do not form a complete set?

If I am not misunderstanding anything: by the spectral theorem, Hermitian operators that act upon finite-dimensional Hilbert space as well as compact Hermitian operators that act upon ...
3
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2answers
148 views

exercise on the closed subspaces of an Hilbert spaces

I have a question regarding exercise 3.1.13 of "Analysis Now" by Pedersen volume 118 of the Springer GTM. The exercise aim to show that any closed subspace $X$ of $L^2([0,1])\cap L^{\infty}([0,1])]$ ...
4
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1answer
161 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
2
votes
2answers
303 views

Uncountable orthonormal system in Hilbert spaces

I need an example of a Hilbert space in which the following does not hold for all $x$: $$ x=\sum_k^{\infty} \langle x,u_k \rangle u_k. $$ That is, there are elements that are not expressible as ...
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1answer
61 views

Properties of ON-basis in Hilbert space

Let $H$ be a Hilbert space with an ON-basis $(e_n)_{n=1}^\infty$ and let $A$ be a bounded linear mapping $A:H\to H$ such that $$\sum_{n=1}^\infty\|A(e_n)\|^2<\infty$$ 1: Show that if ...
3
votes
3answers
89 views

What am I doing wrong? inner product

The general form of an inner product in $\mathbb{C}^n$ is $\langle x,y\rangle=y^{*}Bx$ where B is a Hermitian positive definite matrix. Then for any square matrix $A$ we have $\langle ...
2
votes
1answer
97 views

Bounded linear mappings in Hilbert space preserve orthogonality?

My question is the title of this thread! Assume we have a bounded, linear mapping $A:H\to H$ where $(H,\langle\cdot,\cdot\rangle)$ is a Hilbert space, and two non-zero elements that are orthogonal, ...
4
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0answers
125 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
0
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1answer
63 views

Proving that $T(B(x,2\epsilon))\cap B(y,2\epsilon) \neq \emptyset $

$H$ Hilbert space. $x,y \in H$ and $T\in L(H)$ 1) $T(B(x,\epsilon))\cap B(0,\epsilon) \neq \emptyset $ 2) $T(B(0,\epsilon))\cap B(y,\epsilon) \neq \emptyset $ 3) $T(B(x,2\epsilon))\cap ...
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1answer
43 views

Show that ON-sequence is a base

I have a Hilbert space $H$ and a base $(e_n)_{n=1}^\infty$ and a ON-sequence $(f_n)_{n=1}^\infty$. Given $$ \sum_{n=1}^\infty ||e_n - f_n||^2 < 1 $$ show that $(f_n)_{n=1}^\infty$ is a base. My ...
0
votes
1answer
39 views

$H$ Hilbert, $\ker L \neq H \Rightarrow (\ker L )^{\perp} \neq \lbrace 0 \rbrace$

If $H$ is a Hilbert space on $\mathbb{C}, L : H \rightarrow \mathbb{C} $ is linear and bounded, $\ker L \neq H $ then $ (\ker L )^{\perp} \neq \lbrace 0 \rbrace.$ It seems like a quite easy ...
0
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1answer
46 views

Bound of $\displaystyle ||\ \cdot\ ||_{H_{1}(\Omega)^{n}}$ with $\displaystyle ||\ \cdot\ ||_{H(div)}$

I am reading Brenner and Scott's book on finite elements, chapter 12 at the moment. I have come up to something that seems simple, but that I am having trouble figuring out by myself to do with the ...
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1answer
173 views

Derivative of inner product (taking limit inside)

For each $x \in [a,b]$ let $A_x: H \to H$ be an operator on a Hilbert space. The inner product $(A_xu,v)_{H}$ can be thought of as a function from $[a,b] \to \mathbb{R}.$ I want to say that ...
0
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1answer
64 views

Completions of a vector space with inner product

Assume $(H,(\cdot,\cdot)_H)$, and $(G,(\cdot,\cdot)_G)$ are two vector spaces with inner product. Suppose $A:H\rightarrow G$ is a linear isometry. Let $T(H)_{*}$ be the completion of $T(H)$ with ...
3
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1answer
70 views

Hilbert space - probability measure: st. norm. variables

I am considering the following homework. Let $\Omega=\ell_2$ be the Hilbert space of square summable sequences, $\mathcal A$ the Borel $\sigma$-algebra and $\{e_n: n\mbox{ natural}\}$ the natural ...