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

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1answer
146 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
46 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. ...
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1answer
51 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
243 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
308 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 ...
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1answer
42 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
65 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
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1answer
64 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 ...
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2answers
169 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 ...
2
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1answer
591 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
110 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
228 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
367 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?
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1answer
540 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
56 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
79 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 ...
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1answer
70 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?
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1answer
125 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
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1answer
138 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. ...
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1answer
82 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 ...
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1answer
252 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 ...
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2answers
115 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
115 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 ...
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2answers
133 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 $L^2([0,1])\cap L^{\infty}([0,1])]$ ...
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1answer
129 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 ...
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2answers
206 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
50 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
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3answers
82 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 ...
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1answer
90 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, ...
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0answers
108 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 = ...
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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
41 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 ...
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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 ...
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1answer
41 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
136 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 ...
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1answer
63 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 ...
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1answer
65 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 ...
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1answer
275 views

An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...
2
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1answer
60 views

Corollary to Putnam's theorem

Suppose $T_1$ and $T_2$ are normal operators on Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively. Putnam showed that if $X$ is an operator satisfying $T_2X=XT_1$, then $T_2^*X=XT_1^*$. ...
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1answer
185 views

A problem with a proof of Bessel's inequality, and how to get Parseval's identity from it

I am studying functional analysis, and I think there is a problem with the proof written in my notes for Bessel's inequality. The theorem is: Let $H$ be a Hilbert space and ...
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1answer
115 views

What is a good definition of Hilbert space?

Motivation of my question: in my opinion, in view of the common definition, the statement "$\ell_p$ is a Hilbert space if and only if $p=2$" makes no sense because there is no inner product in the ...
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1answer
82 views

Is the fractional Laplacian on $\mathbb{R}$ of trace class or not?

Is the fractional Laplacian on $\mathbb{R}$ of trace class or not? I don't know the basis of $\mathrm{L}^{2}(\mathbb{R})$ for applying the definition of a trace class operator. Thank you very much.
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1answer
63 views

real eigenvalues for non normal operator

Is there unbounded non normal operator in Hilbert space which has only real eigenvalues? If yes, could you give me an example?
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1answer
110 views

Weighted $\ell^2$ space is Hilbert

This is my exam's question that I could not solve it. Please help me to undrestand how to solve it: let $\{w_n\}$ is a positive real numbers sequence, and let $$\ell^2(w):=\left\{\{x_n\}:x_n \in R ...
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1answer
309 views

Self adjointness of square root operator

Theorem: If $A$ is self adjoint and nonnegative, then $A$ has a unique nonnegative square root $A^{\frac{1}{2}}$. As I understand, thesis of this theorem say only about the existence of ...
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1answer
49 views

How to prove, that the ordering on positive bounded operators agrees with ordering of their ranges?

Hypothesis: Assume, that $A$ and $B$ are positive bounded operators (on some Hilbert space $H$) and $A\geq B \geq 0$. Then ${\rm range}(A) \supset {\rm range}(B)$. The textbook "$C^*$-algebras by ...
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1answer
58 views

Find minimal $\alpha_3$ such that $u\in H^3(\Omega)$ and $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$

My instructor presented me the quiz below but forgot to define key terms such as minimality and $H^3$. Quiz Let $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$. Find the minimal $\alpha_3$ such that $u\in ...
3
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1answer
86 views

Is $B - B'$ self-adjoint provided $B,B'$ are positive operators?

If I have two positive operators $B,B'$ on an arbitrary Hilbert space $\mathcal{H}$ not necessarily over $\mathbb{C}$, how do I know that $B - B'$ is self adjoint? EDIT: Reed and Simon define ...
3
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1answer
49 views

When does an operator commute with another operator given by a series?

Suppose $B$ is a bounded operator on some Hilbert space $\mathcal{H}$, given by a series of the form $$ B = I + \sum^\infty_{k = 1} c_k(I - A)^k $$ where $A$ is a given bounded operator on ...