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

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3
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1answer
33 views

Family of sequences in a Hilbert space with certain property

Suppose $\mathcal{F}$ is a family of sequences on the unit sphere of $l_2$ with the following property: For any sequence $\varepsilon_n\downarrow 0$ but which is not eventually identically $0$, there ...
0
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1answer
13 views

on Limits and sequences proofs of a closed subspace of the hilbert space.

$M$ is a closed subspace of the Hilbert space $H$ and $ x\in H$ My book states these two claims. (1) If $d = \inf_{y \in M} \|x - y \|^2 $then there is a sequence of elements $\{y_n\}$ of $M$ such ...
3
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2answers
31 views

Finding the maximum of an integral of a function with given constraints.

This comes from Rudin's Real Analysis text. The first part of the problem asks us to compute $\displaystyle\min_{a,b,c}\int_{-1}^1|x^3-a-bx-cx^2|dx$ (which I have done). Now it asks us to find ...
2
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3answers
81 views

Hermitian and self-adjoint operators on infinite-dimensional Hilbert spaces

I am a physicist and I am trying to get a grasp on the following terms from functional analysis: As I understand, an operator is Hermitian if it is symmetric and bounded (domains of A and A* don't ...
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2answers
36 views

Spectral Measures: Support vs. Norm

Given a complex Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: $$T:=\int_\mathbb{C}zdE(z)$$ ...
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3answers
47 views

Spectral Measures: Support vs. Spectrum

Given a complex Hilbert space $\mathcal{H}$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: ...
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2answers
31 views

An empty subdifferential

Can you give me an example of function $f$ defined on an Hilbert space, real valued (extended with $+ \infty$), lower semi continuous, convex and proper for which $\operatorname{dom}(\partial f)= ...
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0answers
15 views

Every Hilbert Space has an orthonormal basis [duplicate]

I'd be really grateful if someone could tell me what steps I should take (ie. what books to read) before I can prove the statement in the title. I currently have taken rigorous courses in Linear ...
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1answer
40 views

Identifying a Hilbert space with its dual

Let $H$ be a Hilbert space. Often people say "we identify $H$ with its dual $H^*$ with the Riesz representation theorem". I know there is a map $R:H^* \to H$ which is ismometric and isomorphic. So by ...
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1answer
39 views

$\{ x \in H: x=\sum_{k=1}^{\infty}c_{k}u_{k}$, $|c_{k}| \leq \frac{1}{k}\}$ is compact

Let $H$ be a complex inner product space that is also a complete metric space with respect to the distance induced by the inner product. Assume $\{u_{k}\}_{k=1}^{\infty}$ be an orthonormal set in ...
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1answer
38 views

Showing $||L|| = \sup_{x,\, y\, \in H,\, x,\, y\, \neq 0} \frac{|\langle Ax,y\rangle|}{||x||\cdot ||y||}.$

Prove that, for a Hilbert space $H$ and a linear bounded operator $L:H \to H$ such that the domain of $L$ is $H$, $$||L|| = \sup_{x,\, y\, \in H}_{ x,\, y\, \neq 0} \frac{|\langle ...
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2answers
46 views

ONB: Fourier Series

Given the Hilbert space $L^2([-\pi,\pi])$. Consider the orthonormal system: $$\mathcal{S}:=\{\frac{1}{\sqrt{2\pi}}e^{ikx}:k\in\mathbb{Z}\}$$ This is an ONB. How do I prove this? I guess, I could try ...
0
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1answer
20 views

Norm of $A$ is zer0 when $\Bbb H$ is Complex Hiblert Space

If $\Bbb H$ is a $\Bbb C$-Hilbert space and $A\in \Bbb B(\Bbb H),$i.e. bounded linear operator on $\Bbb H$ such that $\langle Ah,h\rangle=0$ for all h in $\Bbb H$, then $A=0$ For the proof of this ...
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1answer
44 views

Radon-Nikodym: Integrability?

Let $\lambda:\Sigma\to\mathbb{R}_+$ and $\kappa:\Sigma\to\mathbb{R}_+$ be finite measures on $\Omega$. Then by Radon-Nikodym: $$\kappa(E)\leq L\cdot\lambda(E)\quad(\forall ...
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0answers
14 views

Prove that every reproducing kernel is a positive matrix (and vice versa)

Let $\mathcal{H}$ be a functional hilbert space (defined over a set $S$) with a reproducing kernel K. Prove that: a) $K$ is a positive matrix means the queadtric form is positive, i.e ...
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1answer
35 views

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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1answer
22 views

Proving for each seperatble hilbert space exist complete sequence

Let $H$ be a separable Hilbert Space. Prove that exists orthonormal complete sequence and give example for one non-orthonormal sequence. I thought taking orthonormal basis for $H$ denoted by ...
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0answers
30 views

dense subspace of $C(0,T)$

I want to prove that the space H of functions which are continuous in [0,T] with weak derivative in $L^2[0,T]$ and their value in 0 is 0, is dense in the space of continuous functions in [0,T] with ...
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1answer
25 views

Does a Reproducing Kernel Hilbert Space of functions always have a distance defined in it?

Recall that a (reproducing kernel hilbert space) RKHS has two equivalent definitions: 1) Its a Hilbert space of functions $\mathcal{H}$ (i.e. vector space with an inner product $\langle \cdot, \cdot ...
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1answer
56 views

What is the image of operator exponential?

Given a Banach space $V$ and a bounded linear operator $A:V\to V$, the operator $e^A$ is bounded and invertible. When $V$ is finite dimensional, every invertible operator is of the form $e^B$ (one can ...
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0answers
33 views

Show that $H=\oplus_{\alpha \in \sigma(T)}\text{ker}(\alpha I -T)$

Suppose $T$ is an operator on a Hilbert space $H$ such that $\sigma(T)=\sigma_{p}(T)$ (point spectrum of $T$), and for each $\alpha \in \sigma(T)$, the corresponding eigenspace ker$(\alpha I-T)$ is a ...
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1answer
29 views

Product Hilbert Spaces that require completion?

My lecturer said that to make the Hilbert space $(H\oplus H',⟨\cdot,\cdot⟩ )$ we need to (1) make the cartesian product $H \oplus H' = H\times H'$, (2) give it an inner product, and - the confusing ...
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1answer
35 views

Norms for which every subset of closed unit ball containing the open unit ball is convex

It can be shown without much difficulty that any Euclidean norms satisfies the following condition :$$(P) \quad B \subset X \subset B' \Rightarrow X \, \text{is convex}$$ where $B=\{x \in E / \|x\| ...
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1answer
21 views

Adjoint of differential operator in two variables

I would like to find the adjoint of the operator $$L = x \frac{\partial^{2}}{\partial y^{2}} \frac{\partial }{\partial x}.$$ I know the adjoint is the operator $L^{*}$ such that $$(Lu,v) = (u, ...
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0answers
76 views

Comparison of Hilbert space tensor product and wedge product

For Hilbert Spaces: $$(|0\rangle + |1\rangle)\otimes (|0\rangle + |1\rangle) = |00\rangle + |01\rangle + |10\rangle + |11\rangle.$$ where all results are column vectors \begin{eqnarray*} 0 ...
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1answer
11 views

Calculating the form domain of an operator

I am reading the book "Mathematical Methods in Quantum Mechanics" by Gerald Teschl and just came across the concept of a form domain. It is defined for non-negative operators i.e $<\phi,A \phi> ...
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0answers
10 views

dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$?

Please can you help me to understand how the dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$? This is the relevant passage: So far, we have assumed implicitly that the ...
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1answer
40 views

Identity involving the resolvent of an operator.

$\mathcal{H}$ is a complex separable Hilbert space, $D \subseteq \mathcal{H}$ is a dense subspace. $L : D \rightarrow \mathcal{H}$ is a densely defined, symmetric and closed operator. $L$ is not ...
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2answers
49 views

Is a Hilbert space a vector space or a space of functions?

I was learning what Hilbert space was and this is the definition that I have: $\mathcal{H}$ Hilbert Space is a vector space with $\langle \cdot , \cdot\rangle$ inner product and is complete with ...
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1answer
53 views

Uniformly bounded sequence in Hilbert-Sobolev space

Let $\Omega \subset \mathbb{R}$ be a bounded open set with $C%1$ boundary and $H^1(\Omega) = W^{1,2}(\Omega)$ be the Hilbert-Sobolev space. Let ${u_n}$ be a sequence of functions which are uniformly ...
0
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1answer
18 views

$\|x\|=1,\|x-y\|\ge\epsilon\forall x,y\in A,x\neq y$ , is this finite set in hilbert spaace?

$A$ be a subset of a Hilbert Space $H\ni \|x\|=1\forall x\in A$ and there is an $\epsilon\ni\|x-y\|\ge\epsilon\forall x,y\in A,x\neq y$ I need to know whether $A$ is finite or not. Intuitively, It ...
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0answers
19 views

How do you show the connection of reproducing kernels to feature maps?

This question is in the context of Hilbert Reproducing Hilbert Spaces and reproducing Kernels and their relation to feature maps (and machine learning). We have a Hilbert space $\mathcal{F}$ and ...
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1answer
20 views

Help proof regarding sequence in subset of Hilbert space

I'm to prove the following: Let $H$ be a Hilbert space, and let $M$ be a non-empty convex subset of $H$. Suppose that $(x_n)$ is a sequence in $M$ such that $ ||x_n|| \to d$, where $d= \inf ...
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1answer
43 views

The definition of convex body and the Hilbert cube

I currently have a question about the definition of convex body. The formal definition is: a convex body is a convex set which has non-empty interior. By non-empty interior, we meant for a set ...
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0answers
53 views

Proving that the set is closed.

We use the sequential definition to prove a set is closed. So no continuity or closure or anything related to the topology of the set is allowed. Show $A = \{ x \in \ell^2: |x_n| \leq 1/n \}$ is ...
0
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1answer
39 views

Existence of adjoint operator on a Hilbert space

friends! I read that the algebra $\mathscr{L}(H,H)$ of the bounded operators on a Hilbert space $H$ is a $B^\ast$-algebra in the sense defined here . I easily verify all the properties except for the ...
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0answers
23 views

Existance of Unitary Operator which Extends any inner product preserving operator in a Hilbert Space

Suppose $V$ is a finite dimensional Hilbert Space with a subspace $W$ Suppose $T:W\to V$ be a linear operator which preserves inner product i.e $\forall w_1,w_2\in W$ we have $\langle ...
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1answer
22 views

Questions about open balls and convergence in Hilbert space

So, I've started reading about dimension theory and currently am dealing with a lemma which is used in a proof of $\dim(\mathcal{\ell}^{2}_{\mathbb{Q}})=1$. This lemma says that convergence of a ...
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0answers
21 views

product of one dimensional basis functions spanning two dimensional space

Lets assume I have a set of basis functions $h_1(x),h_2(x), ...$ spanning the whole hilbert space of one dimensional square integrable functions. Now I want a basis set that spans the the whole ...
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3answers
77 views

Is the (first order theory) of Hilbert spaces categorical?

Suppose the axioms of a Hilbert space (i.e. vector space, inner product, completeness and separability) are formulated as a first order theory. It can be shown that any infinite dimensional Hilbert ...
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3answers
23 views

When a symmetric densely defined operator is an adjoint operator?

I am wondering if it is possible to say that if a symmetric differential operator is densely defined then the operator is self-adjoint indeed? More Precisely, Let $A:D(A)(\subset H)\to H$ a densely ...
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0answers
29 views

If $y_m \to y$ in $H$ then $|y_m|_H \leq C|y|$ for this sequence?

Let $w_i$ be a basis for a Banach space $V$. We have $V \subset H$ a continuous and dense embedding into a Hilbert space $H$. Define $y_m = \sum_{i=1}^m a_{im}w_i$. We have that $y_m \to y$ in a $H$ ...
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1answer
407 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
77 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
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1answer
60 views

Can we have infinite-dimensional separable Hilbert spaces?

In Wikipedia's Hilbert space article on separability, it says: A Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert ...
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1answer
23 views

The projection operator defined by $(P_n(h) - h, v)_H = 0$

Let $V \subset H$ be separable Hilbert spaces with continuous embedding and suppose $\{v_n\}$ be a (non-orthogonal) basis for $V$. If we let $V_n = \text{span}(v_1, ..., v_n)$ and given $h \in H$ we ...
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1answer
62 views

Spectral Measures: Integration

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. How to define the integral for unbounded measurable functions: ...
3
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1answer
43 views

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is ...
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1answer
75 views

Subalgebra generated by selfadjoint operator $A_0\in\mathscr{L}(H,H)$

Let $\mathscr{L}(H,H)$ be the Banach algebra of bounded operators defined on a complex Hilbert space $H$ and let $B(A_0)$ be the subalgebra generated by the selfadjoint operator $A_0$, i.e. ...
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0answers
40 views

Sum of closed linear subspaces necessarily closed?

Let $H$ be an infinite-dimensional Hilbert space. Let $L_1,L_2 \subset H$ be two closed linear subspaces. If it is also known that $L_1 \perp L_2$ then it is not hard to show that $L_1 + L_2 = \{x_1 ...