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

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59 views

What does it exactly mean that finite linear combination to be dense?

This phrase comes up over and over again when studying Hilbert space, and since I don't have the strongest background in linear algebra, the statement like "finite linear combination of elements in ...
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1answer
37 views

For a normal operator $T$ we have $\sup_{\Vert x \Vert = 1} \mathrm{Re} \langle x, Tx \rangle = \sup_{\lambda \in \sigma(T)} \mathrm{Re} \lambda$

If $(X, \langle \cdot, \cdot \rangle)$ is a complex Hilbert-space and $T : X \rightarrow X$ a normal operator, i.e. an operator such that $T T^\ast = T^* T$ then I'd like to show that: $$\sup_{\...
2
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1answer
28 views

Sequence of operators that commute imply the limit commutes?

Given a sequence of compact operators $A_n\to A$ as $n\ \to \infty$ and $B$ (which has finite rank). $\varphi \in L^2([a,b])$ If $A_nB\varphi = BA_n \varphi$ Am I able to say anything about $A$, i....
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1answer
40 views

Proof of $\hat{\mathrm{O}}$ta's theorem

I'm trying to prove $\hat{\mathrm{O}}$ta's theorem : Let $A$ be a closed operator on a Hilbert space $H$ and $\overline{\mathcal{D}(A)}=H$. Suppose that $A\mathcal{D}(A)\subset \mathcal{D}(A)$ and ...
2
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1answer
105 views

Classification of representations of compact $C^*$ algebras for single operators.

I am looking at Arveson's book, an invitation to $C^*$ algebras. There, it is explained p. 21 ($C^*$ algebras of compact operators) that any representation of a compact $C^*$ algebra can be decomposed ...
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2answers
39 views

An exercise about variational principle

Let $H$ be a Hilbert space. Let $l: H \to \mathbb{R}$ be a continuously linear function. Let $g: H \to \mathbb{R}$ be defined by $$g \left ( x \right )= \frac{\left \| x \right \|^2}{2}-l\left ( x \...
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0answers
26 views

Cartesian Decomposition.

I just read this on some notes written by my professor. It requires $X$ to be a linear map from complex Hilbert space $\mathcal{H}$ to itself, and that the Cartesian decomposition of $X$ is $X = H + ...
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20 views

Writing matrix representation of multiplication operator

For a given $m(x)\in L^2(0,1)$, let's write the multiplication operator $M\colon L^2(0,1)\longrightarrow L^2(0,1)$ as $Mf(x)=m(x)f(x)$. To write the matrix representation of this operator we need a ...
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1answer
40 views

Uniform convergence in Mercer Theorem for bounded kernels

Let $\mu$ be a finite, strictly positive measure on $\mathbb{R}$, and let $k$ be a measurable positive-definite kernel. Assume $k$ is bounded, and let $T:L^2(\mu)\rightarrow L^2(\mu)$ be defined by $$ ...
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1answer
33 views

GNS construction of a weight

In the theory of quantum groups in the operator algebraic setting, one deals with weights (instead of positive linear functionals). Definition: A weight is a function $\phi $ : $A^+ \rightarrow [0, \...
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1answer
21 views

convergence of series in inner product space

let $V$ be some inner product space and $\lbrace {e_i\rbrace }_{i\in\mathbb{N}} \subset V$ be some countable orthonormal set. I am wondering if for any $x\in V$ the series $$\sum\limits_{i=1}^{\infty} ...
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0answers
40 views

Spectral theory of compact, self-adjoint operators.

Let $T$ be a compact, self-adjoint operator on a separable Hilbert space H. Suppose that $f\in H$, $||f|| =1$ and $||(T-3)f||\leq 1/2$. Let P be the orthogonal projection onto the direct sum of all ...
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1answer
16 views

Initial and final sub-spaces of a partial isometry

Let $H$ be a Hilbert space and assume $H_0$ and $K_0$ are two sub-spaces of $H$ with dim$H_0$=dim$K_0$. Question: Is there any partial isometry $u$ whose initial projection is $H_0$ and final ...
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0answers
38 views

Finding the closest function to another in a Hilbert space.

Let H be the Hilbert space L$^2$([0,1)], and let $S$ be the subspace of functions f $\in$ H satisfying $\int^1_0(1+x)f(x)dx=0$. Find the element of $S$ closest to the function $g\in H$ defined by ...
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1answer
30 views

Product of Hilbert bases of $L^2(\mathbb{R}^p)$ and $L^2(\mathbb{R}^q)$ is a Hilbert basis for $L^2(\mathbb{R}^{p+q})$

Let $(\alpha_n)_n$ be a Hilbert basis of $L^2(\mathbb{R}^p)$ and let $(\beta_k)_k$ be a Hilbert basis for $L^2(\mathbb{R}^q)$. I need to show that $(\alpha_n \beta_k)_{(n,k) \in \mathbb{Z}}$ is also a ...
5
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1answer
37 views

Linear span in the intersection of Hilbert spaces

Let $V$ be a vector space. Assume $H_1$ and $H_2$ are subspaces of $V$, and that both $H_1$ and $H_2$ are Hilbert spaces with inner-products $\langle \cdot, \cdot\rangle_1$ and $\langle \cdot,\cdot\...
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2answers
50 views

Showing that a Hilbert Basis $(e_n)_{n \in \mathbb{N}}$ verifies $u= \sum (u,e_n)e_n $

The definition I have been given for a Hilbert Basis in a Hilbert Space $H$ over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ is: A sequence $(e_n)_{n \in \mathbb{N}}$ is an orthonormal basis if it ...
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0answers
38 views

Let $H$ be a Hilbert space and $Φ≤H$ be equipped with a topology. Under which topology on $Φ^*$ is $H^*\ni f\mapsto\left.f\right|_Φ$ continuous?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $\Phi$ be a vector subspace of $H$ equipped ...
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1answer
38 views

Bilinear forms defines inner product on Hilbert Space

I have difficulties understanding the reason why when I have a self adjoint linear operator $T : \mathcal{H} \rightarrow \mathcal{H}$, and know that $A\|f\|^2 \leq \langle Tf,f \rangle \leq B\|f\|^2$ ...
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2answers
33 views

Proof of Anti-Linearity of Hermitian Conjugate

How can I prove that the adjoint operation/ Hermitian conjugate in anti-linear i.e $(\sum_{i} a_i A_i)^\dagger = \sum_{i} a_i^* A_i^\dagger$, where $A$ is any linear operator on a Hilbert space $V$. ...
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0answers
37 views

Is the span of a compact, connected and infinite dimensional subset of $l^2$ open as a subset of the closed span?

Let $K$ be a compact, connected and infinite dimensional subset of $l^2$. Is it possible to prove that $\operatorname{span}(K) \setminus \{0\}$ is open in $\overline{\operatorname{span}(K)}$?
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1answer
61 views

$5$ questions on the definition of the Gelfand triple

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb F\in\left\{\mathbb R,\mathbb C\right\}$, $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\...
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0answers
17 views

Show that the linear functional $\delta_{\lambda}(\varphi)=\varphi(\lambda),\delta: H^2(\Bbb{D})\to\Bbb{R}$ is continuous

Show that the linear functional $\delta_{\lambda}(\varphi)=\varphi(\lambda),\delta: H^2(\Bbb{D})\to\Bbb{R}$ is continuous where $\delta\in \Bbb{D}$, $\Bbb{D}$ is the unit disk, $H^2({\Bbb{D}})$ is a ...
0
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1answer
45 views

Do eigenvectors with pairwise distinct eigenvalues of a bounded, linear, nonnegative, symmetric operator on a Hilbert space build an orthogonal basis?

Let $H$ be a Hilbert space and $Q$ be a bounded, linear, nonnegative and symmetric operator on $H$ with finite trace. By the Hilbert–Schmidt theorem, there is an orthonormal basis $(e_n)_{n\in\...
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0answers
32 views

Is $\delta:C_2[0,1]\to \Bbb{R}, \delta(f)=f(0)$ discontinuous?

Is the functional $\delta:C_{2}[-1,1]\to \Bbb{R}, \delta(f)=f(0)$ with $L_2$ norm discontinuous? I understood I should show whether or not the functional is unbounded, but I can't seem to understand ...
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0answers
39 views

Is there a condition along with Strong convergence which gives convergence in norm?

Let $\{T_n\}$ be a sequence of operators in a Hilbert Space and $T_n$ converges to an operator $T$ in the strong operator topology. Is there any assumption which I can put on $\{T_n\}$ so that the SOT ...
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1answer
25 views

If $(u_n)_n,(v_n)_n$ are seq. in a Hilbert space and $(e_k)_k$ is an ONB, then $\sum_k|\sum_n(u_n,e_k)(v_n,e_k)|≤\sum_n\sum_k|(u_n,e_k)(v_n,e_k)|$

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $(u_n)_{n\in\mathbb N}\subseteq H$ and $(v_n)...
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1answer
38 views

The vector-valued distribution of compact support

Let $\mathcal{H}$ be infinite dimensional Hilbert space and $D(\mathbb{R}^n)$ be the space of smooth complex functions of compact support. Consider the distribution $T: D(\mathbb{R}^n) \to \mathcal{H}...
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1answer
53 views

Functional analysis - check that a closed subspace of a Hilbert space is convex

Suppose that V is a Hilbert space over $F$ and $W$ is a closed subspace of $V$ . Then for every $x \in V$ , there exist unique $y \in W$ and $z \in$ (the orthogonal compliment of $W$) such that $x = y ...
0
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1answer
24 views

Are these two statements involving inf and sup equivalent?

Suppose $H$ is a Hilbert space. Is $$\inf_{h \in H}\sup_{g \in H}\frac{f(h,g)}{|h||g|} \geq C$$ the same as $$f(h,h) \geq C|h|^2\quad \forall h \in H$$ ? Here $f\colon H \times H \to \...
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0answers
58 views

Show that there exists a certain operator in $L(H)$ where $H$ is a separable Hilbert Space.

Given a separable Hilbert Space $H$ and $\sum_{n=1}^{\infty} f_n$ an absolutely convergent series in $H$, I need to show that there is an operator $A \in L(H)$ such that $A(e_n)=f_n$, where $(e_n)...
8
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1answer
140 views

Vector space that can be made into a Banach space but not a Hilbert space

Are there any (real or complex) vector spaces which can be made into a Banach space given a suitable norm, but cannot be given a norm that makes it a Hilbert space? I know that the parallelogram law ...
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2answers
88 views

Polynomials form a Hilbert basis for $L^2$

If you form a set of orthonormal polynomials on $[0,1]$, by applying the Gram-Schmidt process from monomials $\{1, x, x^2, \dots \}$ then what is required to show that this is a basis for $L^2[0,1]$? ...
5
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1answer
75 views

Comparing two sigma algebras in Hilbert spaces

Let $H$ be a non-separable Hilbert space. We denote $B$ by the sigma algebra generated by the norm topology in $H$. We also denote $B_{w}$ by the sigma algebra generated by the weak topology in $H$. ...
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2answers
54 views

Show that $\sum_k\omega_k^2=\infty$ a.e. if $\{\omega_k\}$ is an orthonormal basis

Let $\{\omega_k\}$ be a closed orthonormal system in $L^2[a,b]$. Show that $\sum_k\omega_k^2=\infty$ a.e. Suppose $\exists E\subset[a,b]$, $m(E)>0$ and $\sum_k\omega_k^2\le M<\infty$. I want ...
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2answers
40 views

Isosceles Triangles in Hilbert Spaces and Metric Spaces generally

In what types of metric spaces $\langle X, d \rangle$ is it possible to do the following? Task: For any two points $x, y \in X$ such that $d(x,y) \leq 2\epsilon$, find a third point $z$ such that $...
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0answers
92 views

Hilbert space in Papa Rudin

In Rudin's Real and Complex Analysis, there is a problem in Chapter 4 on a Hilbert space $X = \text{span} \{e^{ist} \, \mid \, s \in \mathbb{R}\}$ with the inner product $$(f,g) = \lim_{T \to \infty} \...
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1answer
37 views

Base of infinity dimensional function space

As I know ,the Sobolev space $H^k(\Omega)=W^{k,2}(\Omega) $ and $L^2(\Omega)$ are Hilbert space. So, they must have orthogonal basis. But I can't find it on my book. Where I can find it ? I In fact ...
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1answer
47 views

Finding the spectral decomposition of $\Delta= \frac{d^2}{dx^2}$ [closed]

What is the spectral decomposition of the operator $\Delta= \frac{d^2}{dx^2}$ in $(L^{2}(\mathbb R), dx)$? Thanks you in advance
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0answers
61 views

Kernel-induced metric

Given a kernel k on input space X defining RKHS (Reproducing kernel Hilbert space) H. Let Φ : X → H denote the corresponding feature map (think of Φ(x) = k(x, .)). Let x, z ∈ X . How can I show that ...
5
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1answer
88 views

Inequivalent Hilbert norms on given vector space

Suppose we have a vector space $X$. Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two different complete norms on $X$ s.th. $X$ equipped with $\|\cdot\|_j, \ j\in\{1,2\}$ is a Hilbert space. Are there ...
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3answers
99 views

Subset of $\ell^2$ is precompact

Suppose we have a sequence of $a_i$ with some restrictions on it. Which restrictions must be to make set $$A= \left\{(x_i) \in \ell_2 \mid \sum\limits_{i\geqslant1} |a_i x_i|^2 \leqslant 1 \right\} $...
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1answer
40 views

Generalization of Bessel inequality

Let $H$ be a Hilbert space and suppose we have a sequence $\{x_n\}\subset H$ s.th. for every $h\in H$ we have $ \sum \left|\langle h, x_n\rangle\right|^2<\infty$. How to prove that there exists ...
3
votes
3answers
61 views

How to show that the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not Hilbert space?

I want to show that the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not a Hilbert space. So I should show that it is not an inner product space. Most likely, The ...
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1answer
95 views

Two (equivalent ?) norms on Hilbert space

Let $H$ be a vector space equipped with two inner products $\langle \cdot,\cdot\rangle_1, \ \langle \cdot,\cdot\rangle_2$, s.th. $(H,\langle \cdot,\cdot\rangle_j)$ is a Hilbert space for $j=1,2$. ...
6
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1answer
45 views

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$, looking at $C_{(2)}[-1,1]$, with $L_2$ norm. I tried to look at a general polynomial $\sum_{i=0}^{98} a_ix^...
1
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1answer
67 views

Direct sum of two closed subspaces of Banach space is not closed

I'm looking for an example of two closed subspace of a Banach space (or even a Hilbert space) whose sum is not closed. We have $l^2$ as Banach space and $A$ and $B$ are closed subspaces of $l^2$ : $...
3
votes
1answer
34 views

Do the holomorphic or meromorphic functions on a domain $D \subseteq \mathbb{C}$ form a Hilbert space $\mathcal{H}$?

In a physics paper I am reading that the meromorphic functions on $\mathbb{C}$ with $f(x) = f(\overline{x})$ form a Hilbert space. $$ \mathcal{H} = \{ f(x) : f(x) = f(\overline{x}) \}$$ Even let $f(...
3
votes
1answer
69 views

Spectrum of right shift operator in weighted $l2$ sequence space

Let $l_2(a)$ be a hilbert space defined with following inner product: $\langle x_n,y_n\rangle = \sum a^k x_k y_k$. (It's a weighted sequence space with the weights $\omega_i = a^i$). It's elements ...
1
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1answer
32 views

Relation betwen dimension of Hilbert space and cardinality of its dense subset

Suppose $H$ is infinite dimensional Hilbert space. Let $A$ be a dense subset of $H$. How to prove that $\mathrm{dim}_{\mathrm{orth.}}\ H \le \mathrm{card}(A)$ ? When we have equality ? I need only ...