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

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Uncountable series without axiom of choice

Consider a sequence of positive real numbers $(\alpha_i)_{i\in I}$ for some (suppose maybe wellordered for now) set $I$. Using axiom of choice, it is easy to see that $\sum_i \alpha_i$ is infinite if ...
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22 views

Zero eigenvalue of an Operator [on hold]

In my functional analysis notes, there is a claim with proof(which I seem to understand, but don't get the point) is the following. Consider the bounded linear operator L: H$ \rightarrow $ H, where H ...
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27 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
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1answer
33 views

Reproducing kernel Hilbert space, why?

Let $K: X \times X \rightarrow \mathbb{C}$ be a positive definite kernel on a set $X$, i.e. for any $x_1, \cdots, x_n \in X$, the matrix $$ [K(x_i, x_j)]_{ij} \in \mathbb{C}^{n \times n} $$ is ...
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17 views

Weak convergence in Hilbert space implies strong convergence of averages for some subsequence

Let $H$ Hilbert Space. Show that if $x_n\rightharpoonup x$ then there exists a subsequence $\{x_{nk}\}$ of $\{x_{n}\}$ such that the sequence $\lim_{m\rightarrow \infty } ...
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1answer
48 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
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1answer
30 views

Proving dense set is core for a self adjoint operator

Let $A$ be a self adjoint operator in a Hilbert space $H$ and $D\subseteq D(A)$ a dense subset such that $$ e^{iAt}:D \to D. $$ How can I show that $D$ is a core for $A$? I need to show that ...
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15 views

Green-Operator for Sturm-Liouville Differential equation compact on Sobolev space?

Let $g$ be Green's Function for a Sturm-Liouville differential equation. Is the operator $G: H_{0}^{1}(0,1) \rightarrow H_{0}^{1}(0,1)$ defined by $(Gf)(x) := \int_{0}^{1} g(x,y)f(y) dy, \quad f \in ...
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1answer
35 views

Is the intersection between two $n$-spheres an $(n-1)$-sphere?

It is true that the intersection between two $n$-sphere in $\mathbb{R}^n$ is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and ...
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1answer
51 views

Recapitulated: Stone's Theorem Integral

This problem grew out from: Stone's Theorem Integral For a definition, a nonexample and a comparison see: Generalized Riemann Integral: Definition Generalized Riemann Integral: Nonexample ...
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11 views

The median in a isosceles triangle is ortoghonal into a hilbert space

how can I prove that if $p$, $q$, $r$ and $o$ are points in a Hilbert space such that $p$, $q$, $o$ are collinear, $\|p-o\|=\|q-o\|$ and $\|p-r\|=\|q-r\|$ then $r-o \perp p-o$?. I think it's a ...
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2answers
30 views

Book for Hilbert spaces.

Which book either on functional analysis or specifically for Hilbert spaces has the best way of explaining with most examples and to the point without much applications. I studied Limaye's book and ...
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0answers
26 views

Subset being orthonormal basis if $\sum_{n\geq 1} \|a_{n}-b_{n}\|^{2} < \infty$ [on hold]

I have a question Supposing $X$ is a Hilbert space and lets suppose that $A=\{a_{n} \ : \ n\geq 1\}$ is an orthonormal basis of $X$. Let $B=\{b_{n} \ : \ n\geq 1 \}$ be an orthonormal subset of $X$. ...
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1answer
20 views

Spectral Measures: Subspace Characterization

Disclaimer This thread is related to: Spectral Measures: Subspace Decomposition It is meant to record. See: Answer own Question It is written as question. Have fun! :) Question Given a Hilbert ...
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1answer
17 views

An example of an unbounded non-orthogonal projection in a Hilbert space

What is an example of an unbounded non-orthogonal projection in a Hilbert spaces? Does it exist? A non-orthogonal projection is an idempotent operator: $T^2=T$. So the question is: can such an ...
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26 views

Orthonormal set is a Hilbert basis $\iff$ Parseval's identity is true

Let $H$ be a Hilbert space and $\{e_k:k\in \mathbb{Z}\}$ an orthonormal set. Prove that the set is a Hilbert basis if and only if Parseval's identity is true. The direct theorem is almost ...
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42 views

Cardinality of dense subset of Hilbert space

If $H$ is an infinite dimensional Hilbert space, I want to show that any total orthonormal family in $H$ has the same cardinality as the minimum cardinality of a dense subset of $H$ but I am ...
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3answers
132 views

If a linear operator has an adjoint operator, it is bounded

This is a question I'm struggling with for a while: Let $H$ be a Hilber space. Let $T,S: H\rightarrow H$ be linear operators (not neccessarily bounded) such that for every $x,y\in H$: $\langle ...
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1answer
54 views

Sequence in a hilbert space.

$M$ is a closed subspace of the Hilbert space $H$, and x $\in H$. Call $d = \inf_{y \in M} ||x - y||^2$ Show that there exist a sequence of elements $y_n$ of M such that $||y_n - x ||^2 \rightarrow ...
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1answer
20 views

Prove the operator on hilbert space is compact

My question is actually the same as the first part of this one, Prove that T is compact which has not been answered. I am thinking about two ways, 1) use a bounded sequence $\{g_n\}$, and try to ...
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1answer
32 views

If $T^{2}$ is a compact operator then $T$ is compact

Suppose $T$ is a bounded , self-adjoint operator on a Hilbert space such that $T^{2}$ is compact. Then prove that $T$ is compact. I proved it by continuous functional calculus but am looking for a ...
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9 views

Relation between RKHS and space of continuous functions

Consider a Mercer Kernel $K\colon \mathcal{X}\times \mathcal{X}\to \mathbb{R}$, $\mathcal{X}$ being a compact subset of $\mathbb{R}^m$, and its (unique) associated Reproducing Kernel HIlbert Space ...
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2answers
59 views

An orthonormal subset of a Hilbert space is closed.

In Rudin Real and Complex Analysis there is an exercise (6, Ch. 4) that asks to show that a countably infinite orthonormal set $\{u_n:n\in\mathbb{N}\}$ in a Hilbert space $H$ is closed and bounded but ...
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1answer
72 views

$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin's (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq ...
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1answer
15 views

Question on Completeness of Derived Inner Product Space

Let $(\mathcal{H},\langle{,}\rangle)$ be a separable, infinite-dimensional Hilbert space. Let $\mathcal{X}''$ denote the space of bounded sequences in $\mathcal{H}$. For a Banach limit $L$, define a ...
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2answers
35 views

Find an approximation of the unit ball as a weak-limit of a sequence in the unit sphere

Let $H$ be an infinite dimensional Hilbert space. It is well known that the weak-closure of the unit ball is the unit sphere. But I want to prove it as basicaly as possible, using the ...
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1answer
38 views

How can I prove that a sequence such that every converging subsequence coverges to the same limit, converges?

I want to claim that if $(x_n)_{n\in N}$ is a sequence, and there is $a$ such that if $(x_{n_k})$ converges, so $\lim x_{n_k} = a$ (it means that all converging subsequences have the same limit), then ...
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1answer
20 views

Some closed subspace of $l_2$?

$(a)$ I was trying to define a continuous linear map $T$ on $l_2$ whose kernel would be the $A$ and can conclude $A=T^{-1}(0)$ and hence closed set? could anyone help me to solve any of one?
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Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$. I've already proved that if $U$ is a closed subspace then $U = (U^{\bot})^{\bot}$. I also ...
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1answer
64 views

Proving this set is dense in $\ell^2$

I found this weirdest question and was wondering how could this be proved. This question is a part of a beautiful semi-constructive built of two dense disjoint convex sets in $\ell^2$, which I find ...
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2answers
24 views

Unique nearest point property

Consider $\mathbb{R}^2$ with a norm defined by $\|(x,y)\| = |x|+|y|$. Define $\mathrm{dist}(K,p) = \inf_{q \in K} \|q-p\|$. Why are there infinitely many points $q \in K$ that satisfy $\|p-q\| = ...
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1answer
53 views

Uniquness and Exisstence of One Theorem

I need a short and nice Proof for Uniqueness and Existence of the following theorem: Suppose (H, <0,0> ) is a Hilbert space, and M is a closed convex set and $x \in H$, then there is a unique ...
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1answer
21 views

is there a convex bounded subset A of H such that A is not norm closed and A∩L is norm closed for every finite dimensional subspace L of H

"Given an infinite dimensional Hilbert space H. Show that there is a convex bounded subset A of H such that A is not norm closed and A∩L is norm closed for every finite dimensional subspace L of H." ...
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11 views

Show that there is a convex bounded subset A of H such that A is not norm closed and A∩L is norm closed for every finite dimensional subspace L of H. [closed]

Given an infinite dimensional Hilbert space H. Show that there is a convex bounded subset A of H such that A is not norm closed and A ∩ L is norm closed for every finite dimensional subspace L of H.
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1answer
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6 views

proof of separability hilbert space H.

I think that a Hilbert space is called separable if it contains a complete orthonormal sequence. Finite dimensional Hilbert spaces are considered separable. I need to prove that the Hilbert space ...
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1answer
20 views

How to see injection and boundedness

Lemma. If $A$ is a bounded linear operator defined on a Hilbert space and $\|Af\| \geq c\|f\|$ and $\|A^*f\| \geq c\|f\|$ for some constant $c$. Then $A$ has a bounded inverse. In the proof of ...
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32 views

Equivalent formulation for compact operators

According to Wikipedia, an operator is compact if it can be written in the form $T(u)=\sum_{n=1}^\infty \lambda_n<f_n, u> g_n$, where $\{f_n\}$ and $\{g_n\}$ are orthonormal sets and ...
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23 views

$L^{2}(\mathbb{R})$ is a separable Hilbert space.

I want to show $L^{2}(\mathbb{R})$ is separable. My idea is $C_{c}(\mathbb{R})$ is dense in $L^{2}(\mathbb{R})$ in $L^2$ norm and polynomials with rational coefficients are dense in $C[a,b]$ in $\sup$ ...
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0answers
12 views

Hilbert space subspace, orthogonal projection

I do not know how to solve this problem, and I do no know why it has to be a closed subspace. Thank you very much.
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1answer
19 views

linear transformation between Hilbert space

By definition, $|T|=\sup|(Tf,g)|, |f|\le1,|g|\le1$ $$||T||\ge(Tf,f)$$ But I can not find an example such that $||T||>(Tf,f)$ for any $|f|<1$. Any suggestion? Thanks in advance~
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Is this following bilinear form coercive?

First of all I want to mention that this is homework, so don't spoil it and reveal all the answer. just some guidenss :) Let $H$ be a Hilbert space, $T:H\rightarrow H$ a bounded linear operator for ...
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1answer
62 views

Inequivalent norms (given by different inner products) on infinite dimensional Hilbert space.

I have this question in reviewing for my exam. Let $H$ be an infinite dimensional Hilbert space. Write down an inner product on $H$ that gives a norm inequivalent with the original norm. Is $H$ ...
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1answer
30 views

Two versions of Lax-Milgram theorem

I'm having some troubles differentiating between two versions of Lax-Milgram theorem, one shown in my class and one that I saw is common on the internet. Let $H$ be hilbert space, $B$ bilinear form ...
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2answers
24 views

Existence Adjoint Operator: Boundedness?

Context This would make the check on the GNS construction much more simple. Problem Given a Hilbert space $\mathcal{H}$. Consider a merely linear operator $A:\mathcal{H}\to\mathcal{H}$. Suppose ...
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1answer
45 views

perturbation by orthogonal projection

Let $G$ be an operator with discrete spectrum on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$. Let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} = G+P$. My ...
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1answer
18 views

Finite sum $\sum_{r,k} p_kP_r(x_k)f(x_k)P_r(x_m)=f(x_m)$

Let $x_0,\ldots,x_n\in\mathbb{R}$ be $n+1$ arbitrary real points and $p_0,...,p_n>0$ be positive real numbers. Let $P_0,P_1,\ldots,P_n$ be polinomials such that $$\sum_{k=0}^n ...
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1answer
29 views

Partial Isometries: Characterizations

Any partial isometry satisfies: $$\Omega\Omega^*\Omega=\Omega$$ From this, one derives projections: $$\Omega^*\Omega,\Omega\Omega^*$$ Conversely, given projections: $$\Omega^*\Omega,\Omega\Omega^*$$ ...
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28 views

Scalar product of $L_2$ with $\mu(E):=\int_E gdx$

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 400 here) that, if we define measure $\mu$ for $E\subset[-1,1]$ by $$\mu(E):=\int_E g(x)dx$$ where the integral ...
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1answer
19 views

Hilber transform on [0,1)

Let $\mathbb{T}=[0,1)$ and $H$ be a Hilbert transform on $L^p(\mathbb{T})$ when $2\leq p< \infty$. If $f$ is $L^p$ and $f_n$ is trignometric polynomial such that $f_n\rightarrow f$ in $L^p$ sense. ...