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

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17 views

Subdifferential of a continuous function is non-empty

Prove that if $X$ $-$ normed vector space, $x_0 \in \text{int }A$ and convex function $f$ is continuous in $x_0$ then $\partial f(x_0) \neq \emptyset$. $\partial f(x_0) = \{x^*\in X^*:\forall x \in ...
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0answers
49 views

Dichotomy for global existence or blow up for solutions of evolution problems.

Consider the problem (Nonlinear Schrödinger equation) \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u\mp u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ ...
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1answer
59 views

Hilbert Spaces: Tensor Product

Problem Given a Hilbert space $\dim\mathcal{H}=\infty$. Construct the algebraic tensor product $\mathcal{H}\otimes\mathcal{H}$. Obviously, it is not complete yet. What if one takes formally: ...
2
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1answer
32 views

Reducing Subspaces: Nonexample?

Given a Hilbert space $\mathcal{H}$. Consider an operator $T:\mathcal{D}(T)\to\mathcal{H}$. Regard a closed subspace $Z\leq\mathcal{H}$. Suppose the subspace reduces the operator: $$TZ\subseteq ...
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0answers
15 views

Limits of trajectory of gradient flow in Hilbert space

I have been studying about gradient flow in Hilbert space of a Morse function $f$. Specifically, let $X$ be a Hilbert space and $f : X\to \mathbb R$ be $C^3$ function. The gradient flow here is ...
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1answer
24 views

How to show that a vector space is closed?

I am trying to complete a proof which requires me to prove that a subspace $H$ of $L^2{(\Omega,\mathcal{F}},\mathbb{P})$ is closed vector space in $L^2{(\Omega,\mathcal{F}},\mathbb{P})$ What do I ...
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2answers
60 views

Dense subspaces, closed subspaces and unbounded operators in Hilbert spaces

Let $\mathcal{H}$ be a Hilbert space, and let $N\subseteq\mathcal{H}$. I found two interesting statements (without proof): if a closed subspace $N$ is such that $N^{\perp}=\{0\}$ (which is ...
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1answer
28 views

$H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$

When $\Omega$ is a bounded open set of $\mathbb{R}^N$ with the help of Poincare inequality, we know that $H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$ is a Hilbert space. ...
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1answer
10 views

If $H$ is a Hilbert space and $T$ an isometric operator, then $\overline{R(T-I)}=H \implies N(T-I)=\{0\}$?

Let $H$ be a Hilbert space. Let $T$ be a linear operator and $R(T)$, $D(T)$, $N(T)$ the range, domain and kernel of $T$, respectively. If $\|Tx\|=\|x\|$ for all $x \in D(T)$, then $T$ is called an ...
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2answers
20 views

Which Sobolev-Space to use to formulate weak biharmonic equation, $H^2_0$ or $H_0^1\cap H^2$?

For the weak formulation of the biharmonic equation on a smooth domain $\Omega$ $$ \Delta^2u=0\;\text{in}\;\Omega\\ u=0, \nabla u\cdot \nu=0\; \text{on}\; \partial\Omega $$ why does one take ...
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2answers
55 views

Operator Tensor Product

Let $S$ and $T$ be bounded operators over a Hilbert space $\mathcal{H}$. Define their tensor product $S\otimes T$ as acting on $\mathcal{H}\otimes\mathcal{H}$ by $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
4
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1answer
80 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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2answers
76 views

Stone's Theorem Integral: Avanced Integral

Reference This problem grew out from: Stone's Theorem Integral: Basic Integral Problem Given the real line as measure space $\mathbb{R}$ and a Hilbert space $\mathcal{H}$. Consider a strongly ...
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2answers
117 views

Stone's Theorem Integral: Basic Integral

Disclaimer This thread has been renewed: Stone's Theorem Integral: Advanced Integral Problem Given a finite Borel measure $\mu$ and a Hilbert space $\mathcal{H}$. Consider a strongly continuous ...
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0answers
27 views

Characterization of Hilbert spaces among Banach spaces

Let $H$ be a complex Hilbert space. I know that $H\simeq \overline{H^*}$ by the Riesz representation theorem, where $\overline{X}$ means the complex conjugate space of $X$. I want to prove the ...
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0answers
22 views

Compact operator space Hilbert

Let $H_1$ and $ H_2$ Hilbert space and $T:H_1\rightarrow{H_2}$ a compact operator. Shows $N(T)^\perp \subseteq H_1$ is a subspace separable of $H_1$. indeed as $N(T)^\perp$ is a closed subspace of ...
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1answer
236 views

Graph of symmetric linear map is closed

A homework problem: Let $H$ be a Hilbert space. Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$). Show that $S$ is bounded. My attempt: I'd ...
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1answer
19 views

double Orthogonal complement is equal to topological closure

So I'm in an advanced Linear Algebra class and we just moved into Hilbert spaces and such, and I'm struggling with this question. Let $A$ be a nonempty subset of a Hilbert space $H$. Denote by ...
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1answer
34 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 } ...
4
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1answer
73 views

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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1answer
27 views

Does Convergence of Maps Evaluated at Points Imply Convergence in Operator Norm?

Suppose that I have $T,T_n \in B_H$, for some Hilbert space $H$. Is the following implication true? $$ \|(T-T_n)x\| \rightarrow 0 \ \forall x\in H \ \Rightarrow \ \|T-T_n\| \rightarrow 0, \ \text{ie} ...
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1answer
1k views

How to prove that an operator is compact?

Consider $T\colon\ell^2\to\ell^2$ an operator such that $Te_k=\lambda_k e_k$ with $\lambda_k\to 0$ as $k \to \infty$ how to prove that it is compact?
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1answer
41 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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1answer
59 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
38 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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1answer
99 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 ...
34
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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
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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18 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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0answers
12 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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1answer
18 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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30 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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43 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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1answer
34 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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3answers
156 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
55 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
43 views

Prove that T is compact

If $H$ is a Hilbert space with basis $\{\varphi_{k}\}^{\infty}_{k=1}$, how do I show that the operator $T$ defined by $T(\varphi_{k})=\frac{1}{k}\varphi_{k+1}$ is compact and has no eigenvectors? ...
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0answers
34 views

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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0answers
10 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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1answer
179 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
68 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
75 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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2answers
38 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 ...
3
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1answer
16 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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1answer
39 views

Orthogonal Complements of polynomials in $L^2[0,1]$

I have two very difficult questions in my home work in function analysis, in which I have two calculate the complements of the following sets, in $L^2[0,1]$: All polynomials in the variable $x^2$ ...
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42 views

Does the orthogonal projection theorem guarantees uniqueness of the projected space?

Given a Hilbert space $H$, and linear map $P:H \to H$ such that $P^2=P$ and for every $x\in H$ : $\|Px\| \le \|x\|$, there is a closed linear-subspace $M$ such that $P=P_M$, the projection on $M$. My ...
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1answer
68 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 ...
2
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1answer
60 views

Proving that if $<Ax,x>=0$ for every $x$, then $A$ is the zero operator

I feel kind of dumb but I needed this little claim as a part of a proof I'm writing, and I figured out that I'd better just ask, since I could not find the correct algebraic manipulation needed in ...
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1answer
42 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 ...
2
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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?