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

learn more… | top users | synonyms

0
votes
2answers
66 views

Stone's Theorem Integral

Given a finite Borel measure $\mu$ and a Hilbert space $\mathcal{H}$. Consider a strongly unitary group $U:\mathbb{R}\to\mathcal{B}(\mathcal{H})$. Introduce for simple vector-valued functions: ...
0
votes
1answer
22 views

Recapitulated: Stone's Theorem Integral

This problem grew out from: Stone's Theorem Integral For a definition and a nonexample: Generalized Riemann Integral: Definition Generalized Riemann Integral: Nonexample The Riemann integral ...
0
votes
0answers
15 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 ...
1
vote
0answers
7 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 ...
0
votes
1answer
17 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 ...
1
vote
2answers
24 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 ...
0
votes
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 $X$. ...
1
vote
2answers
84 views

Spectral Measures: Subspace Decomposition

Attention This thread has been split into this one and: Spectral Measures: Subspace Characterization Problem Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. ...
0
votes
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 ...
2
votes
0answers
22 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 ...
1
vote
0answers
39 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 ...
1
vote
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 ...
5
votes
3answers
113 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 ...
0
votes
1answer
19 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 ...
1
vote
1answer
53 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 ...
0
votes
1answer
37 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? ...
1
vote
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 ...
1
vote
0answers
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 ...
0
votes
1answer
164 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 ...
3
votes
2answers
57 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 ...
1
vote
1answer
70 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 ...
2
votes
2answers
33 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
votes
1answer
11 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 ...
1
vote
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 $x^2$ ...
0
votes
0answers
40 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 ...
1
vote
1answer
63 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
votes
1answer
58 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 ...
1
vote
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 ...
2
votes
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?
0
votes
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\| = ...
0
votes
0answers
37 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 ...
2
votes
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." ...
0
votes
0answers
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.
0
votes
0answers
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 ...
0
votes
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 ...
1
vote
3answers
76 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 ...
4
votes
2answers
96 views

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 ...
2
votes
0answers
31 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 ...
1
vote
0answers
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 $\sup$ ...
1
vote
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~
0
votes
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.
0
votes
1answer
37 views

Compact Operator on Hilbert Space

How do I show that the range of $\lambda I-T$ is all of $H$ (Hilbert Space) if and only if the null-space $\bar\lambda I-T^{\ast}$ is trivial? Thanks!
3
votes
1answer
59 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 $H$ ...
0
votes
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 ...
0
votes
1answer
34 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
1
vote
1answer
44 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 ...
0
votes
2answers
22 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 ...
1
vote
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 ...
1
vote
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: $$\Omega^*\Omega,\Omega\Omega^*$$ ...