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

learn more… | top users | synonyms

2
votes
1answer
472 views

Yet another exercise from Stein's Real Analysis

So I'm stuck at the following result, about compact operators on Hilbert spaces (which I think it's called Fredholm's theorem) from Stein. It's exercise 29 from Chapter 4. Let $T$ be a compact ...
1
vote
2answers
390 views

Proof of Convexity?

Given a positive semidefinite matrix $A$, is $\operatorname{Tr}X^TAX$ a convex function in $X$? Am looking for a proof of convexity or non-convexity, whichever is true.
4
votes
1answer
122 views

Convergence of a series involving inner products

Let $\{A_{j}\}$ be a sequence of bounded operators on a Hilbert space satisfying $\|A_{j}^{\ast}A_{k}\| \leq C_{j - k}$ and $\|A_{j}A_{k}^{\ast}\| \leq C_{j - k}$ where $\sum C_{i} < \infty$. Fix an ...
0
votes
1answer
339 views

Relation between range and kernel of a linear operator

Let $S = I - T$ where $T$ is a compact linear operator on a Hilbert space $H$. Why is it that the range of $S$ is equal to $S((\ker S)^{\perp})$?
3
votes
1answer
324 views

-Almost- self-adjoint bounded operators on Hilbert spaces

I ran into the following question about bounded operators on Hilbert spaces; I could really use some help. It goes like this: Suppose that $\left\{T_{k}\right\}$ is a collection of bounded operators ...
0
votes
1answer
237 views

Common eigenvector of linear operator

Let $T$ and $S$ be two symmetric, compact linear operators on a (separable) Hilbert space $H$ that commute. Why is there at least 1 common eigenvector of $T$ and $S$?
5
votes
1answer
490 views

Exhibiting open covers with no finite subcovers.

How do I exhibit an open cover of the closed unit ball of the following: (a) $X = \ell^2$ (b) $X=C[0,1]$ (c) $X= L^2[0,1]$ that has no finite subcover?
3
votes
2answers
104 views

Limit of Inner Products in Hilbert Space

Let $H$ be an infinite dimensional Hilbert space. Then there exists an orthonormal basis $\{e_{i}\}_{i = 1}^{\infty}$. Suppose we know that $\lim_{k \rightarrow \infty}(f_{k}, e_{j}) = (f, e_{j})$ for ...
2
votes
1answer
145 views

completeness of orthonormal set

I am currently working through some lecture notes on the Geometry of Hilbert spaces, and I am stuck with the following comment: If we are given the inner product space $C([0,1])$ of continuous ...
0
votes
1answer
64 views

Recovering an Operator on $L^2$ Given its Action as a Composition on the Spectrum of $f$

Let $\ell^2 = \ell^2(\mathbb{Z})$ denote the Hilbert space of square summable complex sequences on $\mathbb{Z}$ and suppose that $\sigma:\mathbb{Z} \to \mathbb{Z}$ is a function such that the linear ...
2
votes
1answer
114 views

Characterizations of the form domain for unbounded selfadjoint operators

This question follows from this one and especially from Willie Wong's answer: link. In Reed & Simon's book Methods of modern mathematical physics, vol. I, pag.277, the form domain of a ...
4
votes
0answers
110 views

Relations between spectrum and quadratic forms in the unbounded case

Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
2
votes
1answer
253 views

An orthonormal family in an inner product space

Why does the inner product space $( C[0,1], \| \cdot \|_2$) have an orthonormal family $(e^{\color{red}{2\pi}inx})_{n\in \mathbb{N}}$ ?
1
vote
1answer
128 views

Hilbert space $H$ is strictly smooth

I am trying to show that every Hilbert space $H$ is strictly smooth with modulus of smoothness $\phi_H(t)=\sqrt{1+t^2} -1 $. To show this I think I should show $H$ is uniformly smooth first. ...
12
votes
1answer
629 views

Meaning of “kernel”

In analysis, there are at least three kinds of "kernel" concepts: In probability theory, there is a concept called transition probability, also called probability kernel, from one measure space $X$ ...
2
votes
3answers
164 views

orthonormal basis in $l^{2}$

I need an orthonormal basis in $l^{2}$. One possible choice would be to take as such the sequences $\{1,0,0,0,...\}, \{0,1,0,0,...\}, \{0,0,1,0,...\}$, but I need a basis where only finitely many ...
3
votes
1answer
108 views

Contractive Operator and Realization Theorem

Good morning, I have searched, by using google for a time, a proof of the following theorem : Let $\pmatrix{A&B \\ C&D}\colon H \oplus K \to H\oplus K$ be a contractive operator of a Hilbert ...
5
votes
3answers
442 views

$f$ an isometry from a hilbert space $H$ to itself such that $f(0)=0$ then $f$ linear.

This question was on an exam and I am not sure how to answer it. I mostly tried writing zero in different ways and tried lots of algebra to get something out. I also tried to use the fact that $H$ is ...
2
votes
3answers
624 views

Separability of the space of bounded operators on a Hilbert space

Let $H$ be a (separable) infinite dimensional Hilbert space, and $B(H)$ the space of bounded operators on $H$. Is $B(H)$ separable in the operator norm topology? What about in the strong and weak ...
0
votes
1answer
82 views

Linear dependence

Let $X$ be a Hilbert space and let $f:X\to\mathbb{R}$. Let $M=\{x\in X:f(x)=0\}$ be the nullspace of $f$. Let $M^\perp=\{x\in X:(x,y)=0\text{ for all }y\in M\}$ be the orthogonal complement of ...
12
votes
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?
1
vote
1answer
146 views

Null Space and Range of Particular kind of Operator on Hilbert Space

Let $H$ be the real separable Hilbert space with orthonormal basis $\{e_n\}$ and consider the operator $T:H \times H \to H \times H$ given by $$T(\sum a_ne_n, \sum b_ne_n) = \sum A_n(a_ne_n, ...
3
votes
2answers
836 views

$C[0,1]$ is not Hilbert space

Prove that the space $C[0,1]$ of continuous functions from $[0,1]$ to $\mathbb{R}$ with the inner product $ \langle f,g \rangle =\int_{0}^{1} f(t)g(t)dt \quad $ is not Hilbert space. I know that I ...
2
votes
2answers
461 views

Hilbert Schmidt operators as an ideal in operators.

Let $H$ be a Hilbert space. For $\{e_n\}$ an orthonormal basis of $H$, we call $T\in B(H)$, a Hilbert Schmidt operator if $ \|T\|_2^2:=\sum_n \|Te_n\|^2 <\infty.$ I have seen somewhere before ...
14
votes
3answers
744 views

Intersection between orthogonal complement of a subspace and a set

Consider the normed vector space $E=\mathbb{R}^n$. Define $ P=\{x \in \mathbb{R}^n: x_i \geq 0, \forall i \}$. Let $M$ be a subspace such that $M \cap P = \{0\}$. I want to see that $M^\perp \cap ...
1
vote
0answers
42 views

A bilinear form whose corresponding quadratic form has a larger norm

I know that given a bilinear form $B$ and it's corresponding quadratic form $Q$, assuming that $B$ is complex and bounded, that $\|B\| \le 2\|Q \|$ However, I've failed to manufacture an example of a ...
3
votes
1answer
316 views

Orthonormal basis for product $L^2$ space

Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite measure spaces such that $L^2(X)$ and $L^2(Y)$ . Let $\{f_n\}$ be an orthonormal basis for $L^2(X)$ and let $\{g_m\}$ be an orthonormal basis for ...
2
votes
1answer
176 views

What is the orthogonal complement of this subset of $L^2[0,1]$?

Let $A\subset[0,1]$ be measurable, and let $g\in L^2(A,dx)$. Let $C=\{f\in L^2[0,1]:m\{x\in A:f(x) \ne g(x)\}=0 \}$, that is, the set of functions which are equivalent to $g$ on $A$. Prove that $C$ ...
4
votes
1answer
361 views

Dense subset of a Hilbert space

Let $A$ be a dense subspace of a Hilbert space $H$. Denote $\ell^2$ the Hilbert space of (complex valued) square-summable sequences and denote $\ell^2(H)$ the Hilbert space of $H$-valued ...
9
votes
1answer
297 views

A paradox on Hilbert spaces and their duals

I am making some elementary mistakes here. Could you please help me point out the problems? Thank you very much! Suppose on some space $H$ we have two inner products, which make $H$ after completion ...
7
votes
1answer
208 views

Reproducing Kernel Hilbert Space is dense?

Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. Let $E^*$ be a space of all continuous ...
3
votes
1answer
592 views

Dual of $C[0,1]$, Hilbert space and Riesz representation.

Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. I need help proving the following claim: ...
0
votes
1answer
187 views

Triangle inequality in a Hilbert space

Is there anything wrong with the following: If $\{f_n\}$, $\{g_n\}$ are two sequences of functions in a Hilbert space $H$, then $$\begin{align*} \sqrt{\sum_n |\langle f,f_n \rangle|^2} - ...
1
vote
0answers
89 views

Meaning of $\ker(x)$ when $x$ is an element of a Hilbert space

The follow question arose from the paper: The Hundal Example Revisited We consider a separable Hilbert Space $X$ with countable ortho-normal basis $\{e_n\}_{n=1}^\infty$. The following is an excerpt ...
8
votes
1answer
535 views

What is the use of Spectral Theorem?

Obviously the version for compact and self-adjoint linear operators on Hilbert Spaces is very useful since it decomposes the operators into orthogonal projections. However, the following more general ...
2
votes
1answer
112 views

Extension domains for $W^{1,2}$

I'd like to have some hints for a problem I bumped into some times ago but I was not able to solve (even if I think the most is done...). Before the problem, let me recall some definitions. Let ...
2
votes
1answer
105 views

Hilbert sum of $L_2(X_\nu,\mu_\nu)$ spaces.

Let $\{(X_\nu,\mu_\nu):\nu\in\Lambda\}$ be a family of measurable spaces. Is it true that $\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}$ isometrically isomorphic to ...
4
votes
1answer
533 views

positive invertible operators

I need the following result. I think it's quite obvious but I don't know how to prove that: Let $C, T : \mathcal{H} \rightarrow \mathcal{H}$ be two positive, bounded, self-adjoint, invertible ...
4
votes
1answer
131 views

The union of cyclic subspace is also a cyclic space

Given a separable Hilbert space $H$, $U$ is a unitary operator. A cyclic subspace, denoted as $Z(x)$ for some $x\in H$, is defined as the closure of linear span of $U^nx$, where $n\in \Bbb Z$ is any ...
1
vote
1answer
103 views

The adjoint of an injection

I was googling "Hilbert space" and was reading the associated Wikipedia page when I found this statement confusing : "Let $V$ be a closed subspace of an Hilbert space $H$. Then the inclusion mapping ...
2
votes
1answer
144 views

A question on norm of error vector

Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
4
votes
1answer
160 views

Continuation of Linear Operator in Hilbert spaces

First of all, here is the assignment: Let $X$ be a Hilbert space over $\mathbb{C}$, $V \subseteq X$ be a closed subspace and $f \in L(V, \mathbb{C}) $ a linear continuous operator. Show that ...
1
vote
3answers
435 views

minimum principle in Hilbert space

Minimum principle is following: Let $M$ be a closed convex nonempty subset of Hilbert space. Then there exists $x\in M$ which have a minimum norm. Assume that $M$ is not convex subset. What is a ...
3
votes
2answers
155 views

Minimization problem in Sobolev spaces

This is a homework problem and I don't know how to solve it: Consider the following minimization problem on the real-valued sobolev space $H^{1,2}(\Omega)$ with dimension $n=1$ and $\Omega=(0,1)$: ...
2
votes
1answer
126 views

A question about projection in Hilbert space .

Let $a$ be a non-zero element of an Hilbert space $H$. I try to prove that for every $x\in H$, $$ d(x, \{a\}^{\perp})=\frac{\left|\langle x,a\rangle \right|}{\left\|a\right\|}. $$ So $d(x, ...
6
votes
1answer
139 views

A complete eigenvector basis for the restricted operator

Let $X$ be a (not necessarily bounded) selfadjoint linear operator on a Hilbert space $H$ and let $M$ be a closed subspace such that $X(M) \subset M$. Suppose that $X$ admits an orthonormal basis ...
1
vote
2answers
342 views

Question about limits of weakly convergent sequence in $H^1_0(\Omega)$

Let $H = H_{0}^{1}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ whose boundary $\partial\Omega$ is a smooth manifold. We know that the embedding $$H\hookrightarrow L^s(\Omega)$$ is compact for ...
4
votes
2answers
88 views

Weak convergence as convergence of matrix elements

Let $H$ be a Hilbert space with orthonormal basis $(e_h)_{h \in \mathbb{N}}$ and let $(A_n)_{n \in \mathbb{N}}$ and $A$ be bounded linear operators. We say that $A_n$ converges weakly to $A$ if ...
2
votes
2answers
289 views

Approximating $|x|$ by a linear combination of $1, \cos x, \sin x, \cos 2x, \sin 2x$

Let $\phi(x) = |x|$ for $x \in (-\pi, \pi)$. Suppose we approximate $\phi(x)$ by a linear combination of the functions $\{1, \cos x, \sin x, \cos 2x, \sin 2x\}$. What linear combination of the form: ...
1
vote
0answers
155 views

The “dotplus” $\dotplus$ sign

Any one know what does the dot plus sign $\dotplus$ means? It is used to express a Hilbert space $H= \operatorname{span} \{f\} \dotplus V$, where $V$ is a Hilbert space, and $f$ is an analytic ...