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

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3
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32 views

Showing a C* Algebra contains a compact operator

In my functional analysis class we are currently dealing with C* Algebras, and I just met this problem: Let $ \mathbb{H} $ be a separable Hilbert space, and suppose we have $ A \subset B(\mathbb{H}...
3
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0answers
31 views

Proving/ Disproving that a set is compact in $l^2$

How can I prove or disprove that the following set in the real sequence space $l^2$ ( equipped with the norm $||(X_1,X_2,...)||_2 = \sqrt {\sum_{i=1}^{\infty} X_i^2}$ ) , is compact? $$ A = ( (X_1,...
3
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3answers
45 views

multiplication of finite sum (inner product space)

I am having difficulty to understand the first line of the proof of theorem 3.22 below. (taken from a linear analysis book) Why need to be different index, i.e. $m,n$ when multiplying the two sums? ...
5
votes
2answers
50 views

For a normal operator is it true that $\|T^*T^2\| = \|T^3\|$?

For a normal operator is it always true that $\|T^*T^2\| = \|T^3\|$? See the accepted answer for the case in a Hilbert space Update: how about $\|T^*T^2\| = \|T\|^3$ in a Hilbert space
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0answers
31 views

Is $l^2 \cong l^2 \otimes l^2 \cong l^2 \oplus l^2$?

I'm trying to learn about tensor products of Hilbert spaces and started to wonder if $l^2 \cong l^2 \otimes l^2 \cong l^2 \oplus l^2$? If $(e_n)$ denotes the standard basis, in the first case, it ...
0
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1answer
9 views

Can a span of an orthonormal subset be embedded into $l^2$?

Rudin - RCA p.85 Let $H$ be a Hilbert space and $\beta$ be an orthonormal subset of $H$. Let $T:H\rightarrow l^2(\beta)$ be the continuous linear transformation such that $T(x)(v)=(x , v)$ for $x\in ...
2
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0answers
35 views

Hilbert Space, showing a sequence in Cauchy

Suppose $X$ is a Hilbert space, $M\subset X$ is a closed subspace and $y\notin M$. Let $d = \inf\{ \|x-y\|:x\in M\}$ show that if $\{x_n\}_{1}^{\infty}$ and $\lim_{n\rightarrow \infty}\|x_n - y\| = d$ ...
1
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1answer
22 views

Operator Sum: Selfadjoint

Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$A:\mathcal{D}A\to\mathcal{H}:\quad A=A^{**}$$ Does it follow that: $$S:=\overline{A+A^*}:\quad S=S^*$$ (Rigorous proof?) Densely ...
0
votes
1answer
21 views

Limit of sesquilinear forms is a sesquilinear form

Suppose $P_n$ is a monotone sequences of orthogonal projections in a complex Hilbert space $\mathcal{H}$. I want to show that the limit of the sesquilinear forms defined by: $\Gamma_n(x,y)=(x,P_n y)$ ...
3
votes
2answers
25 views

Monotone sequence of orthogonal projections on a complex Hilbert space

Suppose $P_n$ is a monotone sequence of orthogonal projections on a complex Hilbert space $\mathcal{H}$, i.e. $V_n= Im(P_n)$ is a decreasing or increasing sequence of subspaces and $P_n^\star=P_n$ and ...
1
vote
0answers
16 views

Characterization of noncommutative $L^2$-spaces as ordered vector spaces

If $M$ is a von Neumann algebra and $\tau\colon M_+\to[0,\infty]$ is a normal, semi-finite, faithful trace, the associated GNS Hilbert space is the completion of $\{x\in M\mid \tau(x^\ast x)<\infty\...
0
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1answer
20 views

Orthonormal basis for Hilbert space

Usually, an orthonormal basis for Hilbert space means an orthonormal subset which unconditionally spans the whole space. However, I'm curious whether there exists an orthonormal basis for every ...
1
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0answers
20 views

Separable Hilbert space convergence problem

Suppose that $\{x_j \}_{1}^{\infty}$ is a sequence of separable Hilbert space $X$ and that $\|x_j\| \leq 1$ for all $j$. Show that there is a subsequence $\{x_{j_k} \}_{k=1}^{\infty}$ such that for ...
0
votes
0answers
29 views

Does $S^{\perp\perp}=\overline{S}$?

I'm trying to prove that $S^{\perp\perp} = \overline{S}$, where S is a subspace of a Hilbert Space, where $S^{\perp\perp} = \{f: <f,g> = 0, \forall g\in S^\perp\}$ and $\overline{S}$ is the ...
2
votes
1answer
34 views

Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_n$, what's the second derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle f,...
1
vote
1answer
25 views

Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_{n\in\mathbb N}$, what's the derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle f,...
0
votes
3answers
36 views

If $S$ is a subset of a Hilbert space $H$ then $S^\perp$ is closed?

Is this a true statement? (I found it as a theorem in a paper) If $S$ is a subset of a Hilbert space $H$ then $S^\perp$ is closed. If it were true then $(S^\perp)^\perp$ would be closed, that is ...
0
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0answers
14 views

Gaussian Hilbert spaces indexed by a Hilbert space

Let $H$ a real Hilbert space. Then, there is a real Gaussian Hilbert space $G$ indexed by $H$. I know this result is a consequence of Kolmogorov Extension Theorem, but I have not idea of how begin....
0
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0answers
35 views

Adjoint of differential operator

I would like to find the adjoint of the operator $T_a$ ($a\in \mathbb{C}$) on $ \mathcal{H}=L^{2}(\mathbb{R}^{2},dxdy)$ with $(u,v)=\int \int u(x,y)\overline{v(x,y)} dx dy$ $$T_a(u)=ia(y\partial_xu-x\...
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0answers
48 views

The space of sequences which are eventually zero in $l^2$ is not a Hilbert space.

Define $V$ to be the space of sequences which are eventually zero, i.e. $$V=\bigcup_{N=1}^{\infty}\{(x_n)_{n=1}^{\infty}\in l^2: x_n=0 \; \text{for}\; n\ge N\}.$$ Is $V$ a Hilbert space with ...
1
vote
1answer
29 views

How do outer products differ from tensor products?

From my naive understanding, an outer product appears to be a particular case of a tensor product, but applied to the "simple" case of tensor products of vectors. However, in quantum mechanics there ...
0
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1answer
32 views

Finite Dimensional Hilbert Space

A while ago someone asked this question. I really like what the accepted answer is trying to do. But, I am having trouble figuring out his justification for the first line in the proof: $$\bigcup_{x \...
1
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1answer
30 views

Definition of inner product space

In the definition, we defined linearity in the first argument, Hermitian symmetry. And these two imply anti-linearity in second argument. Is it equivalent, if I cancel the Hermitian symmetry and only ...
2
votes
1answer
28 views

If $(U,〈\;⋅\;,\;⋅\;〉),H$ are Hilbert spaces, $W\in U$, $Y\in H$, $Z\in L(U, H)$ and $f\in L(H,L(H,\mathbb R))$, then $〈Y,fZW〉=〈ZW,fY〉$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be Hilbert spaces $W\in U$, $Y\in H$ and $Z\in\mathfrak L(U, H)$$^1$ $f\in\mathfrak L\left(H,\mathfrak L\left(H,\mathbb R\right)\right)$ How ...
0
votes
2answers
29 views

Let $C$ be a convex closed nonempty subset of a Hilbert space $H$. Show that there is a unique element in $C$ with the minimum norm.

Let $H$ be a Hilbert space and $C \subset H $ be a convex, closed and nonempty subset of $H$. Prove that there exists a unique $x_0\in C$ with minimum norm among the elements of $C$. I don't know ...
1
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1answer
49 views

Sequence in hilbert space, mutually orthogonal vectors

Let $y_1,y_2,\cdots$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1,\cdots,y_n\}$. Assume that $||y_{n+1}||\leq ||y-y_{n+1}||$ for all $y\in V_n$ for $n=1,2,3,\cdots$. Show ...
2
votes
1answer
40 views

If $H$ is a Hilbert space, are we able to identify the derivative ${\rm D}f(x)$ at some $x\in H$ of a differentiable $f\in H'$ with an element of $H$?

I'm confused about some equation I've seen in a book and want to write down some thoughts. I would appreciate, if somebody could tell me whether I'm terribly mistaken or not: Let $(H,\langle\;\...
1
vote
1answer
34 views

$L^2$ convergence of Taylor series of a holomorphic function

I am reading Otto Forster's book "Lecture on Riemann surfaces" and on pages 109-110, he introduces the space $L^2(D,\mathcal{O})$ of holomorphic square-integrable functions $f:D\to \mathbb{C}$ (where $...
1
vote
1answer
16 views

positive operator, projection on Hilbert,$Q|T|Q \ge |QTQ|?$

Let $T$ be an operator on a Hilbert space $H$. And $Q$ be a projection. Whether $$Q|T|Q \ge |QTQ|?$$ Obviously, if $T$ is positive, then $Q|T|Q = |QTQ|$. Also, there are some $T$ such that $QTQ=0$ ...
0
votes
1answer
33 views

Lipschitz Homeomorphism and $\mathcal{l_p}$ spaces

While solving a homework that asked me to find two metric spaces that are not lipschitz homeomorphic it crossed my mind the question (assuming to have in $\mathcal{l^n}$ the metric $|x|$=$\sqrt[n]{\...
0
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2answers
19 views

summing inner product of orthonomal basis

I need some help with some very basic linear algebra when doing calculations in inner product space. Here is a line I got lost when reading a book... \begin{align*} (x,e_m)=\left(\sum_{n=1}^k\...
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0answers
24 views

Fredholm index in Calkin Algebra

Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space, let $\mathcal{B}\left(\mathcal{H}\right)$ be the Banach algebra of bounded linear operators and $\mathcal{K}\left(\mathcal{H}\right)...
1
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1answer
30 views

Compact Operator with Infinite rank Doesn't have a Closed Image

Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space. Claim: A compact operator $T$ which has infinite-rank has an image that isn't closed. I'm trying to prove this claim but I'm ...
0
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0answers
120 views

Several questions about the proof of this lemma

I am reading this paper. I have Several questions about the proof of the lemma 3.3. How to Prove 1,2,3,4?
0
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2answers
37 views

Prove that $(A^\bot)^\bot = \overline{\operatorname{span}A}$

I proved that $(A^\bot)^\bot \supset \overline{\operatorname{span}A}$. To prove the reverse inclusion, let $x \in (A^\bot)^\bot$, then $\langle x, y\rangle = 0$ for all $y \in A^\bot$. How can I ...
0
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0answers
14 views

Parameterizing the set of subquotients of a Hilbert space

If you want to parameterize the set of subspaces of a finite-dimensional Hilbert space $V$, and naturally induce a norm derived from $V$ on the resulting moduli space, the classical approach is to ...
2
votes
2answers
39 views

Spectrum of an unbounded operator

Consider a densely defined unbounded operator $A_0:D(A_0)(\subset{H})\to H$ which has the following properties: 1- Symmetric, $\langle A_0x,y\rangle=\langle x,A_0y \rangle$ 2- Positive, $\langle ...
1
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1answer
20 views

spectral projection

Let $T$ be a self-adjoint operator on a Hilbert space $H$. $P$ is a projection on $H$. Let $E^{|PTP|}(1,\infty)$ be a spectral projection of $|PTP|$. My question is: whether $E^{|PTP|}(1,\infty) \le P$...
0
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1answer
25 views

If $U,H$ are Hilbert spaces, $Q$ is an operator on $U$ and $U_0:=Q^{\frac 12}U$, find an expression for the norm of Hilbert-Schmidt operators $U_0→H$

Let $U$ and $H$ be Hilbert spaces $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for ...
1
vote
1answer
35 views

If $Q$ is an operator on a Hilbert space $U$ and $(e_n)_{n\in\mathbb N}$ is an ONB of $U$ with $Qe_n=λ_ne_n$, then $Q^{-1}e_n=\frac 1{λ_n}e_n$

Let $U$ be a Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for all }n\...
3
votes
1answer
43 views

completion of $C^{\infty}\left(S\right)$ is $L^2(S)$?

I have the space of infinitely derivable functions, i.e. $C^{\infty}\left(S\right)$ with the following inner product $$\left\langle f,g\right\rangle =\intop_{0}^{2\pi}\intop_{0}^{2\pi}\left(\overline{...
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0answers
23 views

Tensor products in separable Hilbert spaces

In a variety of scientific publications, I have come across the use of tensor products of random Hilbert-Schmidt operators defined on separable Hilbert spaces. Let us introduce some notations. Define ...
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0answers
44 views

Self adjoint operators on Hilbert spaces are bounded

I think I have a proof that if $A: H\rightarrow H$ is a self adjoint operator on a Hilbert space $H$, then $A$ is bounded: We can use the closed graph theorem. Let $x_n \rightarrow x$ and $Ax_n \...
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0answers
31 views

Eigenfunction of a selft-adjoint operator?

Let $A = \int_{0}^{\infty} \lambda dE(\lambda)$ be the spectral decomposition of a selft-adjoint operator $A$ on a Hilbert space $H$. Then the restriction operator $P_{\lambda}$ for $A$ is defined by $...
0
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0answers
16 views

Positive definite functions coming from finite dimensional representations

Let $G$ be a topological group, let $\mathcal{H}$ be a complex Hilbert space, let $v\in\mathcal{H}$ be a nonzero vector, and let $\rho:G\rightarrow \mathcal{U}(\mathcal{H})$ be a unitary ...
3
votes
1answer
71 views

What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
0
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0answers
28 views

$\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert space

I want to describe $\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert spaces; $\mathbb{C}^n\otimes \mathbb{C}^m$ is endowed with the scalar product $\langle x\otimes y, x'\otimes y'\...
0
votes
0answers
17 views

Validity of: if $H$ is pre-Hilbert and $V$ is a f.d. vector subspace, then $f^*$ exists and is unique

We were recalling some basic stuff about Hilbert spaces in class, and the professor gave the remark: If $H$ is a Hilbert space and $V$ is a finite dimensional vector subspace of $H$, then for all $f\...
3
votes
1answer
42 views

Why is it true that the multiplication operator in a reproducing kernel Hilbert space is always continuous?

In my functional analysis I was met with this seemingly trivial theorem on RKHS If $ \mathbb{H} $ is a reproducing Kernel Hilbert Space and we have a multiplier $ \phi $ meaning it satisfies $ \...
1
vote
1answer
50 views

Spectrum of a positive operator

We know that if $A$ is a self-adjoint unbounded operator on a Hilbert space $(H;\left<.,.\right>)$ then $\sigma(A) \subset \mathbb R$. Now, how it can be shown that if $A$ is more positive i.e. ...