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

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1answer
16 views

some detail calculation on the proof of equivalence of norms

We say that two norm $\|x\|_1$ and $\|x\|_2$ on a vector space $X$ are said to be equivalent if there exists $K>0$ and $M>0$ such that $$ K\|x\|_1\le \|x\|_2\le M\|x\|_1 $$ Prove that on a ...
3
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3answers
72 views

Why are function spaces generally infinite dimensional

The other day, I was trying to explain some concepts in Fourier analysis and wavelets to my girlfriend (an electrical engineering student) and obviously, the concept of Lebesgue integration came up in ...
0
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1answer
44 views

Approximate point spectrum of a normal operator

how can I show the following theorem? Let $H$ a Hilbert space and $T:H \to H$ a linear, continuos and normal operator. Then for every $\lambda \in \sigma(T)$ there exists a sequence $(x_n)_{n \in ...
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1answer
25 views

Spectral projections, additivity

Let $K$ be a positive operator on a Hilbert space $H$. $Q_1$ and $Q_2$ are projections such that $Q_1\perp Q_2$. Is $$ E^{Q_1K Q_1} (1,\infty) + E^{Q_2K Q_2} (1,\infty) =E^{Q_1K Q_1 +Q_2K Q_2} ...
1
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0answers
20 views

Two definitions of the operator $\exp(x)$ in $L^2(\mathbb R)$

The operator $x$ acts on a dense subspace of $L^2(\mathbb R)$ and is not bounded. So if we define $\exp(x)$ via the power series $\sum_{n=0}^\infty \frac {x^n}{n!}$, convergence will not follow in the ...
3
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0answers
34 views

Show that for the triples $V \subset H \subset V^{*}$, the following are true

Let $H$ be a Hilbert space equipped with scalar product $(,)$ and the corresponding norm $|\cdot|$. Let $V \subset H$ be a linear subspace that is dense in $H$. Assume that $V$ is a Banach space for ...
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1answer
26 views

Explicit example of two non-isomorphic Hilbert spaces with the same algebraic dimension [duplicate]

I´m wondering if there exist a vector space A and inner products: $\langle\cdot{,}\cdot\rangle_1$ and $\langle\cdot{,}\cdot\rangle_2$, such that: $\big( A,\langle\cdot{,}\cdot\rangle_1 \big)$ and ...
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0answers
29 views

Calculate the trace of $T_nL$ where $L\in L(H)$, $T\in L(H,L(H))$ and $T_n:=\langle T,e_n\rangle_H$ for some ONB $(e_n)_n$ of a Hilbert space $H$

Let$^1$ $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb R$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $T\in\mathfrak L\left(H,\mathfrak L\left(H\right)\right)$ ...
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0answers
20 views

precompact operators in a Hilbert space [functional analysis]

I've linked to a Theorem (from H&N's Applied Functional Analysis) whose proof I'm trying to understand (I asked a question about the previous chunk of the proof yesterday). The theorem is ...
1
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1answer
22 views

Operator groups

In $H := L^2(\mathbb{R}, \lambda)$ Hilbert-space, the following two, one-variable operator groups are given: $$(U_s f)(x):=f(x-s)$$ $$(V_s f)(x):=e^{is x} f(x)$$ $f \in H, s \in \mathbb{R}$. a, ...
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0answers
40 views

What would be an arrow in category of Hilbert space?

Let $H,K$ be Hilbert spaces. Let $S$ be a Hilbert basis for $H$. (Which means that $S$ is orthogonal and the span of $S$ is dense in $H$.) For each arrow $S\rightarrow K$, there exists a unique ...
4
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0answers
50 views

Operator continuity on Hilbert space

Let $A: H \to H$ be a linear operator on Hilbert space $H$, and let $\{\alpha_n\}_{n = 1}^{\infty} \subset \mathbb{R}$ converges to nonzero number. Prove that if the series $\sum_{n = 1}^{\infty} ...
0
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1answer
17 views

If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional?

If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional, for any $\lambda$ in A's spectrum? P.S: What I know now is that the spectrum of A is discrete.
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0answers
11 views

Example for injective and surjective bounded and unbounded operator

I am looking for some bounded and unbounded densely defined operators on a real Hilbert space $H$, let say $A:D(A)(\subset H)\to H$, that are one-to-one but they are not onto. I am wondering whether ...
3
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0answers
25 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 ...
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 = ( ...
3
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3answers
38 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
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2answers
49 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
25 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 ...
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0answers
33 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\| = ...
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
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1answer
20 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
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2answers
21 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
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0answers
15 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 ...
0
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1answer
19 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
14 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 ...
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0answers
25 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
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1answer
27 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 ...
1
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1answer
19 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 ...
-2
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0answers
13 views

Some property of a sequence in Hilbert space [duplicate]

Let $y_1, y_2, \cdots$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1, y_2, \cdots, y_n\}.$ Assume that $||y_{n+1}|| \leq || y -y_{n+1}||$ for each $y \in V_n$ for $n = 1, 2, ...
0
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3answers
31 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
13 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 ...
0
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0answers
32 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 $(u,v)=\int \int u(x,y)\overline{v(x,y)} dx dy$ ...
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0answers
39 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
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1answer
27 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
30 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
29 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 ...
1
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1answer
26 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
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2answers
27 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
46 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
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1answer
37 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 ...
1
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1answer
29 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
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1answer
15 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
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1answer
28 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 ...
0
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2answers
18 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*} ...
0
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0answers
19 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 ...
1
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1answer
22 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
98 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
33 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 ...