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

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78 views

Hilbert space question

Let $\{x_n\}$ be a sequence of pairwise orthogonal vectors in a Hilbert space $H$. Prove that the following are equivalent: a) $\displaystyle\sum_{n=1}^\infty \|x_n\|^2<\infty$ b) ...
2
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1answer
449 views

Bounded linear operator on a Hilbert space

I am having a bit of difficulty with the following homework problem. Let $\{x_n\}$ be an orthonormal basis in a Hilbert space $V$ over $\mathbb{C}$ and let $\{c_n\}_{n \in \mathbb{N}}$ be a fixed ...
3
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1answer
84 views

Function space in QM

I need to understand how one can think of a function as a vector (in Hilbert space, more specifically) so I can follow along QM texts. I've read this question's answers, but they were uninspiring to ...
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1answer
431 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
2
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1answer
100 views

A question about the proof in functional analysis

I'm now reading Pazy's book about the semi-group operator. To prove the existence of the solution of KdV equation. He define the Hilbert space $H^s(\mathbb{R})$ $$ \Vert ...
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3answers
1k views

Question on weak convergence ( Example).

Can anybody tell me why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$. I can't see how $\sin(nx)$ can converge? Explanation with any other example will be nice as well.
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1answer
240 views

Orthonormal basis implies orthogonal basis!

If $\{\frac{f_{n}}{\|f_{n}\|}\}_{n\in I}$ is an orthonormal basis for a separable Hilbert space $H$, and $\{f_{n}\}_{n\in I}$ is a complete and orthogonal set in $H$, is it true that $\{f_{n}\}_{n\in ...
4
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2answers
888 views

How to show a compact, closed-range operator on an infinite-dimensional Hilbert space has finite rank, without using the open-mapping theorem?

If $H$ is an $\infty$-dimensional Hilbert space and $T:H\to{H}$ is a compact operator with closed range, how do I show that $T$ has finite rank, without using the open-mapping theorem? (The ...
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1answer
216 views

Why are only Sobolev spaces with certain exponents Hilbert Space?

I would like to know why $W^{k,2} (\Omega) $ is a Hilbert space , why is it impossible to define inner product in other Sobolev spaces, ie exponent $\ge2$ . Here $||u||_{W^{k,2} (\Omega)} $ = ...
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1answer
471 views

Poincaré inequality in unbounded domain

Help me please, how can I to show that Poincaré inequality in unbounded domain doesn't holds? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
3
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2answers
304 views

The image of orthonormal basis under compact operator

I need a help to prove that statement: if $\{e_n\}$ an orthonormal basis in Hilbert space $H$ and $A$ is a compact operator from $H$ to $H$, then $Ae_n\rightarrow 0$. Thx for any help.
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1answer
400 views

Square root of compact operator

I'm trying to solve a functional analysis problem A self-adjoint non-negative operator $A$ on a Hilbert space $H$ is compact if and only if its $\sqrt{A}$ is compact.
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0answers
120 views

Is inversion sequentially continuous in SOT?

Let $A_n \overset{SOT}{\to} A$ where $A$ is invertible. Does $A_n^{-1} \overset{SOT}{\to} A^{-1}$? Does $A_n^{-1} \overset{WOT}{\to} A^{-1}$? EDIT: Forgot to mention $\{A,A_n\}\in\mathscr{B(H)}$ ...
3
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2answers
118 views

closedness of image of closed, unbounded operator

I want to prove the following: Suppose $\mathcal{H}_1$ and $\mathcal{H_2}$ are Hilbert spaces and let $T: \mathcal{D} \rightarrow \mathcal{H}_2$ be a closed operator, where $\mathcal{D} \subset ...
2
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1answer
333 views

Derivative of Convex Functional

Suppose that $H$ is a real Hilbert space and that $f:H \to \mathbb{R}$ is differentiable in the Frechet sense. Then we can think of the derivative as a function $f': H \to H^* = H$. Suppose that this ...
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1answer
91 views

Changing of integration and operator

I have a question which maybe looks very simple: Let $T$ be an orthogonal projection on a Hilbert space $H$. If $g(x,u)\in H$, for all $u\in \mathbb R$, and the inner product is defined by $$\langle ...
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3answers
251 views

Maximal Value of Integral

Calculate the maximal value of $\int_{-1}^1g(x)x^3 \, \mathrm{d}x$, where $g$ is subject to the conditions $\int_{-1}^1g(x)\, \mathrm{d}x = 0;\;\;\;\;\;\;\;$ $\int_{-1}^1g(x)x^2\, \mathrm{d}x = ...
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1answer
317 views

Compact Self Adjoint operator on a Hilbert Space

Let $T$ be a continuous and bounded self-adjoint compact operator on a Hilbert space H. I want to prove that if $T^2=0$, then $T=0$. Is there any thing wrong with the following: $T^2$ = $TT^*=0$ ...
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1answer
144 views

Checking axioms for inner product

I'm going through a question checking that an inner product satisfies the inner product axioms. I have a Hilbert space $H=C[-1,1]$ and for $f,g\in H$ the inner product is defined as $$\langle ...
4
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3answers
88 views

$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ for functions in $C([0,1])$?

Why does the following hold for continuous functions on $[0,1]$? $\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$
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3answers
391 views

$C[0,1]$ is NOT a Banach Space w.r.t $\|\cdot\|_2$

I'm trying to find a cauchy sequence in $C[0,1]$ that converges under $\|\cdot\|_2$ to a limit which isn't continuous. Any ideas?
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1answer
750 views

Operator Norm and Hilbert-Schmidt

I am trying to prove $\|T\| \leq \|T\|_{HS}$ I understand everything up until the following two lines, could somebody please explain why $\| Tx \| \leq \|T \|_{HS}$ $\|x\|$ implies that $\| T ...
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1answer
387 views

Proof by induction of triangle inequality in Hilbert space.

I've made proof by induction over $n$ for triangle inequality : $\left \| x+y \right \|_{e}\leq \left \| x \right \|_{e}+\left \| y \right \|_{e}$ ,where $\left \| x \right ...
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0answers
230 views

Concept of Hilbert triple

I am trying to understand the Hilbert triple $V \subset H \subset V^*$, where $V$ and $H$ are Hilbert spaces and the star denotes the dual space. Eg: $H^1 \subset L^2 \subset H^{-1}.$ If $V \subset ...
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1answer
444 views

Hilbert Spaces and Closed Subspaces

Let $H$ be a Hilbert Space, and $M$ a closed subspace. Is it true that $H = M \bigoplus M^{\perp}$ Does this hold if $M$ is not closed? Or only if $H$ is finite/infinite dimensional?
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1answer
86 views

Prove that the space is not complete

Let $X$ be a separable space with infinite dimension, let $(\cdot,\cdot)$ and $\|\cdot \|$ be the scalar product and the norm of $X$, and $\{e_n\}_n$ be an orthonormal basis of $X$. We define ...
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0answers
154 views

Is there a deeper connection between the two Riesz's Representation Theorems?

I have been reading Kreyszig's Functional Analysis when I encountered two versions of Riesz's Representation Theorems: (1) Every bounded linear functional $f$ on a Hilbert space $H$ can be ...
2
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1answer
182 views

Operator Norm $\| T\|$

Would somebody mind explaining why if $T$ is a continuous and bounded operator on a Hilbert space $H$, we have $$\|T\| = 1 \;\;\;\Rightarrow \;\;\;\|Te_n \| = \|e_n\|\;\;\mbox{for all }\;\;x\in H$$ ...
2
votes
2answers
162 views

Is $L^2(D)$ separable?

Let $D$ be a bounded connected open subset of $R^n$ and $μ$ is a finite measure on $D$, say the Lebesgue measure. Is $L_2(μ)$ separable? Is a bounded sequence $\{f_k\}$ of $L^2(μ)$ pre-compact?
4
votes
1answer
392 views

Weak convergence

Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $V,W$ be two closed subspaces. For $x_0\in H$ we may define the sequence of projections $$x_{2n+1}=P_W(x_{2n}), \qquad ...
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2answers
308 views

Hilbert space operators, relation between trace, rank and range

If $A\colon H\to S$ is a bounded operator on a Hilbert space $H$, and $S\subset H$. It is known that $\operatorname{trace}(A)=\sum_{n} \langle Af_n,f_n\rangle$ for any orthonormal basis $\{f_{n}\}$. ...
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2answers
173 views

$L^{2}$ functions

Let $f(x)$ be a continuous function for all $x\in \mathbb R$, such that $f\in L^{2}(\mathbb R)$ (i.e., $\int_{-\infty}^{\infty}|f(x)|^{2}dx<\infty$), and define $$f_{o}(x):=\sup_{|x-y|\leq ...
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1answer
658 views

Is the right shift operator bounded?

I was reading my lecture notes for functional analysis when I came across the following statement: Let $(e_{n})$ be a total orthonormal sequence in a separable Hilbert space H. The right shift ...
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1answer
323 views

The exponent of self-adjoint operator

If $X$ is a Hilbert space and $A$ is an unbounded self-adjoint operator on $X$, is it necessarily that $A^2$ is self-adjoint as well?(admittedly, $A^2$ is densely defined)
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2answers
228 views

Why is the numerical range of a self-adjoint operator an interval?

I was reviewing for a test for functional analysis when I came across the following statement: Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. Then the numerical range of it is ...
3
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1answer
82 views

comparison between spaces

There a lot of function spaces and would be nice if somebody can correct me if I am wrong in comparing a few. I want to compare $C^2,L^2,W^{2,2}$ (continuous up to third derivative, Hilbert space of ...
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1answer
51 views

Dimension of a set and its closure are equal in an Inner-product space?

I want to show that, given a subset $M$ of an Inner Product space $X$. If $M$ is a total set then, $M^\perp=\{0\}$. Which I have shown using the completion of $X$, which will be a Hilbert Space. And ...
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1answer
354 views

Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.

I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces. I was wondering what is the physical ...
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1answer
169 views

how to show $f$ attains a minimum?

Let $H$ be a Hilbert space and let $f\colon H\rightarrow \mathbb{R}$ be a continuous convex function such that $f(x_n)\rightarrow\infty$ whenever $\lVert x_n\rVert\to\infty$. We need to show that $f$ ...
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0answers
146 views

Frames and completeness

Let $H$ be a separable Hilbert space. It is known that if $\{f_{n}\}$ is a frame then it is complete, but the converse is not true. In which cases the converse will be true, i.e. When a complete ...
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2answers
1k views

The direct sum of two closed subspace is closed? (Hilbert space)

I know that if $X$ is a Banach space, then, the direct sum of two closed subspace $X_1$ and $X_2$ is not necessarily closed. But what if $X$ is Hilbert? I assume there is something to do with the ...
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1answer
55 views

Finding a vector in a n.l.s.

Let $X$ be a normed linear space and $Y$ a closed proper subspace. Prove that for all $\varepsilon > 0$, there is an $x \in X$ with $\|x\| = 1$ and such that $\|x − y\| ≥ 1 − \varepsilon$ for all ...
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2answers
162 views

Is it true that $\|f \|_{L^{p-1} }\leq \|f\|_{L^{p}}$?

Is this true? $$ \|f \|_{L^{p-1} }\leq \|f\|_{L^{p}}\;\; $$ Specifically I know $\;\;\|f\|_{L^{2}} \leq \|f\|_{L^{\infty}}$ $\;$ but I can't figure out why?
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1answer
80 views

Finding an ON basis of $L_2$

The set $\{f_n : n \in \mathbb{Z}\}$ with $f_n(x) = e^{2πinx}$ forms an orthonormal basis of the complex space $L_2([0,1])$. I understand why its ON but not why its a basis?
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2answers
599 views

Why is Parseval's Equality and Bessel's Inequality Different?

Bessel's Inequality: $\sum_n |\langle x, e_n \rangle |^2 \leq \|x\|^2$ Parseval: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\sum_n |\langle x, e_n \rangle |^2 = \|x\|^2$
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1answer
471 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 ...
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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.
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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 ...
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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 $S((\ker S)^{\perp})$?
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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 ...