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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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
votes
1answer
337 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 ...
1
vote
1answer
92 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 ...
3
votes
3answers
262 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 = ...
1
vote
1answer
320 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: $T^2$ = $TT^*=0$ ...
1
vote
1answer
146 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
votes
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 $[0,1]$? $\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$
2
votes
3answers
394 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?
0
votes
1answer
792 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 ...
1
vote
1answer
412 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 ...
1
vote
0answers
232 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 ...
1
vote
1answer
457 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?
1
vote
1answer
87 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 ...
3
votes
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
votes
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
396 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 ...
1
vote
2answers
309 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 $\{f_{n}\}$. ...
5
votes
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 ...
7
votes
1answer
665 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 ...
1
vote
1answer
342 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)
4
votes
2answers
231 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
votes
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 ...
0
votes
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 ...
8
votes
1answer
370 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 ...
2
votes
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 $f$ ...
2
votes
0answers
148 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 ...
10
votes
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 ...
0
votes
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 ...
1
vote
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?
2
votes
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?
2
votes
2answers
617 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: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\sum_n |\langle x, e_n \rangle |^2 = \|x\|^2$
2
votes
1answer
496 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
397 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
345 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
326 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
241 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
492 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
105 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
147 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
116 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
123 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
259 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
634 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
110 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
455 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 ...