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

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Integration over subsets of the complex plane.

Original Problem: Let $\Omega\subset \mathbb{C}$ be an open set and let $f:\Omega\to\mathbb{C}$ be holomorphic such that $f\in L^{2}(\Omega)$. Show that if $B(z,r)$, the ball of radius $r$ ...
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2answers
57 views

Exercise 23 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 23 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 198). Any help will be much appreciated. Thank you in advance. Suppose $\{T_k\}$ is a collection of bounded ...
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1answer
21 views

Existence of minimum norm solution to linear equation $Tx =y$

Let $T: X \to Y$ be a bounded linear map between Hilbert spaces $(X, \langle \cdot , \cdot \rangle_X)$ and $(Y, \langle \cdot , \cdot \rangle_Y)$ (the Hilbert spaces may be complex or just real ...
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1answer
20 views

Extending mappings on simple tensors

Consider the following situation: Let $H, K$ be Hilbert spaces and let $\Phi$ be some mapping defined on simple tensors in $H\otimes K$ taking values in $B(H\otimes K)$ with the property that each ...
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1answer
54 views

Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
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37 views

Inner Product in Hilbert Space

Let $H$ be a Hilbert space and $\phi_{1}, \dots, \phi_{n} \in H$ are linearly independent vectors. How can we construct the inner product on $H$ such that $\phi_{1}, \dots, \phi_{n}$ become orthogonal ...
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1answer
44 views

Prove that $S$is a closed subspace of $H^2$ invariant under multiplication by $z$. Find the inner function $F$ such that $S=FH^2$

Let ${\alpha_n}$ be a sequence of points in the open unit disc such that $\sum(1-|\alpha_n|)<\infty$. Let $S$ be the set of all functions $f$ in $H^2$ spaces such that $f(\alpha_n)=f'(\alpha_n)=0$ ...
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15 views

Is a complex function really just an infinite dimensional matrix?

I have recently sort of come to the understanding that integrating two functions multiplied together is a sort of infinite dimensional dot product, and I only know this from taking an undergraduate ...
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2answers
76 views

Exercise 34 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 34 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 201): Let K be a Hilbert-Schmidt kernel which is real and symmetric. Then, as we saw, the operator $T$ whose ...
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Problem 8 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein and Shakarchi's Real Analysis

The following is problem 8 from chapter 4 ("Hilbert Spaces: An Introduction") of Stein and Shakarchi's Real Analysis. Suppose $\{t_k\}$ is a collection of bounded operators on a Hilbert space $H$. ...
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41 views

“Almost” Hilbert spaces

This question is a bit (very?) vague. Is there some notion of how "close" a Banach space is to being a Hilbert space? What I have in mind is something like a real or complex valued function on ...
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1answer
33 views

Are there any interesting Hilbert spaces that do not present as function spaces?

I was pondering this question in class earlier: All separable, infinite dimensional Hilbert spaces are isometrically isomorphic. Thus, in particular, any such space is isometrically isomorphic to ...
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1answer
154 views

The span of the orthorgonal projections is norm dense in $B(H)$

This is a question in my functional analysis book. How to use the spectral theorem to prove that the span of the orthogonal projections is norm dense in $B(H)$?
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23 views

Relation between two spectra

This seems like an easy enough computation but I'm stuck! Let $X \in B(H)$ for a Hilbert space $H$ such that $X^{2}=0$, but $X\neq 0$. With respect to the decomposition $H=\text{ker}X \oplus ...
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1answer
39 views
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18 views

Show that the set $\{e^{\pm i (n-1/4)t}: n=\pm 1,\pm2,\pm3,\ldots\}$ is not a basis for $L^2[\pi,\pi]$

Show that the set $\{e^{\pm i (n-1/4)t}: n=\pm 1,\pm2,\pm3,\ldots\}$ is not a basis for $L^2[\pi,\pi]$. (HINT: The series $$\sum_n c_n e^{i\lambda_n t}$$ with $\lambda_n=n-1/4$, diverges in ...
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1answer
29 views

Inner product in Besicovitch space

Besicovitch space is a space constructed in the following way: We take the closure (with respect to the uniform convergence topology) of a linear span: ...
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1answer
21 views

Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 ...
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2k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
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0answers
36 views

Distance to a closed subspace of a Hilbert space in terms of inner product with the unit normal

Let $M$ be a closed subspace of a Hilbert space $H$, and suppose $x_0\in H$ Show that: $$\min(\|m-x_0\|, m\in M)=\max(|\langle x_0,n\rangle|, n\in M^\perp ,\|n\|=1)$$ I know that $|\langle ...
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1answer
35 views

Non-separable Hilbert spaces in duals

A topological space $X$ satisfies the countable chain condition if every family of pairwise disjoint open sets in $X$ is countable. I am looking for a reference to the following fact: Suppose that ...
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1answer
30 views

Prove that the space P is a Hilbert Space.

Prove that the space P of all entire functions of the form $$f(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi(t) e^{-izt} dt,$$ is a Hilbert Space, where $\varphi\in L^2[-\pi,\pi]$. The inner product of ...
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40 views

Invertibility of an operator involving inner product

Let $H$ be a Hilbert space with basis $b_i$. For all $t$, let $f(t;\cdot,\cdot)$ be an inner product on $H$. For each $j$, is $$\int_0^T \sum_{i=1}^\infty f(t,b_i,b_j)x_j(t)=0$$ uniquely solvable for ...
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1answer
37 views

Show an operator is compact if $\sum \|Te_n\| < \infty$

Let $H$ be a separable Hilbert space, define a bounded linear operator $T:H \rightarrow H$, show it is compact if $\sum \|Te_n\|_H < \infty$. My attempt: We show that $T(B)$ is totally ...
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13 views

Does a set of 'm' linearly independent continuous functions constitute a Hilbert Space

If I have a Sobolev space $\mathcal{H}^m[a,b]$ of functions $f : [a,b]\rightarrow\mathbb{R}$ where for all $f \in\mathcal{H}^m[a,b]$, $f$ and all derivatives up to order $m-1$ are absolutely ...
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1answer
29 views

Exercise about an operator (adjoint and spectrum)

Let $y\in c_0$ and define the operator from $l^2 \rightarrow l^2$ as the following $$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$$ I have shown that the operator is continuous, compact and ...
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2answers
43 views

Spectrum of self-adjoint operator on Hilbert space real

My book says that a self-adjoint bounded linear operator $A:H\to H$ on a complex Hilbert (not sure if separability is needed) space has a real spectrum. I guess that the key is in the fact that any ...
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2answers
50 views

How to show that $\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$

Show that $$\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$$ where $a,b,c$ are in some Hilbert space $(H,\langle\cdot,\cdot \rangle)$? I see that we have $$\|a+b\|^2\leq2\|a\|^2 +2 \|b\|^2$$ due to the ...
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1answer
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Continuous Quadratic Form $\implies$ Continuous Sesquilinear Form

Given a Hilbert space $\mathcal{H}$. Consider a quadratic form $q:\mathcal{H}\to\mathbb{C}$. Define its inducing sesquilinear form: $$s:\mathcal{H}\times\mathcal{H}\to\mathbb{C}: ...
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1answer
37 views

Isomorphism between Euclidean space and its conjugate

I know that, if $H$ is a Hilbert space, for any continuous linear functional $f\in H^{\ast}$ there is a unique element $x_0\in H$ such that $\forall x\in H\quad f(x)=\langle x,x_0\rangle$. Moreover, ...
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1answer
16 views

Infinite-dimensional version of Gram matrix is invertible

We all know that a Gram matrix (a matrix with entries that are inner products of basis functions) is a invertible. Suppose I have $a_{ij} = (h_i, h_j)_H$ where the $h_j$ are basis functions of a ...
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3answers
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Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
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1answer
38 views

Eigenfunctions and spectrum of $T:H \to H^*$ where $H$ is a Hilbert space

Let $H$ be a Hilbert space with dual $H^*$. Suppose $T:H \to H^*$ is a linear bounded symmetric operator. (We probably don't want to identify $H$ with $H^*$). Can we talk about the ...
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1answer
28 views

The strong operator limit of a sequence of unitary operators

If $\mathcal H$ is a Hilbert space and $U_n \in B(\mathcal H)$ is a strong-operator convergent sequence of unitary operators, say $U_n\rightarrow U$, is it true that $U$ is unitary? More explicitly, ...
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An idempotent operator is compact if and only if it is of finite rank

Would you help me to solve this problem. Show that an idempotent operator on hilbert space is compact if and only if it has finite rank.
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What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
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3answers
48 views

Selfadjoint Operator: Empty Spectrum

Can a selfadjoint operator have empty spectrum? (As far as I remember, yes; but just to be sure.) The point is that if so then the closure of its spectrum cannot equal the convex hull of its ...
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0answers
88 views

Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
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2answers
31 views

Can we show that $E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$

Let $X,Y,Z$ be some random elements on some Hilbert space $(H,\langle\cdot,\cdot\rangle)$. Can we show that $$E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$$ I can clearly see that $$E\|X-Y\|^2 \leq ...
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1answer
86 views

Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact

Is my proof of this proposition correct ? And is this proposition well known? Proposition: Let $C$ be a closed, bounded, and convex set in a separable Hilbert space $H$. Let $L : H \to \mathbb{R}^n$ ...
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2answers
39 views

Is the distance between disjoint closed convex subsets of a Hilbert space positive? Is it attained?

Let $H$ be an infinite dimensional and separable Hilbert space. Let $A,B$ be infinite, closed and convex subsets of $H$. If $A$ and $B$ are disjoint and if at least one of them is bounded, is the ...
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1answer
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Commutant of a set of operators and norm topology.

In the references I have it's remarked that the commutant $S'$ of a set $S$ in $B(H)$, where $H$ is a Hilbert space, is closed in the weak operator topology. And this is true because if ...
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34 views

Hilbert subspaces of $B(\mathbb{R}^n)$

Apart from the one-dimensional subspaces, what are the Hilbert subspaces of $B(\mathbb{R}^n)$? I'm not even sure if such subspaces exist.
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Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
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1answer
49 views

Inequalities with $\|x-y\|$, $|\langle x,y\rangle|$, and $\sqrt{\|x\|^{2}+\|y\|^{2}}$ in a Hilbert space

Let $H$ be a Hilbert space, and let $\|x\|$ denote the norm of $x\in H$, and $\langle x,y\rangle$ denote the inner product of $x,y\in H$. For $x,y\in H$ let us denote $\alpha(x,y)=\|x-y\|$, ...
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0answers
42 views

A matrix as eigenvalue?

I wonder if some work has been developed on operators in Hilbert space that have the property of having matrices instead of numbers as eigenvalues (the matrices do not necessarily act on vectors in ...
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1answer
27 views

Kernel of the Extension of a Bounded Linear Operator

Suppose $T\colon E\to F$ is a bounded linear operator between Banach spaces. Moreover let $i\colon E\to E’$ be a dense, compact inclusion of $E$ into some other Banach space $E’$. Finally assume that ...
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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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Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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1answer
37 views

If a compact operator satisfies $T^nx\to0$ weakly for all $x$, then $\|T^n\|\to0$

Let $H$ be a real Hilbert space, $T:H\to H$ be a compact operator. Suppose that for every $x\in H$, sequence $(T^n x)_{n\in \mathbb{N}}$ converges weakly to $0$. How to prove that $ ...