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

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4
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2answers
210 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
4
votes
1answer
127 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...
4
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1answer
55 views

Norms arising from all representations of *-algebras

It is common that in order to obtain a $C^*$-algebra from a $^*$-algebra $A$ one defines a norm on $A$ by $$\|x\|=\sup\{\|\pi(x)\|\,|\,\pi\ \text{is a }^*\text{-representation of }A\}.$$ However, I ...
4
votes
1answer
247 views

Prove or disprove existence of a sequence converging weakly to $0$ in an infinite dim Hilbert space

This is a problem on an old analysis qual, the prompt is: "Prove or give a counter example: if $H$ is an infinite dimensional Hilbert space and $0$ is the zero vector in $H$, then there exists a ...
4
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1answer
70 views

Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?

Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
4
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1answer
138 views

Multiplication operator and trace class

Suppose we work in $H=l^2(\Bbb{N})$ and suppose the multiplication operator $T_f$ such that $T_f\psi=f\psi$ and $f:\Bbb{N}\rightarrow \Bbb{C}$. We denote by $B_1(H)$ the trace class of operators. ...
4
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1answer
263 views

Showing the basis of a Hilbert Space have the same cardinality

I am trying to show that if we have two orthnormal families, $\{a_i\}_{i\in K}$ and $\{b_j\}_{j\in S}$ and these are the basis of some Hilbert Space H then they have the same carnality. So If I ...
4
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1answer
56 views

Help with proof of closedness of a set

Let $u_n$ be a sequence in Hilbert space such that $\|u_n\|=1$ for all $n$, and $\langle u_n|u_m\rangle=0$ whenever $n\neq m$. Why is the following set closed: $\{0\}\cup \{u_n \mid n\geq 1\}$? Thanks ...
4
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1answer
122 views

Showing a representation is irreducible by showing that a degenerate subspace has codimension one.

Throughout $\phi$ be a continuous character from a locally compact abelian group $G$ to the circle. I'm trying to understand this implication. Basically we want to show that a certain representation ...
4
votes
1answer
398 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 ...
4
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1answer
141 views

An interesting condition for the completeness of an orthonormal system in $ L^2([0,1]) $

Let $\{u_n\}$ be an orthonormal system in $L^2([0,1])$, prove that $\{u_n\}$ is complete iff $$ \sum_{n=1}^\infty \intop_0^1 \left|\intop_0^x u_n(t)\;dt\right|^2 dx = 1/2.$$ It should be noted that ...
4
votes
2answers
507 views

Orthonormal basis in Hilbert space

I an reading a book about functional analysis and there is one thing i really don't understand. Let $\mathcal{H}$ be a Hilbert space. And $U \subset \mathcal{H}$ a closed subspace. Is it possible to ...
4
votes
3answers
2k views

How to prove that square-summable sequences form a Hilbert space?

Let $l^2$ be the set of sequences $x = (x_n)_{n\in\mathbb{N}}$ ($x_n \in \mathbb{C}$) such that $\sum_{k\in\mathbb{N}} \left|x_k\right|^2 < \infty$, how can I prove that $l^2$ is a Hilbert space ...
4
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1answer
33 views

Isomorphism between Hilbert spaces

I want to show that the function $$ L^2(\Omega,\mathcal{O})\longrightarrow L^2(\widetilde{\Omega},\mathcal{O}) \colon f \longmapsto f|_{\widetilde{\Omega}}$$ is a isomorphism, where ...
4
votes
1answer
31 views

Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
4
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1answer
225 views

functions orthogonal to the exponential Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
4
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1answer
164 views

Is $L^2(\Omega)$ dense in $H^{-1}(\Omega)$?

Is it true that $L^2(\Omega)$, identified with its own dual, is dense in $H^{-1}(\Omega)$? $H^{-1}(\Omega)$ is the dual of $H^1_0(\Omega)$ and $H^1_0(\Omega)$ is the $H^1$-closure of smooth functions ...
4
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1answer
227 views

An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...
4
votes
1answer
209 views

How to determine a operator norm?

How to solve following: In Hilbert space $W_2^1=\{f:[0,1]\rightarrow \mathbb{C}|f\in AC[0,1], f'\in L^2[0,1]\}$ with scalar product $(f,g)=\int_0^1 f\overline{g}dx+\int_0^1 f'\overline{g'}dx$ is ...
4
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2answers
345 views

Reproducing Kernel Hilbert Spaces for Dummies

I am in the middle of some machine learning paper that states that for function $f$, imposing the norm constraint, $\|f \|=1$, corresponds to an orthogonal projection onto the direction selected in ...
4
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1answer
409 views

Isomorphic Hilbert spaces

As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
4
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1answer
400 views

Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
4
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3answers
649 views

Question about example of non-separable Hilbert space

I have come across the following example of a non-separable Hilbert space: Example 2.84. Let $I$ be a set, equipped with the discrete topology and the counting measure $\lambda_{\text{ count}}$ ...
4
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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 ...
4
votes
1answer
213 views

Set of all compact operators $K(H)$ is the unique ideal in $B(H)$?

I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
4
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0answers
45 views

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} ...
4
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0answers
103 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
4
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0answers
100 views

Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
4
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0answers
86 views

Ultraweak topology on Banach spaces

If $X$ and $Y$ are Banach spaces with $Y$ reflexive, then the space $\mathcal{B}(X,Y)$ of bounded operators from $X$ to $Y$ is the dual of the projective tensor product of $X$ and $Y^{*}$. As in the ...
4
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1answer
37 views

Is this a correct argument for why this space of sequences is a Hilbert space?

I am assigned the following problem (a piece of Exercise 6.5 in Brezis's book): Let $(\lambda_n)$ be a sequence of positive numbers such that $\lim_{n\to\infty}\lambda_n=+\infty$. Let $V$ be the ...
4
votes
1answer
71 views

In a separable Hilbert space, can you write an operator from $\mathcal H$ to $\mathcal H$ as a column-finite matrix?

In this question, we are representing an operator $T$ as a matrix with respect to an orthonormal basis $\left\{e_n : n \in \mathbb{N}\right\}$. To do so, we let $t_{ij} = \langle T(e_j),e_i\rangle$. ...
4
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0answers
92 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
4
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0answers
119 views

Is this projection operator onto a subspace of a Hilbert space bounded?

(I copy and paste and edit from Is this operator bounded? Hilbert space projection, my question is almost the same) Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense and ...
4
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0answers
61 views

Differentiating an infinite series in Hilbert space

Suppose $H$ is separable Hilbert space and $w_j$ is a basis. Suppose we have $h=\sum a_j(t)w_j$ an infinite sum where the coefficients are functions of $t$. The sum makes sense in the sense that the ...
4
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0answers
158 views

Hilbert spaces - equivalent norm

Let $H$ be a Hilbert space with a norm $\| \cdot \|_1$. Let $\| \cdot \|_2$ be another norm on $H$ which is equivalent with $\| \cdot \|_1$. It is easy to see that $(H, \| \cdot \|_2)$ is a Banach ...
4
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0answers
90 views

Concerning unbounded linear operators on a Hilbert space

Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
4
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0answers
187 views

Inverse of Identity plus Volterra operator

consider the following operator or $L_2(0,1)$, $(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial. I am trying to construct the inverse of this ...
4
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0answers
143 views

What is the dual space of $C([0,T];X)$ ($X$ Hilbert space)?

What is the dual space of $C([0,T];X)$, where $X$ is a Hilbert space? Is it $\operatorname{BV}([0,T]; X^*)$? As we know, for $C([0,T])$, the dual space is $\operatorname{BV}([0,T])$, but when it is ...
4
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1answer
133 views

Canonical field of Hilbert spaces in Dixmier; Plancherel Theorem

I'm working through the proof of Plancherel's Theorem in $C^{*}$-algebras by Dixmier, section 18.8. For the most part, I'm happy with it, although I have one problem. From Dixmier, I have the proof ...
4
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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 ...
4
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0answers
144 views

When functions, analytically continued, carry over certain properties

Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
3
votes
1answer
2k views

Volterra Operator is compact but has no eigenvalue

Volterra operator is defined as operator $V:L^2[0,1]\rightarrow L^2[0,1]$ by \begin{eqnarray} (V)(f(x))=\int_0^xf(y)dy \end{eqnarray} Would you help me to prove that this operator is compact but has ...
3
votes
2answers
376 views

A continuity condition for a bilinear form on a Hilbert space

Let $H$ be a real Hilbert space, and let $B : H \times H \to \mathbb{R}$ be bilinear and symmetric. Suppose there is a constant $C$ such that for all $x \in H$, $|B(x,x)| \le C \|x\|^2$. Must $B$ be ...
3
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1answer
182 views

Proving Inner Product Space

Let $E=C^1 [a,b]$ be the space of all continuously differentiable functions. For $f,g \in E$ define $$ \langle f,g \rangle \ = \ \int_a^b f'(x) \ g'(x) \ dx$$ Is $\langle f,g \rangle$ an inner product ...
3
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2answers
51 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 ...
3
votes
2answers
59 views

If $x_n \to x$ in Hilbert space, does $|x_n| \leq C|x|$?

If $x_n \to x$ in a Hilbert space $X$, is it true that $|x_n| \leq C|x|$ for all $n$ for some constant $C$? It is true for $n$ big enough. But not sure about all $n$.
3
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3answers
523 views

What is a Hilbert space?

I've just seen a question about Hilbert Subspaces. This made me wonder what a Hilbert space is. Can anyone explain in layman's terms?
3
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2answers
689 views

$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
3
votes
1answer
125 views

Dense subspace of $\ell^2$

Is the set \begin{align} A=\left\{a=(a_1,a_2,\dots)\in\ell^2 \ \ \lvert \ \ \sum_{k=1}^\infty \frac{a_n}{n}=0 \right\}\subset\ell^2 \end{align} dense in $\ell^2$ Is the following argument ...
3
votes
1answer
46 views

When does an operator commute with another operator given by a series?

Suppose $B$ is a bounded operator on some Hilbert space $\mathcal{H}$, given by a series of the form $$ B = I + \sum^\infty_{k = 1} c_k(I - A)^k $$ where $A$ is a given bounded operator on ...