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

learn more… | top users | synonyms

5
votes
1answer
1k views

How to prove this integral operator is compact?

$T_kf=\int K(x,y)f(y)dy$ where $K(x,y)=\frac{\phi(x)\phi(y)}{|x-y|^{n-\alpha}}$ $\phi(x)$ is a smooth function on a compact support. $f$ is defined on $R^n$ and $K$ is defined on $R^n\times R^n$ ...
5
votes
2answers
210 views

Orthogonality checking in Kreyszig exercise

Let $H$ be inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $\lVert \cdot\rVert$. Let $x,y \in H$. Would you help me to prove that $\langle x,y\rangle=0$ if and only if ...
5
votes
1answer
570 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?
5
votes
1answer
352 views

Bounded operator from a Hilbert space to $\ell^1$ is compact

Let $H$ be any Hilbert space. How can we prove that any bounded linear operator $T\colon H \to \ell^1$ is compact? If we use the fact that the space $\ell^1$ has Schur property (norm and weak ...
5
votes
1answer
442 views

Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces? For the sum we have the notion of a direct integral, here.
5
votes
1answer
168 views

Addition of Unbounded Operators

Let $H$ be a (separable complex) Hilbert space and let $A$ and $B$ be two densely-defined, maximally-defined linear operators on $H$ with domains $D(A)$ and $D(B)$ respectively. (By maximall-defined, ...
5
votes
1answer
344 views

Is this functional weakly continuous?

Take a $C^1$ function $G \colon \mathbb{R}\to \mathbb{R}$ and define a functional $$\mathcal{G}(u)=\int_0^1G(u(t))\, dt, \quad u \in H^1(0, 1).$$ We then have $\mathcal{G}\in C^1\big(H^1(0, 1)\to ...
5
votes
1answer
63 views

Inequivalent Hilbert norms on given vector space

Suppose we have a vector space $X$. Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two different complete norms on $X$ s.th. $X$ equipped with $\|\cdot\|_j, \ j\in\{1,2\}$ is a Hilbert space. Are there ...
5
votes
1answer
94 views

Why do dagger categories supposedly capture the structure of a Hilbert space?

A dagger functor is a contravariant endofunctor $(\;)^\dagger$ satisfying $X^\dagger = X$ on objects and $f^{\dagger\dagger}$ on morphisms. It is supposed to model adjoint maps on Hilbert spaces, and ...
5
votes
1answer
211 views

Proof involving strongly continuous semigroups.

Let $ (T(t))_{t \geq 0} $ be a $ C_{0} $-semigroup on a Hilbert space $ X $ with an infinitesimal generator $ A $, and let $ \rho \in (0,1) $. I want to prove that $ \displaystyle \sup_{t \geq 0} \| ...
5
votes
3answers
187 views

Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
5
votes
1answer
480 views

The sup norm on $C[0,1]$ is not equivalent to another one, induced by some inner product

Let $\mathrm{C}[0,1]$ be the space of continuous functions $[0,1]\rightarrow \mathbb{R}$ endowed with the norm $||x||_{\infty}=\mathrm{max}_{t\in [0,1]}|x(t)|$. It is easy to verify that this norm is ...
5
votes
2answers
1k views

Every Hilbert space has an orthonomal basis - using Zorn's Lemma

The problem is to prove that every Hilbert space has a orthonormal basis. We are given Zorn's Lemma, which is taken as an axiom of set theory: Lemma If X is a nonempty partially ordered set with the ...
5
votes
2answers
1k views

Inner product on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$, respectively. Then $H_1\otimes H_2$ is at least a pre-Hilbert space (we are ...
5
votes
2answers
113 views

Prob. 8, Sec. 3.5 in Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications

Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications Prob. 8, Sec. 3.5 $\DeclareMathOperator{\span}{span}$Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$, and let $M = ...
5
votes
1answer
294 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 $L : H \to \mathbb{R}^n$ ...
5
votes
1answer
313 views

Show that a subspace of l2 is not complete

I would like to know if this exercise is correct. Let $\Bbb R^\infty=\{x:\Bbb N\rightarrow \Bbb R: \exists n \text{ such that}\quad x(k)=0 \quad \forall k\geq n\}$. Show that $(\Bbb R^\infty, \| ...
5
votes
1answer
107 views

Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1 $ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
5
votes
1answer
666 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 ...
5
votes
1answer
485 views

Bounded operator and Compactness problem

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a ...
5
votes
1answer
73 views

Space of Jordan curves

The space of square-integrable functions $f:[0,1]\rightarrow\mathbb{R}$ is well conceivable: it's essentially an $\infty$-dimensional Euclidean space (the Hilbert space $L^2$) with well interpretable ...
5
votes
1answer
840 views

Showing the sum of orthogonal projections with orthogonal ranges is also an orthogonal projection

Show that if $P$ and $Q$ are two orthogonal projections with orthogonal ranges, then $P+Q$ is also an orthogonal projection. First I need to show $(P+Q)^\ast = P+Q$. I am thinking that since ...
5
votes
2answers
607 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 ...
5
votes
1answer
27 views

Linear span in the intersection of Hilbert spaces

Let $V$ be a vector space. Assume $H_1$ and $H_2$ are subspaces of $V$, and that both $H_1$ and $H_2$ are Hilbert spaces with inner-products $\langle \cdot, \cdot\rangle_1$ and $\langle ...
5
votes
1answer
66 views

Showing a certain subspace of Hilbert space is dense

Let H be the Hilbert space of square-summable sequences of reals. A few years ago I thought I had proved that the subspace Z of real sequences with only finitely many nonzero terms, such that they ...
5
votes
3answers
230 views

Is this space a Hilbert Space?

I have a space of continuously differentiable functions on [a, b] with the dot product defined in this way: $ x \cdot y = \int_a^b \! [x(t)y(t) + x'(t)y'(t)] \, \mathrm{d}t. $ Is this space a Hilbert ...
5
votes
1answer
91 views

Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
5
votes
1answer
167 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
5
votes
2answers
197 views

Does a symmetric operator on a Hilbert space have a symmetric adjoint?

Suppose we have a linear operator $T$, densely-defined on some Hilbert space. If $T$ is symmetric (i.e., $T^*$ extends $T$: notationally, $T\subseteq T^*$) does it follow that $T^*$ is also symmetric ...
5
votes
1answer
234 views

A baby version of the Stein-Cotlar almost-orthogonality lemma

The following is an exercise from Stein and Shakarchi's Real Analysis. Suppose $\{T_k\}$ is a collection of bounded operators on a Hilbert space $H$, each with norm at most $1$. Suppose also that ...
5
votes
1answer
84 views

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
5
votes
3answers
1k 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 $\lambda_{\text{ count}}$ ...
5
votes
2answers
1k 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 ...
5
votes
2answers
174 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 ...
5
votes
1answer
50 views

$5$ questions on the definition of the Gelfand triple

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb F\in\left\{\mathbb R,\mathbb C\right\}$, $\left\|\;\cdot\;\right\|$ be the norm induced by ...
5
votes
1answer
72 views

Equivalent formulations: pure contraction

I want to prove the following equivalence: let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. TFAE: $\|Tx\|<\|x\|$ for each $x\in H\setminus\{0\}$ $\|T\|\leq1$ and ...
5
votes
1answer
178 views

Example: operator injective, then the adjoint is NOT surjective

Let $T: V \rightarrow W$ be a bounded operator on normed spaces $V,W$. Now, there is a unique adjoint operator $T': W' \rightarrow V'$ defined by $T'(\alpha) = \alpha \circ T$. In finite dimensional ...
5
votes
1answer
295 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
5
votes
1answer
173 views

Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
5
votes
1answer
599 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
5
votes
2answers
319 views

Brownian Motion Covariance: max instead of min

It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
5
votes
1answer
185 views

Show $T$ is compact

$H$ and $K$ are Hilbert Spaces, $(u_n)$ and $(v_n)$ are sequences in $H$ and $K$ respectively. $\sum_{n=1}^{n=\infty} \|u_n\|\|v_n\| $ converges. $T\colon H\rightarrow K$ is defined by ...
5
votes
1answer
625 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 ...
5
votes
1answer
63 views

Every complete orthonormal set in a Hilbert space $H$ is an orthonormal basis, if and only if $H$ is finite dimensional.

Show that any orthonormal set in a Hilbert space $H$ is linearly independent, and use this to show that $H$ is finite dimensional if and only if every complete orthonormal set is an orthonormal basis. ...
5
votes
0answers
74 views

If the Fourier transform of a measure is zero then the measure is zero

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ be such that $$\hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle} \Bbb d \mu _{(y)} = 0, \ \forall ...
5
votes
0answers
49 views

Progressed : Convergence problem in Hilbert Space and necessity of inner product

******** PROGRESS : so thanks to Ian's great comment I can get by the proof and that completeness is necessary but I need to know does this hold for general Banach spaces that are not Hilbert spaces? ...
5
votes
0answers
57 views

Over ZF does “every non-seperable Hilbert space has an orthonormal basis” imply “there exists a non-Lebesgue measurable set”?

I know from this question that it's an open problem whether or not the existence of a dense orthonomral basis for every real or complex Hilbert space $(\text{B}_\text{orth})$ implies the axiom of ...
5
votes
0answers
66 views

An inner product on the dual space of a non-complete inner product space?

As is well known, for any Hilbert space $V$, there is a natural inner product on the continuous dual. (the space of all continuous linear functionals). Is there a way to endow an inner product on ...
5
votes
0answers
66 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
5
votes
0answers
83 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...