0
votes
1answer
28 views

Positive-definite function on a group function on a group

I have quite a hard time understanding the definition of positive-definite functions that is based on Hilbert spaces, the one that I read from Wiki; it does not exactly specify that how $H$ relates to ...
1
vote
1answer
74 views

Weak convergence in $C[0,1]$

For a uniformly bounded sequence $(f_n)$ in $C[0,1]$, show that $f_n$ converges weakly to $0$ $\iff $ $\lim \limits_{n \to \infty} f_n(y) =0$ for all $y \in [0,1]$ Is the equivalence true if we do ...
0
votes
1answer
30 views

Is this subspace dense in $L^{2}(\Omega,\mu)$

Let $(\Omega,\mu)$ be a measure space, and let $X=L^{2}(\Omega,\mu)$ be the complex Hilbert space of square-integrable complex measurable functions on $\Omega$. (Each $f \in L^{2}$ is an equivalence ...
0
votes
0answers
9 views

Integral representation of joint projection valued measures

Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...
0
votes
1answer
30 views

When $M$ is closed $M^\perp$ is one-dimensional vector space

If $M=\{ x: Lx=0\}$, where $L$ is continuous linear functional on $H$ (Hilbert Space). Prove that $M^\perp$ is vector space of one-dimensional unless $M= H$. I know $M$ is closed so that $M^\perp$ is ...
5
votes
1answer
70 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 ...
0
votes
0answers
25 views

Bounded $L^1$ functions subgroup of $L^2$

I'm currently looking at statistics and the characteristic function. And the claim is that the characteristic function must exist for every probability distribution since every probability ...
5
votes
1answer
68 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
1
vote
1answer
32 views

Is every closed set $K\subseteq \mathbb{C}$ the essential range of a measurable function?

For a complex-valued function $h$ on a measure space $(S,\Sigma, \mu)$, the $\textit{essential range}$ of $h$ is the set of all $\lambda \in \mathbb{C}$ such that for all $\epsilon >0$ the ...
4
votes
1answer
42 views

Finding an orthonormal basis from an existing one in a Hilbert space

Suppose we are given a separable Hilbert space $H$ with countable orthonormal basis $\{e_n\}$. Suppose we are given an orthonormal set $\{f_n\}$ such that $\sum\|e_n-f_n\| < 1$. How do we prove ...
2
votes
2answers
66 views

Convergence of sums using Hilbert space techniques [duplicate]

Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_nb_n < \infty$ for any sequence $b_n$ satisfying $\sum_{n=1}^{\infty}b_n^2 < \infty$. Prove that ...
0
votes
1answer
32 views

I want to show one norm is less than or equal to another norm on C([0,1])

Let $|| \ ||_1$ be the norm on $C([0,1])$ defined by $||f||_1 = \int_0^1|f(t)|dt$. a) Show that $||f||_1 \le ||f||_{[0,1]}$ b) Are $|| \ ||_1$ and $|| \ ||_{[0,1]}$ equivalent? For part a) I think ...
0
votes
1answer
25 views

If $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$

Suppose $H$ is a Hilbert space. Is it true that if $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$ for any fixed $h\in H$? Certainly if $x_n\to ...
1
vote
0answers
39 views

Finding the norm of a linear operation.

I am reading A course in real analysis by John McDonald, on page 530, it says "it is easy to show $|||J|||=1$" where $J$ is the linear operation $J:C([0,1])\rightarrow C([0,1])$, defined by $J(f)(x) = ...
1
vote
0answers
53 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
2
votes
3answers
127 views

help with showing completeness

Let $\left\{H_n\right\}_{n=1}^\infty$ be a sequence of Hilbert spaces and let $H=\left\{\left\{x_n\right\}:x_n\in H_n, \sum ||x_n||^2<\infty \right\}$. Define the inner product as ...
2
votes
0answers
86 views

The support of Gaussian measure in Hilbert Space $L^2(S^1)$ with covariance $(1-\Delta)^{-1}$

Let $\mu$ be Gaussian measure defined on Hilbert space $\mathcal{H}=L^2(S^1)$ ($S^1$ - circle) by formula $$ \int e^{(f,g)} d\mu(f) = e^{-\tfrac{1}{2}(g,C g) }. $$ The covariance operator $C$ is ...
1
vote
0answers
41 views

Operators on a Hilbert space question

For a Borel measure $\mu$ define $\langle S_\mu x,y\rangle=\int_H\langle x,z\rangle \langle y,z\rangle \mu(z)$. An exercise in my book that I am reading says that I could find a $\mu$ s.t. $S_\mu$ ...
1
vote
1answer
70 views

Is every Hilbert space an $L^2$ space

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you
0
votes
1answer
85 views

what is the advantage of having countable dense subset?

what is the advantage of having countable dense subset (for example of the set $L^2([0,1])$, if i have to prove weak convergence ? edit: to prove is that every sequence $(f_n)_n$ with ...
3
votes
1answer
60 views

Hilbert space - probability measure: st. norm. variables

I am considering the following homework. Let $\Omega=\ell_2$ be the Hilbert space of square summable sequences, $\mathcal A$ the Borel $\sigma$-algebra and $\{e_n: n\mbox{ natural}\}$ the natural ...
4
votes
2answers
122 views

Showing $\mathcal{H}$ is a hilbert space.

So this is an early exercise in Conway's A Course In Functional Analysis. I'm trying to get to grips with this upto open mapping and closed graph to see if I want to do any more functional analysis. ...
2
votes
2answers
192 views

Stone Weierstrass Overkill in the Measurable Setting?

If $\mu$ is Lebesgue measure on the Borel sigma algebra $\mathcal{B}$ of $[0,1]$. Establishing that the linear span in $L^{2}([0,1]\times[0,1],\mathcal{B}\otimes\mathcal{B},d(\mu \times \mu))$ of the ...
3
votes
1answer
125 views

Is my proof that a function is measurable correct?

Let $V$ be separable and Hilbert. Let $\mathcal V = L^2(0,T;V)$. Assume for each $t \in [0,T]$, $$a(t;\cdot,\cdot):V \times V \to \mathbb{R}$$ is continuous and bilinear. Or equivalently, we have ...
5
votes
0answers
158 views

Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...
4
votes
2answers
152 views

about weak convergence in $L^{2}(0,T;H)$

I am trying to do an exercise and if the affirmation below is true, my exercise is done . This is the affirmation : Affirmation : Let $H$ a Hilbert space and suppose $u_k$ converges weakly to $u$ ...
2
votes
1answer
76 views

How to prove that the set $e^{inx}$ is closed with respect to this measure?

Why is the set $\{e^{inx}\}$ closed in $L^2(d\nu)$, where $d\nu(x)=(1+|h(x)|)dx+|ds(x)|$? $d\nu$ is defined in this proof I am struggling to understand, where $\mu$ is a complex Borel measure on ...
2
votes
2answers
170 views

Proof of the Riesz Representation Theorem

Theorem: Let $F$ be a continuous linear functional on the Hilbert space $H$, then $\exists !$ (exists one and only one) $y \in H$ such that $F(x) = (x,y)$ for $x\in H$. Proof: Uniqueness: ...
3
votes
0answers
164 views

Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory, Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
2
votes
0answers
130 views

Testing whether a finite measure is absolutely continuous with respect to Lebesgue measure using wavelets

I've been working through Fundamentals of Stochastic Filtering (Bain, Crisan) and am a little perplexed by the following (initially) seemingly straightforward exercise and its given solution. We are ...
6
votes
1answer
183 views

Measure on a separable Hilbert space

Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have $$ ...
1
vote
1answer
104 views

Mutually orthogonal subspaces of $L^{2}(X,\mu)$

Let $(X,\mathcal{M},\mu)$ be a measure space. If $E\in\mathcal{M}$, we identify $L^{2}(E,\mu)$ with the subspace of $L^{2}(X,\mu)$ consisting of functions that vanish outside $E$. If $\{E_{n}\}$ is a ...
2
votes
1answer
295 views

Conditions on the weight function for Hermite polynomials' completeness

Hermite polynomials form a complete orthonormal basis of the weighted $L^2(\mathbb R, w \; dx)$ space, with inner product $$ \langle f, g \rangle_w = \int_\mathbb R f(x) g(x) \; w(x) dx. $$ A short ...
4
votes
0answers
141 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 ...
3
votes
1answer
153 views

Compute spectral/projection-valued measures explicitly?

Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following: ...
2
votes
1answer
83 views

Is it ok to switch the limits in $L_2$?

Let $(X,B,\mu)$ be a probability space and let $U$ be a unitary operator on $L_2(X,B,\mu)$. Suppose that $g_n$ is a convergent sequence in $L_2(X,B,\mu)$, $g_n\rightarrow g$. Suppose also that there ...
2
votes
1answer
86 views

Proving $L^2$ convergence (application of dominated convergence?)

For any $f\in L^2(\mathbb{R}^d)$ prove \begin{align}\left\lVert \int_{\mathbb{R}^d} e^{i |x-y|^2}f(y) dy-\int_{\mathbb{R}^d}e^{i |x-y|^2} e^{-|y|^2/a}f(y) dy \right\rVert_{L^2} \rightarrow 0\ \ \ ...
4
votes
3answers
618 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 $\lambda_{\text{ count}}$ ...
1
vote
1answer
140 views

Hilbert spaces other than $L^2$

From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ ...
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?
3
votes
1answer
307 views

Orthonormal basis for product $L^2$ space

Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite measure spaces such that $L^2(X)$ and $L^2(Y)$ . Let $\{f_n\}$ be an orthonormal basis for $L^2(X)$ and let $\{g_m\}$ be an orthonormal basis for ...
2
votes
1answer
105 views

Hilbert sum of $L_2(X_\nu,\mu_\nu)$ spaces.

Let $\{(X_\nu,\mu_\nu):\nu\in\Lambda\}$ be a family of measurable spaces. Is it true that $\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}$ isometrically isomorphic to ...
9
votes
4answers
770 views

Measure on Hilbert Space

On $\mathbb{R}^n$, we of course have the usual Lebesgue meausre. In many ways, separable, infinite-dimesional Hilbert space is the most natural generalization of $\mathbb{R}^n$ to ...
8
votes
2answers
468 views

Haar's base for $L^2[0,1]$

$\newcommand{\span}{\operatorname{span}}$ Define $e_{0,0}\equiv 1$, and for all $n\in \mathbb{N}$ $$e_{n,k}=\begin{cases} 2^{n/2} &\text{if } \frac{k-1}{2^n}\leq x\lt \frac{k-\frac{1}{2}}{2^n}\\ ...
7
votes
1answer
245 views

Existence of the Pettis integral

This is related to a question of MO: A question on the integral of Hilbert valued functions. I'm sure it's easy, but I cannot think right now, so I thought I'd ask. Let $f:[0,1]\rightarrow H$ be a ...
1
vote
3answers
177 views

Why is it useful to express PDE solutions as $L^2$-convergent series?

The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...
2
votes
3answers
632 views

Orthonormal basis in $L^2(\Omega)$

In the one dimension case, where $\Omega\subseteq{\bf R}$ is a bounded domain, for example $\Omega=[0,2\pi]$, one can find the orthonormal basis for $L^2(\Omega)$: $$\{e_n\}_{n\in {\bf Z}}$$ where ...