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

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8
votes
1answer
135 views

Is $\mathcal{C}([0,1])$ homeomorphic to a Hilbert space?

Let $\mathcal{C}([0,1])$ the Banach space of continuous functions from $[0,1]$ to $\mathbb{C}$. The norm on $\mathcal{C}([0,1])$ is $f \mapsto \| f\|_{\infty}= \sup_{x \in [0,1]} |f(x)|$. Is it ...
2
votes
1answer
101 views

Is my proof correct? I want to show if $V \subset H$ is dense, then $L^2(0,T;V) \subset L^2(0,T;H)$ is dense too.

I want to show that if $V \subset H$ is a dense embedding then $L^2(0,T;V) \subset L^2(0,T;H)$ is dense too. Everything is a Hilbert space. Let $h \in L^2(0,T;H)$. Then $h(t) \in H$ for each $t$. By ...
0
votes
2answers
143 views

Extending bilinear form from subspace to whole space

Let $X$ be a linear subspace of a Hilbert space $Y$. Let $a(\cdot,\cdot):X \times X \to \mathbb{R}$ be bilinear. Suppose I know what $a$ is on $X$. Is there some theorem or other that tells me that ...
2
votes
0answers
45 views

Prove that the sequence is in $\ell^{2}$. [duplicate]

Let $(a_{n})$ be a sequence of complex numbers such that for every $(b_{n})\in \ell^{2}$the series $\sum_{1}^{\infty}a_{n}b_{n}$ converges. Prove that $(a_{n})\in \ell^{2}.$ What I've tried so far is ...
0
votes
1answer
134 views

Intersection of affine subspaces of finite codimension in Hilbert space

I'm wondering whether the following assertion is true: Any two affine subspaces of the same finite codimension in a ($\infty$-dimensional) Hilbert space either are parallel or have nonempty ...
5
votes
2answers
624 views

Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces

Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e: $$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in ...
2
votes
3answers
106 views

The Kernel of unbounded operator in Hilbert space

If $T$ is a densely defined operator from a subspace of a Hilbert space $H$ to a Hilbert space $K$, how to prove that $\mbox{Ker}(T)=\mbox{Ker}(T^*T)$?
1
vote
1answer
57 views

Does a cofinite dimensional subspace of a subspace remain cofinite dimensional upon taking closures?

Let H be a separable, infinite dimensional Hilbert space. Let X and Y be (not necessarily closed) subspaces such that X is a cofinite dimensional subspace of Y. Let X′ be the closure of X and Y′ the ...
7
votes
1answer
108 views

Tight Probability on Hilbert space

I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$ And for any ...
2
votes
2answers
230 views

Unbounded operator $T $ is bounded below when $\overline T$ is bounded

How to prove the following? A densely defined symmetric operator $T$ in Hilbert space $H$ has a closure $\overline T$ which is bounded iff both $T,-T$ are bounded below (there exist constants $c,c' ...
1
vote
0answers
461 views

Closed unit ball in infinite dimensional normed linear space

I have to prove that in any infinite dimension normed linear space we have that the closed unit ball is not compact. I know that I have to construct a sequence such that $||x_n||=1$ and ...
2
votes
1answer
150 views

inner product space and injective -surjective

Let $V$ and $W$ be two finite-dimensional inner product spaces over the same field and let $T\in \mathcal{L}(V,W)\ $ be a linear transformation. Show that $T$ is injective if $T^*$ is surjective.
2
votes
1answer
150 views

Is this set dense in $H^1(\Omega)?$

Is $$V_1 = \{v \in H^1(\Omega) \;:\;f(v) = 0 \text{ on } \partial \Omega\}$$ dense in $H^1(\Omega)$ with the same norm as $H^1(\Omega)?$ Here $f$ is some linear functional so that $V_1$ is also ...
2
votes
1answer
137 views

Are WOT/SOT topologies hereditarily separable?

Just out of curiosity, Are weak and strong operator topologies on $B(H)$ hereditarily separable? In other words, if $S$ is a subset of $B(H)$, where $H$ is a separable Hilbert space, is $S$ ...
2
votes
1answer
96 views

Two inequalities related to norm

We have some difficulties in the following problem: Let $H$ be a real Hilbert space. Find $\alpha>0$ such that $$ \langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq ...
1
vote
1answer
235 views

Point spectrum in Hilbert spaces

Let $H$ be a Hilbert space and and $T\in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e the set of all eigenvalues of T) and let $E$ denote the spectral measure. I'm trying to ...
3
votes
1answer
152 views

proof for a basis in $L^2$

I know, correct me if I am wrong, that the functions $H_n(x)\exp(-x^2/2)$ form a complete basis in $L^2(\mathbb{R},dx)$, where $H_n(x)$ is the $n$th Hermite polynomial. This must be true also for ...
2
votes
0answers
55 views

Find a bounded function with a supporting point

Given, $g(Z)=\operatorname{tr}\phi(Z)$, where $\phi(Z)= Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right) Z$ where $Z$ is a real rectangular matrix with more rows than columns (tall and ...
1
vote
1answer
69 views

A basic result about operators on Hilbert space.

I am studying following result. Let $H$ and $K$ be Hilbert spaces and an operator $A \in B(H, K)$, which has closed range. The spaces $H$ and $K$ have the following orthogonal decompositions: $H = ...
1
vote
0answers
260 views

Orthogonal Projection on hilbert spaces

I found this exercise on a book, I guess it's not hard but don't know what to do. Let $H$ be a Hilbert space and let $P:H \rightarrow H$ be linear. If $P$ is a projection, i.e $P^2 =P$, and ...
2
votes
0answers
114 views

Completeness proof.

I'm getting stuck showing a space is a Hilbert space. For $\Omega$ an open, connected and bounded set in $\Bbb R^2$ with regular boundary $\partial \Omega$, let $V=\{v \in H^1(\Omega)\ ;\ ...
5
votes
1answer
552 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 $U\subset\mathbb{R}^n$ ...
1
vote
1answer
171 views

It is possible to generalize the “real” line to be able to embed $\omega_1$ or any uncountable ordinal into a finite segment of it?

This question is motivated from a previous question, but is in itself independent of it. So, I understand that it is not possible to embed $\omega_1$ or any uncountable ordinal into the real line, ...
1
vote
1answer
727 views

Dual Space as a Hilbert Space

I have this problem: Let $(X, \langle\cdot,\cdot\rangle)$ a Hilbert Space on $\mathbb{R}$ with Riez map $\mathcal{R}:X^{\prime}\rightarrow X$, define $[\cdot,\cdot]:X^{\prime}\times ...
1
vote
1answer
53 views

Question on notation: What does $0 \leq M \leq 1$ mean for a bounded operator $M$?

Let $\mathcal{H}$ be a Hilbert space and let $M\colon \mathcal{H} \rightarrow \mathcal{H}$ be bounded linear operator. I am working through a paper by Roger Godement from the 1950's. In one section ...
5
votes
4answers
2k views

Orthogonal complement of a Hilbert Space

I have this problem: Let $S$ be a subset of a Hilbert $H$ and let $M$ be the closed subspace generated by $S$. Show that $M^{\perp} = S^{\perp}$ $M = (S^{\perp})^{\perp}$ if $V$ is a subspace of ...
6
votes
2answers
1k views

Isomorphisms of inner-product spaces

I think I understand why all finite-dimensional vector spaces over a field $\mathbb{K}$ are isomorphic to $\mathbb{K}^n$. Any linear map $T: V \rightarrow W$ between finite-dimensional vector spaces ...
2
votes
1answer
267 views

Pde problem with Neumann BC's

Let $U \subset\mathbb{R}^n$ be a bounded open set with smooth boundary $\partial U$. Consider the Neumann boundary problem $$-\Delta u +u=f, \quad \left.\frac{\partial u}{\partial ...
2
votes
2answers
77 views

Inner product? Yes or no?

I define an "inner product" on $H_0^2(U)$ where $U \subset R^n$ is bounded open set: $$\langle u,v\rangle = \int_U \Delta u \Delta v dx.$$ I need this when trying to find a weak solution for my PDE ...
2
votes
0answers
192 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 ...
4
votes
0answers
231 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 ...
2
votes
1answer
815 views

PDE weak solution problem

My professor grades really strictly (details). I would be very happy if you could help me with this problem: Let $U \subset R^n$ be a bounded set. Consider $ \Delta^2 u = f$ on $U$ and ...
1
vote
1answer
54 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 $f(\alpha_n)=f'(\alpha_n)=0$ ...
2
votes
0answers
96 views

A question on weakly convergence and norm convergence.

Let $2 \le p<\frac{2n}{n-2}$. Suppose that a sequence $\{u_k\}_k\subset H^1(\mathbb{R}^n)$ weakly converges to $u \in H^1(\mathbb{R}^n)$, and hence weakly converges to $u$ in $L^p(\mathbb{R}^n)$. ...
7
votes
0answers
449 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
6
votes
1answer
269 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
403 views

Is this sum of convex and concave functions a convex function?

Is this a convex function in $X$, where all the entries are real and $Y,\beta$ are constants where $X,Y$ are rectangular matrices and $\beta$ is a constant vector and $A,B$ are constant p.s.d ...
1
vote
1answer
158 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
114 views

Operator norm estimate

Let $H$ be a Hilbert space with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$. Furthermore, let $B\colon H\rightarrow C[a,b]$ be a bounded operator. According to the Riesz-Frechet theorem there is ...
3
votes
0answers
81 views

Estimate finite-rank operator

I have the the following problem. Let $H$ be a Hilbert space with orthonormal basis $(e_{j})_{j\in \mathbb{N}}$. Let $x\in [a,b]$, for all $h\in H$ $$ (Bh)(x) = \langle h,k_{x} \rangle,$$ with ...
1
vote
1answer
86 views

must a continuous function into hilbert space with all differences perpendicular be constant?

If $H$ is a Hilbert space and $f:[0,1]\rightarrow H$ is a continuous function such that $f(x)-f(y)\perp f(y)-f(z)$ whenever $x<y<z$, does $f$ have to be constant? By Pythagoras's theorem, the ...
1
vote
0answers
98 views

Confused about Bessel's inequality

I know that if $H$ is a Hilbert space and $(e_{j})_{j\in\mathbb{N}}$ is an orthonormal system in $H$ and $f\in H$. Then one has Bessel's inequality $$\sum_{j=1}^{\infty}|\langle f,e_{j}\rangle ...
1
vote
1answer
56 views

Particular series on Hilbert Space

Let $(H, \langle\cdot,\cdot\rangle)$ a Hilbert space and consider a sequence $\{x_n\}_{n\in\mathbb{N}}$ of $H$ such that: $$\langle x_n,x_m\rangle\ =\ \delta_{mn}\ =\ \left\{\begin{array}{ll}1, & ...
5
votes
2answers
257 views

Show that linear Operator on $\ell^2$ is unbounded

Currently, I am preparing for a next semester course and trying to figure out some basic concepts in functional analysis. Let $T:\mathcal{D}(T)\to \ell^2$ be defined by ...
8
votes
1answer
244 views

Is my statistician friend right/wrong on metric spaces and norms?

I was talking to a statistician friend of mine who said that instead of minimizing this function $\sum_{i,j}W_{ij}d_{ij}^2(X)$ over $X$ it would be better to solve an analogous minimization problem ...
1
vote
1answer
124 views

Weak to strong mapping

Let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be strongly monotone if there exists $\alpha>0$ such that $$ \langle F(u)-F(v), u-v\rangle\geq \alpha \|u-v\|^2, \quad ...
3
votes
1answer
420 views

From weak convergence to strong convergence

Let $H$ be a real Hilbert space and $F:H\rightarrow H$ be a mapping such that $$ (A)\qquad\qquad(u_n\rightharpoonup u_*, F(u_n)\rightarrow F(u_*))\; \Longrightarrow\;(u_n\rightarrow u_*) $$ We are ...
5
votes
1answer
432 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 ...
2
votes
3answers
70 views

Solving for positive semidefiniteness

Given a real matrix M, is there a matrix function f(M) such that $f(M)-M$ is guaranteed to be positive semidefinite, other than the idea of multiplying $M$ with its transpose and apart from the ...
1
vote
1answer
238 views

Norm of oblique projector and angle between subspaces

Take $V$ and $W$ closed subspaces of $H$ a Hilbert space with $V\oplus W=H$ (we'll assume this holds in the sequel, it may not be required everywhere but in the context of interest, it is always ...