1
vote
0answers
36 views

Any finite-dimensional subspace of a Hilbert space is closed: easier proof?

A noted theorem is that a finite-dimensional subspace of a Hilbert space must be topologically closed. I have seen some proofs of this theorem which are less simple than this, but what is wrong with ...
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 ...
0
votes
1answer
22 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
0
votes
2answers
64 views

Dense Countable basis on Hilbert space

Let say that I have a $H$ hilbert space and linear independent countable set $\beta =\{ \beta_1 , \beta_2, \beta_3... \}$ such that $span(\beta)$ is dense set in H. does $span(\beta-\beta_1) =span( ...
1
vote
1answer
47 views

A question about closed balls in Hilbert Space.

Let $H$ be a separable and infinite dimensional Hilbert Space and let $B$ be a closed ball of $H$ whose diameter is some positive real number. Is every covering of $B$ by closed bounded subsets of ...
1
vote
1answer
50 views

Tychonoff vs. Hilbert

Let $(\mathscr H_n,\langle\cdot,\cdot\rangle_n)_{n\in\mathbb N}$ be a sequence of Hilbert spaces. Let $$\mathscr H\equiv\bigoplus_{n\in\mathbb N}\mathscr ...
5
votes
1answer
149 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
4
votes
0answers
85 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 ...
1
vote
0answers
31 views

Continuity of certain projections in the weak topology.

I'd like to prove that: Given a Hilbert space H and S a closed subespace, $S \subseteq H$, the projection $P_{S}:H \to S$ is continuous in the weak topology. I have tried the following. ...
2
votes
1answer
124 views

Relationship between different topologies of bounded operators on a Hilbert space

I am self-studying functional analysis. Given that $B(H)$ are the bounded operators on a Hilbert space, $H$. I would like to ask how to formally prove that the weak topology is weaker than the ...
0
votes
1answer
71 views

Properties of subspaces of Normed Vector Spaces

How does it follow that a subset of a normed vector space cannot be open if it does not contain an open ball $B_{\epsilon}(0)$ where $\epsilon > 0$? I just want to confirm also that for normed ...
0
votes
1answer
39 views

$V=L^2(\Omega,Z)$ is path connected

Let $V=L^2(\Omega,Z)$. Prove that V is path connected by paths of class 1/2 Holder. I would appreciate it if anyone could give me a suggestion. Thank you in advance.
1
vote
2answers
112 views

What is known about this space of parametrised Hilbert spaces?

For each $s \in [0,\infty)$, let $H(s)$ be a Hilbert space. Let us suppose for simplicity that $H(s) = L^2(\Omega_s)$, where $\Omega_s$ is some nice domain that depends on $s$ in a nice way. Define ...
0
votes
1answer
61 views

Proving that $T(B(x,2\epsilon))\cap B(y,2\epsilon) \neq \emptyset $

$H$ Hilbert space. $x,y \in H$ and $T\in L(H)$ 1) $T(B(x,\epsilon))\cap B(0,\epsilon) \neq \emptyset $ 2) $T(B(0,\epsilon))\cap B(y,\epsilon) \neq \emptyset $ 3) $T(B(x,2\epsilon))\cap ...
1
vote
1answer
74 views

closedness a subset of a Hilbert space

Let $H$ be a Hilbert space that admits a countable orthonormal basis $\{e_i\}$. I know this means that $H$ is separable and so is $S$ (as a subset of it, defined below). Show that $S$ is a closed ...
0
votes
1answer
18 views

A doubt in a proof concerning Hilbert spaces.

In this document about Hilbert spaces, I'm confused in the proof of Theorem 1.6 given at the bottom of pg.3. The theorem says that If $K$ is a closed convex set in a Hilbert space $H$ and $h\in ...
5
votes
3answers
189 views

Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
5
votes
0answers
102 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
1
vote
1answer
159 views

Bounded linear operator in weak topology

Let $B$ be a bounded linear operator on $H$. Prove $B\colon (H,w)\to (H,w)$ is continuous. $(H,w)$ is a Hilbert space with its weak topology.
7
votes
1answer
111 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
136 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
2answers
114 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 $S$ ...
0
votes
1answer
204 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 ...
8
votes
1answer
227 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 ...
2
votes
1answer
141 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 $B(H)$?
3
votes
2answers
101 views

Show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ is not closed in $\ell^2$

How to show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ where $e_n=(\delta_{nk})_{k\in\mathbb N}$ is not closed in $\ell^2$?
2
votes
2answers
131 views

Uncountable union of separable spaces is separable?

If $V(x)$ is a separable Hilbert space, is $\bigcup_{x \in X}V(x)\times\{x\}$ separable when $X$ is an uncountable set? How to make it separable if it's not? What assumptions do I need?
0
votes
1answer
432 views

Density and closedness of $C[0,1]$ in $L^\infty[0,1]$ in norm and weak-* topologies

With results: "For convex subsets of a locally convex space, a, originally( strongly) closed equals weakly closed, and b, originally (strongly dense equals weakly dense." Could you help me solve this ...
0
votes
0answers
41 views

About the problem 20 chap 3 (functional analysis, Walter Rudin) [duplicate]

Possible Duplicate: I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed. Let $\{u_1,u_2,u_3,\dots \}$ be sequence of pairwise orthogonal unit vectors in Hilbert ...
2
votes
1answer
128 views

The closed unit disk in an infinite dimensional Hilbert space has a closed subspace homeomorphic to $\mathbb R$

Let $V$ be a Hilbert space, $D^{\infty}$ is the closed unit disk in an infinite dimensional Hilbert space $V$. Prove that $D^{\infty}$ has a closed subspace homeomorphic to $\mathbb{R}$. I've found ...
3
votes
2answers
206 views

Span of functions dense in $L^2$

This is an exercise from Werner's Funktionalanalysis. I have to show that the linear span of the functions $f_n(x)=x^ne^{-x^2/2}, n\geq0$ is dense in $L^2(\mathbb{R})$. The book gives the hint to ...
4
votes
1answer
55 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 ...
2
votes
0answers
95 views

Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...
4
votes
5answers
772 views

Subspaces of Hilbert Spaces of finite dimension

Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ ...
2
votes
2answers
120 views

proving “$C^1([−1,1])$ is dense in the given space with given norm”

Define $$E = \left \{ f \in W^{1,2} (-1,1) \; | \; \| f \|_E := \left( \int_{-1}^1 (1-x^2 ) | f' (x) |^2 dx + \int_{-1}^1 | f(x) |^2 dx \right)^{\frac{1}{2}} < \infty \right \}.$$ Then how can I ...
2
votes
2answers
148 views

I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed.

Let $x_n$ be a sequence in a Hilbert space such that $\left\Vert x_n \right\Vert=1$ and $ \langle x_n,\ x_m \rangle =0 $, for all $n \neq m$. Let $ K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\} $. ...
2
votes
3answers
608 views

Separability of the space of bounded operators on a Hilbert space

Let $H$ be a (separable) infinite dimensional Hilbert space, and $B(H)$ the space of bounded operators on $H$. Is $B(H)$ separable in the operator norm topology? What about in the strong and weak ...
3
votes
2answers
183 views

Distance of functions defined on a Hilbert Space

In our Topology class, we touched on Hilbert spaces for a couple of weeks. I've been studying various problems around the topics we covered, and I came across this one on a list of supplemental ...
8
votes
2answers
471 views

Is compactness a stronger form of continuity?

Let $H$ be a Hilbert space. We say that a linear operator $T \colon H \to H$ is compact if it maps bounded sets to precompact ones, that is, if for every bounded sequence $(a_n)$ in $H$, $(Ta_n)$ has ...