2
votes
1answer
51 views

Maximal subspace on which an operator is bounded

Consider the Banach space $X=C[0,1]$ of real continuous function on $[0,1]$ equipped with the supremum norm. Consider the operator $A:D(A)\to X$, $Af=f'$ for each $f\in D(A)=C^1[0,1]$. We can see that ...
3
votes
2answers
60 views

Properties of reflexive Banach spaces

I just want to see the importance of reflexive Banach spaces and what is special about them compared to other Banach spaces. What kind of properties hold in reflexive spaces that do not necessarily ...
2
votes
2answers
67 views

Properties of a set in $\ell^2$ space

Let $\ell^2 = \{x= (x_1,x_2,x_3,\ldots): x_n\in \mathbb C\text{ and } \sum_{n=1}^\infty |x_n|^2 < \infty\}$ and $e_n \in \ell^2 $ be the sequence whose $n$-th element is 1 and all other elements ...
1
vote
0answers
38 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
67 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
51 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
53 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
156 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
33 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
136 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
72 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
77 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
19 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
193 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
160 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
113 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
139 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
117 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
213 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
230 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
146 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
103 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
448 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
42 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
219 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
96 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
785 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
149 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
628 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
185 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
480 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 ...