0
votes
1answer
15 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
1
vote
0answers
26 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
0
votes
0answers
18 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
0
votes
0answers
17 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
0
votes
0answers
10 views

Hilbert-Schmidt theorem

In the Hilbert-Schmidt theorem what it means : $A e_n=\lambda_n e_n$ ? Thank you .
0
votes
2answers
32 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
1
vote
0answers
30 views

Question about Morse index

in general the Morse index of a critical point $p$ is the suprimum of the dimensions of sub spaces where $f''(p)$ is negative definite but whene $f''(p)=I-T$ ($f''(p)$ is a compact perturbation of ...
4
votes
2answers
35 views

Orthogonal representation of finite operator

I would like to know if my proof is correct. Statement: Let $T$ be a finite rank operator on a Hilbert space $\mathscr{H}$. Show that $\forall \, h \, \in \mathscr{H}, \, T(h)$ can be written as ...
1
vote
0answers
20 views

about a theorem of weakly lower semicontinuous functions

I am studying the proof of the following theorem Theorem: Let $E$ a Hilbert space and suppose that $\varphi :E \rightarrow R$ is a weakly lower semicontinuous functional. Suppose that $\varphi$ is ...
2
votes
1answer
51 views

Question on a derivative on a Hilbert space

I have this functional $J(u)=\frac12 \|u\|^2+\int_0^1 F(t,Ku(t))dt$ where $F(t,u)=\int_0^u f(t,\xi) d\xi$,$\displaystyle Ku(t)=\int_0^1 G(t,s)u(s) ds$ with $G(t,s)=\begin{cases} s(1-t),&0\leq s ...
0
votes
0answers
20 views

Proof Check: Closed range then bounded below

Statement: Given a Hilbert $\mathscr{H}$, and $T \in \mathscr{B}(\mathscr{H}, \mathscr{H})$, where $T$ has closed range. Prove that for all $h \in N(T)^\perp$ then $\exists \, m>0 \, \mbox{s.t.} \, ...
2
votes
3answers
119 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 ...
0
votes
1answer
25 views

Question about surjective continous operator being right invertible

I am reading a proof that a surjective continuous linear operator $T$ on a Hilbert space $H$ is right invertible. I have a question about the proof. The proof (up to the point where I have a question) ...
4
votes
1answer
94 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 ...
0
votes
0answers
22 views

Palais-Smale condition

I have this function: $$\tilde{f}(x)=f(x)+p(||x||)(x_0,x)$$ where $p\in C^2([0,\infty),\mathbb{R}) $ satisfy $0\leq p\leq 1,|p'(t)|\leq \frac{4}{\delta},$ and $$ p(t)= \begin{cases} 1& t\in ...
0
votes
1answer
29 views

Question id a derivative on a Hilbert space

On a Hilbert space $H$; i have this function: $\tilde{f}(x)=f(x)+p(||x||)(x_0,x)$ where $x_0\in H, p\in C^2([0,\infty),\mathbb{R}),f\in C^2(H,\mathbb{R})$ i want to caculate $\tilde{f}', ...
6
votes
1answer
45 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
0
votes
1answer
76 views

Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
1
vote
0answers
25 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. ...
1
vote
1answer
107 views

In a Hilbert space, every bounded and closed set is weakly relatively compact.

My aim is to prove that in a Hilbert space, any sequence has a weakly convergent subsequence. To prove this, I'm trying to prove that: ...
0
votes
1answer
36 views

Proving Density of Subset of Hilbert Space

Suppose we have a subspace, $M$, of Hilbert space $H$. Prove the first statement implies the second statement: 1) If $<f,g> = 0$ for any $g\in M$, then $f=0$ in $H$. 2) $M$ is dense in $H$. I ...
0
votes
0answers
36 views

Prove that a give sequence of function is a base of $L^2([0,1])$

Consider $(\phi_k)_{k \geq0} \in \mathcal C^{\infty}([0,1])$ with $\phi_k \not\equiv 0 $ such that $$\int_0^1 \phi_k(s) ds = 0, \quad \forall k\geq 1$$ and $$\sup_{ t \in [0,1]} \left | \frac{d}{dt} ...
1
vote
0answers
42 views

How to find a hilbert basis of a given subspace considering a given inner product

Let $X$ be the space of continuous functions on $[-1;1]$ to $\mathbb{R}$ with the inner product: $$\langle f,\ g\rangle = \int_{-1}^{1} \! f(x)g(x) \, dx$$ and let $U$ be a subspace of $X$ with $U := ...
1
vote
0answers
58 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
1
vote
3answers
205 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
2
votes
0answers
52 views

An inner product on a space of linear maps

Let $V$ and $H$ be two complex Hilbert spaces. We suppose $V$ to be finite-dimensional. I'd like to understand the structure of Hilbert space on the space of linear mappings $\mathrm{Hom}(V,H)$. ...
2
votes
1answer
47 views

a question on decreasing sequence of subspaces

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
0
votes
0answers
45 views

Why is a self-dual Hilbert $\text{B}$-module a conjugate space?

From p.$455$ of Inner Product Modules over $B$-Algebras by W.L. Paschkle: Proposition 3.8. Suppose that $\text{B}$ is a Von Neumann algebra and $X$ is a self-dual hilbert $\text{B}$-module. Then ...
1
vote
1answer
39 views

distance between a convex set and a point

Let's look at the following famous theorem: Let $\mathcal H$ be a Hilbert space and let $C< \mathcal H$ be a (proper) closed CONVEX set. If $x_0\in\mathcal H\setminus C$ and $\eta:=d(x_0, ...
1
vote
1answer
201 views

Operator self-adjoint

I have this paragraph : "Let M be a Hilbert-Riemannian manifold. $f \in C^2(M,R), p \in K$ is called a nondegenerate critical point, if $d^2 f (p)$ has a bounded inverse. Since $A = d^2 f (p)$ is a ...
1
vote
0answers
61 views

closest point property of subset of Hilbert space - what are the conditions for existence of inf?

I'm proving the closest point property of a subset of a Hilbert space, ie: $$H$$ is a Hilbert space with a norm generated by the inner product and so on. $$h\in H$$ is a point in H $$M\subset H$$ M ...
2
votes
1answer
104 views

Computing an explicit solution to an integral equation via the Neumann Series.

I am hoping that someone can provide guidance for solving the integral equation $$u=f+\lambda Au$$ where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
1
vote
1answer
81 views

Orthogonal family in Hilbert Space

Let $(x_k)_1^\infty$ be an orthogonal family of points in X a Hilbert space. Then $\sum_{i=1}^\infty x_i$ converges if and only if $\sum_{i=1}^\infty ||x_k||^2$ converges. Also need to show that ...
1
vote
1answer
48 views

Is this function in the space $L^1$?

I have this function $$f(x)=\frac{1}{\vert x-y\vert^2(1+\vert x\vert^2)^s}$$ with $x\in\mathbb{R}^3$ and $y$ a fixed point. I have to study for which values of $s>0$ it belongs to ...
7
votes
1answer
104 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 ...
1
vote
0answers
191 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 ...
1
vote
1answer
173 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 ...
2
votes
0answers
52 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 ...
2
votes
0answers
101 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)\ ;\ ...
6
votes
0answers
374 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 ...
1
vote
1answer
93 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 ...
2
votes
1answer
45 views

On the existence of a bounded linear functional

Let $\mathcal{H}$ be a Hilbert space. By the Riesz Representation Theorem, we have that any bounded linear $\psi \in \mathcal{H}^{*}$ is of the form $\psi(h) = \langle h, g \rangle$ for some $g \in ...
0
votes
2answers
66 views

Existence of a certain bouned linear functional in the dual of a Hilbert space

For any vector $h \in H$, where $H$ is a Hilbert space, show that $\exists$ a bounded linear functional $\psi \in H^{*}$ such that: $$\|\psi\| = 1 \ \text{and} \ \psi(h) = \|h\|$$ Can anyone ...
1
vote
0answers
61 views

If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
4
votes
2answers
91 views

A counterexample on the existence of some sequence in Hilbert space

I want to find a uniformly bounded sequence $\{x_n\}$ in $l^2(\mathbb{C})$ such that $x_n$ does not converge to zero in weak topology, i.e., $\exists ~y\in l^2(\mathbb{C}),$ such that $\langle y, ...
1
vote
2answers
102 views

Conclusion in the proof that every Hilbert space has an orthonormal basis regarding countable index set

I am working through a proof that every hilbert space has a orthogonal basis which lies dense in that Hilbert space. In the proof the following is done: Let $v$ be a vector, and $E \subseteq I$ an ...
5
votes
1answer
144 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 ...
6
votes
3answers
262 views

Why isn't it a Hilbert space

Let $X$ be the vector space of all the continuous complex-valued functions on $[0,1]$. Then $X$ has an inner product $$(f,g) = \int_0^1 f(t)\overline{g(t)} dt$$ to make it an inner product space. But ...
1
vote
1answer
180 views

Exponential operator on a Hilbert space

Let $T$ be a linear operator from $H$ to itself. If we define $\exp(T)=\sum_{n=0}^\infty \frac{T^n}{n!}$ then how do we prove the function $f(\lambda)=exp(\lambda T)$ for $\lambda\in\mathbb{C}$ is ...
0
votes
1answer
91 views

Show that $U \subset V \Leftrightarrow V^\bot \subset U^\bot$ for $U,V$ subspaces in a Hilbertspace

Let $(\mathcal H, \langle\cdot,\cdot\rangle)$ be a Hilbertspace, $U,V \subset \mathcal H$ are closed subspaces. I want to show $$U \subset V \Leftrightarrow V^\bot \subset U^\bot$$ $\Rightarrow$ is ...