# Tagged Questions

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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., $\left\| ... 0answers 26 views ### Invertible iff Bounded below and dense range Statement: Given a Hilbert space$\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$is bounded below ... 0answers 18 views ### Convex Feasibility problem on Hilbert space Let$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$be the ... 0answers 17 views ### Projection and Pseudocontraction on Hilbert space Let$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$be the ... 0answers 10 views ### Hilbert-Schmidt theorem In the Hilbert-Schmidt theorem what it means :$A e_n=\lambda_n e_n$? Thank you . 2answers 32 views ### showing$\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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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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 $U := ... 0answers 58 views ### Nilpotents of order 2 are dense in the strong operator topology I need some help with this homework problem. (The Hilbert space in question is$\ell^2(\mathbb{N})$.) I let$T \in \mathcal{B}(\mathcal{H})$and looked at a basic nbhd in the strong operator topology ... 3answers 205 views ### From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ... 0answers 52 views ### An inner product on a space of linear maps Let$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)$. ... 1answer 47 views ### a question on decreasing sequence of subspaces Let$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$is a finite dimensional ... 0answers 45 views ### Why is a self-dual Hilbert$\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 ... 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, ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...