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

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Hermitian and self-adjoint operators on infinite-dimensional Hilbert spaces

I am a physicist and I am trying to get a grasp on the following terms from functional analysis: As I understand, an operator is Hermitian if it is symmetric and bounded (domains of A and A* don't ...
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54 views

Chain of closed subspaces in a Hilbert Space

Let $H$ be a separable Hilbert Space (WLOG, we may assume $H=\ell_2(\mathbb{N})$ is the space of square summable sequences). Can there exist an uncountable chain of closed subspaces? In other words, ...
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37 views

Showing $P^2=P$ for $P(v)=\frac{\langle v,w\rangle }{||w||^2}w$

My book asserts that for fixed $w$ where $w\neq 0$ that $P^2=P$ for $P(v)=\frac{\langle v,w\rangle }{||w||^2}w$ My book has a general corralary that $v\to P(v)$ is a bounded linear transformation and ...
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135 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 ...
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388 views

Weak convergence in a Hilbert Space

What does it mean for a sequence $\{f_n\}_{n=1}^\infty\subseteq H$ to converge weakly? I know it means that it converges in the weak topology and I've read a few definitions of weak topology which all ...
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68 views

Understanding problems of space

I've been trying to understand the concept of space for some time now, but I still can't grasp the essence of it. In high school math we've been using 2D- and 3D- Euclidean space. Now that I am ...
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100 views

Is $L^2(\Omega)$ the only $L^p$ hilbertian space?

I've started today studying Hilbertian spaces, and all of the examples seen in class were about the space $L^2(\Omega)$, where $\Omega$ is a limited domain in $\mathbb{R}^N$ $(N \geq 1)$. Online I ...
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108 views

Are two hilbert spaces with the same algebraic dimension (their hamel bases have the same cardinality) isomorphic?

We know that two hilbert spaces tat have orthonormal bases of the same cardinality are isomorphic (as an inner product spaces). my question is what can we say when we know that their hamel bases ...
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110 views

If $V \subset H$ compact, is $L^2(0,T;V) \subset L^2(0,T;H)$ compact too?

As the question states, if we have the compact embedding of Hilbert spaces $V \subset H$, is $L^2(0,T;V) \subset L^2(0,T;H)$ compact too? If not true in general, is it true for $V=H^1(\Omega)$ and ...
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72 views

Hilbert space on line bundle

Suppose that $L$ is a complex line bundle on a manifold $M$ with measure $\mu$, How can we prove, $L^2(M,L,\mu)$ is Hilbert space?
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263 views

Continuity of scalar product

In a Hilbert space $H$ with inner product and associated norm, why would if $\|x-x_n\| \longrightarrow 0$ and $\|y-y_n\| \longrightarrow 0$ also $\langle x_n,y_n\rangle \longrightarrow\langle ...
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48 views

Prove that there $B : H\to H $ bounded such $ B^n = A $.

Let $ A : H\to H $ a compact self-adjoint operator. Suppose $ A $ is positive. let $ n \geq 2 $. Prove that there is $B : H\to H $ bounded such $ B^n = A $.
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88 views

True or False; Functional Analysis

Given $T: V \to W$ with $V,W$ being Hilbert Spaces. We always have $\| T^ *\| = \| T \|$. I think it is true because of Riesz' Theorem, but I am not sure if a proof is necessary. EDIT: In case ...
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66 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 ...
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134 views

Does $e_n(x)=\exp\left( \frac{i \pi n}{N}x \right)$ define an orthonormal basis of $L^2(-N,N)$?

We know that the Fourier system is complete, i.e. that $\lbrace e_n: ~ n \in \mathbb{N} \rbrace$ defined by \begin{equation} e_n(x)=\frac{1}{\sqrt{2 \pi}}\exp(inx), ~~~ n \in \mathbb{Z} \end{equation} ...
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135 views

Linear operator and extension of its inverse

Let $K:H_1 \to H_2$ be a linear operator between Hilbert spaces that may not be bounded. $K$ is bounded below. So $K$ has an inverse $K^{-1}:\text{Range}(K) \to H_1$. $K^{-1}$ extends by ...
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102 views

If $Lat(\mathcal{A})$ is trivial then $\mathcal{A}'$ consists of scalars.

This is related to Exercise 3 of Section 2.5 of Arveson's book on spectral theory. For those who are interested, we were asked to show the following $\mathcal{A}$ is a Banach *-algebra. ...
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1answer
83 views

Is it ok to switch the limits in $L_2$?

Let $(X,B,\mu)$ be a probability space and let $U$ be a unitary operator on $L_2(X,B,\mu)$. Suppose that $g_n$ is a convergent sequence in $L_2(X,B,\mu)$, $g_n\rightarrow g$. Suppose also that there ...
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251 views

Completion of pre-Hilbert space in H. Brezis' Functional Analysis

I'm trying to solve the problem 5.12 of Harim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations; but I'm stucked understanding the statement which comes as follows: ...
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168 views

Equivalents norms in Sobolev Spaces

I know that this is classical but I have never do the calculations to show that the norms in the sobolev space $W^{k,p}(\Omega)$ \begin{equation} \|u\|_{k,p,\Omega}= \Bigl(\int_{\Omega} ...
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188 views

Perturbation theorem of Weyl

Does anyone know where to find something about the perturbation theorem of Weyl, preferably on the internet. The theorem I'm talking about states: let $A$ be a self-adjoint operator on a Hilbert ...
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1answer
216 views

The commutator subgroup of the group of bounded invertible linear operators

I am curious to know what the commutator subgroup of the group of (bounded) invertible linear operators on a complex Hilbert space is? Note that by "commutator subgroup" I mean the subgroup ...
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704 views

Riesz Lemma to the Riesz Representation Theorem

Let $H$ be a Hilbert Space and let $H^*$ be the dual space of $H$. The Riesz Lemma states that for each $T\in H^*$, there is a unique $y_T\in H$ such that $T(x)=(y_T,x)$ $\forall x\in H$. Also, ...
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1k views

Hilbert spaces, square integrability etc

(Someone may please change the title if they can think of a better one) We have a Hilbert Space $\mathcal{H}$ that consists of all functions $\psi(x)$ such that $\int_{-\infty}^{\infty} |\psi(x)|^2 ...
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24 views

Prove there cannot be an inner product which turns $l^p$ into an inner product space?

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be ...
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60 views

Proving that if $<Ax,x>=0$ for every $x$, then $A$ is the zero operator

I feel kind of dumb but I needed this little claim as a part of a proof I'm writing, and I figured out that I'd better just ask, since I could not find the correct algebraic manipulation needed in ...
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120 views

Exercise 34 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 34 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 201): Let K be a Hilbert-Schmidt kernel which is real and symmetric. Then, as we saw, the operator $T$ whose ...
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48 views

“Almost” Hilbert spaces

This question is a bit (very?) vague. Is there some notion of how "close" a Banach space is to being a Hilbert space? What I have in mind is something like a real or complex valued function on ...
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47 views

Is the distance between disjoint closed convex subsets of a Hilbert space positive? Is it attained?

Let $H$ be an infinite dimensional and separable Hilbert space. Let $A,B$ be infinite, closed and convex subsets of $H$. If $A$ and $B$ are disjoint and if at least one of them is bounded, is the ...
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98 views

Can Hilbert spaces generalize non-Euclidean geometry by having the sum of the angles of a triangle not be equal to pi?

I am an amateur mathematician learning new things. Let A and B be vectors in a Hilbert space. The three vectors A, B and A-B form a triangle. The idea of the angle between two vectors can be ...
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52 views

Closed subspaces of $L^2(0,1)$

I would like to prove that the almost-everywhere constant functions, and the functions whose integral is 0 are closed subspaces of $L^2(0,1)$. It's readily seen that they are subspaces. I'm finding ...
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56 views

Implying a positive definite operator

If we are given that $A:V \rightarrow V$ is an operator where $V$ is a real Hilbert space. If we are given that $A$ is bounded, strictly positive $\big(\langle Au,u \rangle > 0$ for all $u \neq ...
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65 views

Weak convergence of partial sums

I recently came across an interesting problem on weak convergence in $\ell^2 (\Bbb N)$. Suppose that we have canonical basis $\{e_i\}$ in $\ell^2 (\Bbb N)$. We need to prove that the sequence ...
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69 views

Prove that this space is not Banach

Let $\Omega\subset\mathbb{R}^n$ be an open, bounded set with boundary $\partial\Omega$ of class $C^1$. $$\mathcal{A}:=\{u\in C^2(\bar\Omega):u=0\text{ on }\partial\Omega \}$$ endowed with the scalar ...
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59 views

Weak and Norm convergence in Banach Space

I know (and have proven) that in a Hilbert space, $\mathscr{H}$, if a sequence $z_i\overset{w}{\to}z$ and $\|z_i\|\to\|z\|$, then $\|z_i-z\|\to0$. I'm trying to find a counterexample in a Banach ...
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43 views

Alternating projections on a Hilbert space

Let $P_1, P_2$ be the orthoprojections onto $S_1, S_2$, closed subspaces of a Hilbert space $H$. It is straightforward to show that if $(P_1P_2)^nx \to z$ then $z \in S_1 \cap S_2$ (I can post a quick ...
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41 views

Density result in Hilbert space

Assume that $b\in \mathbb{C}$ such that $0<\vert b \vert <1$. We consider the familly $f_{p}=\{1,b^{p},b^{2p},b^{3p},b^{4p},...,b^{np},...)$. How can one prove that $\operatorname{Span}(f_{p}, \ ...
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67 views

On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
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48 views

Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$ \lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t} $$ exists then $\psi$ ...
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66 views

Superspace as the Hilbert Space for Quantum Gravity

This is a question I've asked in physics.stackexchange, but have obtained no answers: Let $\mathcal{A}$ be the Ashtekar connection. Since $^{(3)}g_{AB}=i\frac{\delta}{\delta\mathcal{A}^{AB}}$ (see R. ...
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Proof of an equivalence in Hilbert spaces

Let $H$ be a Hilbert space. Prove that the following are equivalent: a) the algebraic dimension of $H$ is finite; b) each closed, not empty subset $C$ has an element of minimum norm (that is the ...
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326 views

Orthogonal Projections in Hilbert space

I am stuck with the following exercise about projections in Rudin 12.26. Let $H$ be a Hilbert space $P,Q\in B(H)$ self-adjoint projections (A projection has the property that $P^2=P$), then the ...
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396 views

Dual and adjoint operator

Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual ...
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65 views

$\lambda_k \to 0$ implies $T$ is compact?

I am doing an exercise which asks to show that if $\{\varphi_k\}$ is an orthonormal basis in a Hilbert space with $T$ a bounded operator satisfying $T\varphi_k = \lambda_k \varphi_k$, then $\lambda_k ...
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1answer
137 views

How to show that time-dependent norm is continuous (please verify my proof)

For each $t \in [0,T]$, let $H_t$ be a Hilbert space. Suppose for each $t$, the operator $T_t:H_0 \to H_t$ is a linear homeomorphism with inverse $T_{-t}:H_t \to H_0$ also linear homeomorphism. ...
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180 views

Projection of vector on subspaces in a Hilbert space

This may be a vague title and I think that this question must have an easy answer. Let $\mathcal{H}$ be a weighted $\ell^2$ space of complex sequence $\{x(n)\}_{n \geqslant 1}$ such that $$\|x\|^2 = ...
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121 views

Want to show an operator is compact

With $V=L^2(0,T;H^1(\Omega))$, let $A:V \to V^*$ with $$\langle Au,v \rangle = \int_0^T \int_{\Omega} \nabla u(t) \cdot \nabla v(t).$$ I want to show that $A$ is a compact operator. So, one way to ...
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65 views

Why is closeness of an ideal useful?

In the GNS-construction for an $C^*$-algebra $\mathcal A$ (see this script on page 30) one starts with a state $\phi:\mathcal A\rightarrow \mathbb C$ (positive linear functional with $\|\phi\|=1$). ...
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430 views

Direct sum of orthogonal subspaces

I'm working on the following problem set. Let $\mathcal{H}$ be a Hilbert space and $A$ and $B$ orthogonal subspaces of $\mathcal{H}$. Prove or disprove: 1) $A \oplus B$ is closed, then $A$ and $B$ ...
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29 views

If $u_m \to u$ and $v_m \to v$, does $b(u_m,v_m) \to b(u,v)$?

In a Hilbert space $H$, if $u_m \to u$ and $v_m \to v$, does $b(u_m,v_m) \to b(u,v)$ if $b$ is a bounded bilinear form on $H$?