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

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Regular representation, representability of the fiber functor, and hom-distributivity for Hilbert spaces

I've culled together a slick proof of $\Bbb C[G]\cong\bigoplus_{V\in\widehat{G}}{\rm End}(V)$ (Peter-Weyl decomposition) for finite groups using the fact that the fiber functor (that is, the forgetful ...
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24 views

Step function integral inequality

I would like to prove the following inequality: $$\langle f,Id \rangle^2 \leq \langle f,1 \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i \mathbb{1}_{I(i)}(s)$, ...
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1answer
39 views

Spectral Measures: Analytic Elements

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the convergence radius by: ...
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2answers
28 views

Dense subspace of dense subspace is dense

Let $H$ be a Hilbert space and $V$, $W$ to linear subspaces such that $V\subset W\subset H$ with $V$ dense in $W$ and $W$ dense in $H$. Does this implie that $V$ is also dense in $H$? I think you can ...
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0answers
28 views

Is $H^1(\Omega, S^2)$ a Hilbert manifold?

I'm considering the topology of the function space $H^1(\Omega, \mathbb{S}^2)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain. Obivously it is not a vector space, but is it a Hilbert ...
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1answer
28 views

Spectral Measures: Nelson

Problem Given a Hilbert space $\mathcal{H}$. Consider a symmetric operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad T\subseteq\overline{T}\subseteq T^*$$ Denote the convergence radius by: ...
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2answers
31 views

Why is the additive category of Hilbert spaces not abelian

As an answer to this post Additive category that is not abelian it was said that the additive category of Hilbert spaces is not abelian. Why is that? Also what category of Hilbert spaces is this? ...
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1answer
20 views

Reducing Subspaces: Characterization

Given a Hilbert space $\mathcal{H}$. Consider an operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad\mathcal{D}:=\mathcal{D}(T)$$ Regard a subspace: ...
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1answer
40 views

Reducing Subspaces: Domain

Problem Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad T=\overline{T}$$ Regard a closed subspace: ...
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1answer
28 views

Reducing Subspaces: Preliminary

Given a Hilbert space $\mathcal{H}$. Consider a dense domain: $$\mathcal{D}\leq\mathcal{H}:\quad\overline{\mathcal{D}}=\mathcal{H}$$ Regard a closed subspace: ...
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1answer
10 views

Linear functionals over a non dense subset in $\ell^2$

I have come across the following statement a couple of times, but cannot figure out quite how to justify it: If $A$ is not a dense subset of $\ell^2$ then its closure is not all of $\ell^2$ so ...
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1answer
13 views

The norm of a bounded linear functional on a Hilbert space is the norm of the vector?

If $L$ is a bounded linear functional on a Hilbert space $H$, then we know that $$Lx=(x,y),\quad \forall x\in H,$$ for some $y\in H$. Is it true that $\|L\|=\|y\|$? We have by Cauchy-Schwartz that ...
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1answer
38 views

Can an inner product on a vector space be negative?

This may be a noob question but I recently read a definition that an inner product on a complex vector space is said to be a positive-definite sesquilinear map. Doesn't positive definite mean that ...
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23 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
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21 views

Self adjoint operator with countable eigenvalues

It has been explained here that any self adjoint operator on a seperable Hilbert space inhabits at most countable many eigenvalues. Now I wonder, can one construct an operator $$ T : L^2([0,1]) \to ...
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1answer
20 views

Difference between hermitian and conjugate linear sesquilinear form

I am trying to work out the difference between hermitian and conjugate linear sesquilinear form. Let me elaborate on my confusion: Let $H$ be a Hilbert space. One definition (see e.g. here page 49) ...
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27 views

Is there a name for these sets of functions of several complex variables (most not analytic)?

For each $n \in \mathbb{N}$, I came up with the following sets that I found interesting; at least I've never seen them in the literature before. $S_n = $span{$z_1z_2 \cdots z_n, \bar{z}_1z_2\cdots ...
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1answer
18 views

The definition of continuity for linear functionals

I am trying to prove that a linear functional is continuous on the space $H^1(0,l)$, and I have a couple of different definitions. The one that I want to use is that $f$ is continuous if $f$ is ...
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38 views

Weak* topology on Hilbert space

I am a little confused about the weak* topology on Hilbert space $H$. Beyond doubt, the weak* topology on $H^{**}$ is $\sigma(H^{**},H^*)$. Suppose $\tau$ is the natural embedding from $H$ onto ...
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1answer
20 views

Isolated eigenvalues of a self adjoint operator

If $X$ is a separable Hilbert Space and $T : X \to X$ selfadjoint and bounded, then the point spectrum $$ \sigma_p(T) $$ is only countable as explained here. I have the following three questions: ...
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2answers
81 views

Every bounded sequence has a weakly convergent subsequence: salvage this proof?

I tried to prove the following theorem and was wondering if someone could please tell me if my proof can be fixed somehow... Theorem: Let $H$ be a Hilbert space and $x_n\in H$ a bounded sequence. ...
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0answers
10 views

Step function scalar product inequality

I would like to prove the following inequality: $$\langle f,\frac{|N.+1|}{N} \rangle^2 \leq \langle f,. \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i ...
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1answer
20 views

Question about the following proof in Hilbert space

I started reading the book "Mixed Finite Element Technologies" by Peter Wriggers and Carsten Carstensen, and I have a question about the following. Here is the setup: Then, the authors prove the ...
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0answers
33 views

Is a cyclic subspace of a compact unitary representation finite dimensional?

Let $K$ be a compact Lie group and let $\rho_k: H \rightarrow H$ be a (strongly continuous) unitary representation of K on a Hilbert space H. Why does the orbit, $\rho(K)v$ ,of any $v\in H$ generate a ...
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1answer
29 views

Mourre Theory: Resolvent Formula

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its resolvent by: $$z\in\rho(H):\quad R(z):=(z-H)^{-1}$$ Introduce its ...
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2answers
25 views

Is the Norm of the Square Root of an Operator equal to the Square root of the Norm of the Operator

Suppose we have a positive operator $A \in \mathcal{B}(\mathcal{H})$, does it follow that $$\|A\|^{1/2} = \|A^{1/2}\|?$$ If not, is there some relation between these quantities?
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36 views

What are the negative-dimentional n-sphere and n-cube?

The generalized formula for the volume and surface area of n-sphere allows to evaluate volumes and areas of negative-dimentional n-spheres. $$\begin{array}{ll} S_{n-1}(R) &= ...
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2answers
17 views

Why $\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$?

Let $f,g\in L^2$ with Lebesgue measure. and $K:L^2\to L^2$ be some linear and continuous operator. Show that $$\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$$ where $h\in H\subset L^2$.
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1answer
25 views

Question about bilinear form on Hilbert space

I am trying to verify the following Let $H$ be a Hilbert space, and let $a(\cdot,\cdot)$ be a real continuous bilinear form on $H$ Then, define the operator $A:H-> H'$ as $Au(v) :=a(u,v), v\in H$ ...
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2answers
49 views

Question about dual of dual of Hilbert space

Let $H$ be a real Hilbert space and let $H'$ be the set of continuous linear functionals on $H$. Then, I know by the Riesz Theorem that for every $L(\cdot) \in H'$, there exists a unique $u\in H$ so ...
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1answer
66 views

Existence and uniqueness of the minimizer of Moreau-Yosida approximation

Let $f:H\to\mathbb{R}$, where $H$ is a Hilbert space, be a function that is bounded below, convex ($f(tx+(1-t)y)\leq tf(x)+(1-t)f(y) \text{ for all } x,y\in H \text{ and } 0\leq t\leq 1$), and lower ...
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1answer
50 views

Inner product in Hilbert spaces

Considering a sequence $\{\boldsymbol{v}_k\}_{k=1}^\infty$ in a Hilbert space $\mathcal{H}$, and let $\{c_k\}_{k=1}^\infty \in \ell^2(\mathbb{N})$. Then for all $\boldsymbol{v}\in\mathcal{H}$ $$ ...
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1answer
20 views

Adjoint operators in Hilbert space

Consider the linear and bounded operators $X$ and $Y$on a Hilbert space $\mathcal{H}$ with inner product $\langle \cdot,\cdot \rangle$. How can I show that $$ \langle XY \boldsymbol{v}, ...
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1answer
45 views

Functions so that image of min (resp. max) is a positive definite kernel

I am trying to determine the functions $\phi : \mathbb{R}^+ \to \mathbb{R}$ such that: Pb 1: $K(s, t) = \phi( \mathrm{min} (s,t))$ is a positive definite kernel on $\mathbb{R}^+$. Pb 2: $K(s, t) = ...
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40 views

If $u_n \rightharpoonup u$ in $L^2(0,T;L^2)$ and $u_n$ bounded in $L^\infty(0,T;L^2)$, does $u_n(t) \rightharpoonup u(t)$ in $L^2(\Omega)$ a.e. $t$?

Let $u_n$ converge weakly to $u$ in $L^2(0,T;L^2(\Omega))$ and let $u_n$ be bounded in $L^\infty(0,T;L^2(\Omega))$. Is it true that $u_n(t) \rightharpoonup u(t)$ in $L^2(\Omega)$ (weakly) for a.a. ...
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0answers
28 views

M bounded if and only if $\sup\{|\langle x,y \rangle|:y\in M \}<\infty$

Let $H$ be a Hilbert space. Show that $M\subset H$ is bounded if and only if $\sup\{|\langle x,y \rangle|:y\in M\}<\infty$ for every $x\in H$ My attempt: Since $H$ is a Hilbert space any set is ...
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1answer
50 views

Boundness of funcions in $L^2(0,T;H)$

Let $H$ be a Hilbert space and $u_{k} \rightharpoonup u$ in $L^2(0,T;H)$ (the $\rightharpoonup$ means "weakly convergent to") Assume one has the uniform bounds $$\mathrm{essential~sup}_{0\leq t\leq ...
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1answer
64 views

Norm of the product of an isometry and a bounded operator

Let $A$ be a bounded operator and $V$ a linear isometry, both defined on a complex Hilbert space $H$ (infinite dimensional). I could easily prove that $\|VA\|=\|A\|$. But, I just couldn't prove that ...
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1answer
22 views

computation with polar decomposition of bounded operator on hilbert space

I am trying to prove the following homework problem: Let $T \in B(H)$ (so $T$ is a bounded operator on a Hilbert space $H$), and let $T = U|T|$ be the polar decomposition of $T$. Prove that if $T$ is ...
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26 views

Properties of functional calculus

Suppose we have a self-adjoint bounded operator $S$ on a Hilbert space $\mathscr{H}$ with the property that $||Sx||<||x||$ for each $x\in\mathscr{H}\setminus\{0\}$. Now assume that ...
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1answer
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Semigroups: Product Rule [closed]

Given a Banach space $E$. Consider C0-semigroups: $$S,T:\mathbb{R}_+\to\mathcal{B}(E)$$ Then the product rule holds: $$(TS)'(t)x=T'(t)S(t)x+T(t)S'(t)x$$ How to prove this from scratch?
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25 views

Help with inverse mapping theorem in Hilbert Spaces?

The question I'm trying to answer goes as follows: Let $X$ and $Y$ be banach spaces and $T:X \to Y$ be a bounded linear operator which is surjective. Let $K$ be the closed subspace $\ker T$ and let ...
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1answer
17 views

How to show that if $u$ is a partial isometry then $u = u u^\ast u$?

Let $H, H'$ be Hilbert spaces, $u \in B(H,H')$ and $u^\ast $ its adjoint. I am trying to show that if $u$ is a partial isometry then $u = uu^\ast u$. My idea was to write $H = \ker u \oplus (\ker ...
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1answer
19 views

Does $\|u\|=\|u^\ast\|$ imply $\|uh\| = \|u^\ast\|$?

Let $H$ be a Hilbert space and $u \in B(H)$ and let $u^\ast$ denote its adjoint. I know that $\|u\|=\|u^\ast\|$. But now I am wondering: Does $\|u\|=\|u^\ast\|$ imply $\|uh\| = \|u^\ast h\|$ for ...
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24 views

spectrum, hilbert space, symmetric operator

I am having trouble with the following... Let $A$ be a symmetric operator on a Hilbert space which is not self adjoint. Show that $\sigma(A)=\mathbb{C}$ or $H^+=\{\mu+iv \mid v\geq 0\}$ or ...
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11 views

Scalar product inequality

I would like to prove the following inequality: $$\langle f,1 \rangle \sum_{i=1}^N \sum_{j=1}^i \langle f,\mathbb{1}_{I(j)} \rangle \leq \frac{N}{2} \langle f,1 \rangle^2$$ where $f$ is a step ...
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0answers
27 views

Modulus: Invariant Domain

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$M:\mathcal{H}\to\mathcal{K}:\quad \|M\|=1$$ Regard dense subspaces: ...
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1answer
28 views

Show $\mathcal{H}_\eta = L^2([a,b], \eta)$ is a Hilbert space when $\eta$ is positive, not necessarily continuous

Exercise $8$ of Stein and Sharkarchi's Real Analysis asks first to show that the space of measurable $f$ on $[a,b]$ such that $$\int_a^b |f(t)|^2 \eta (t)dt < \infty $$ denotes $\mathcal{H}_\eta = ...
3
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1answer
33 views

$\langle f, \phi_n \rangle = 0 \implies f = 0$ is equivalent to the definition of orthonormal basis

Is there an "easy" way to see that if $\{\phi_n\}_{n=1}^\infty$ is a set of orthonormal functions in a Hilbert space then showing $\langle f, \phi_n \rangle = 0$ for all $n$ implies $f = 0$ is ...
3
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0answers
65 views

Spectral Measures: Helffer-Sjöstrand

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard almost analytic extensions: $$f_E\in\mathcal{C}^\infty_0(\mathbb{C}):\quad ...