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

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93 views

Weak limits and subsequences

Let $S:X \to X$ be a (nonlinear) map between a Hilbert space $X$. I want to show that $S$ is weakly continuous, so if $x_n \rightharpoonup x$, then $S(x_n) \rightharpoonup S(x)$. To do this, I have ...
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3answers
72 views

Show the Cauchy-Schwarz inequality holds on a Hilbert space

How would one go about showing this? Its a question in one of the workbooks but it doesn't provide an answer. Any help would be appreciated.
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1answer
84 views

Positive compact operator has unique square root.

Let H be a hilbert space and T be a compact positive operator so that by the spectral decomposition theorem, $T=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,e_{n}\rangle e_{n}$ where the $e_{n}$ are the ...
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1answer
56 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 ...
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2answers
26 views

Show that with Orthonormal system in a Hilbert space

Okay so I completed part (i) and I got some help on part (ii) so I am fine with that now. I'm stuck on part (iii) though and don't really understand. Any help would be appreciated
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24 views

Proving a system is linearly independent in a Hilbert Space

Okay, I've done part (i) but I'm stumped on part (ii) and how I can show that. Any help would be appreciated please.
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37 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.} \, ...
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3answers
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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1answer
69 views

Spectral theory and sequences: is this fact a general truth or does it depend on the operator?

Let $\lambda\in\mathbb{R}\setminus\{0\}$, $\textbf{i}$ the imaginary unit, $H$ a Hilbert space, $T:D(T)\subset H\to H$ a invertible densely defined linear operator such that $T^{-1}$ is bounded, ...
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3answers
70 views

If $\langle x,y\rangle = \langle x,z\rangle$ for all $x \in H$ then $y=z$

My professor made the following claim in the body of a proof and verbally explained why it was justified. I thought I understood his explanation at the time but on reviewing my notes it's not as ...
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1answer
120 views

Conditional expectation as an orthogonal projection to what subspace?

Given a random variable $X$ and a sub sigma algebra $N$ of its sampling space, it is often said that $E(\dot \, \mid N)$ is an orthogonal projection, since $X-E(X\mid N)$ and $E( X\mid N)$ are ...
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58 views

Show these operators converge to a particular limit

Let $H$ be a Hilbert space, and $T$ be a operator on $H$ of the form $T=\sum_{n=1}^{\infty}{\lambda}_{n}<x,e_{n}>e_{n}$ where $e_{n}$ are the eigenvectors of $T$ and an orthonormal basis of H ...
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1answer
48 views

Naturality of Riesz' Representation

What does it mean precisely in the context of category theory when somebody says that Riesz' representation is canonic resp. every Hilbert space is naturally antiisomorphic to its dual?
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1answer
56 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 ...
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1answer
15 views

If $u_n \rightharpoonup u$ and $Au_n \rightharpoonup b$, does it follow that $\limsup_{n \to \infty}(Au_n, u_n) \leq (b, u)$ if $A$ is continuous?

Let $H$ be a Hilbert space and let $A:H \to H$ be a (nonlinear) continuous map. If $u_n \rightharpoonup u$ and $Au_n \rightharpoonup b$, does it follow that $$\limsup_{n \to \infty}(Au_n, u_n) \leq ...
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70 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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51 views

Product of strong and weakly converging sequences

Consider a sequence of functions $\{u_n\}\in L^2([0,T],L^2(\Omega)) $ which converges strongly to a function $u\in L^2([0,T],L^2(\Omega))$. Then $u_n \rightarrow u \;\; a.e. $ in ...
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2answers
81 views

Checking whether an operator is self-adjoint. Problem with domain of an operator.

I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The ...
2
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1answer
27 views

Proving an operator $D: L^2[0,1]\rightarrow C'$, $Df(t)=\int^t_0 f(s) ds$ is unitary

Let $C'\subseteq C[0,1]$ be the space of all absolutely continuous function such that $f(0)=0$ and $f' \in L^2[0,1]$. Define an inner product on $C'$ as $\langle f,g \rangle = \int^1_0 f'(t)g'(t)dt$. ...
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31 views

Abstract Wiener space and integration related to a trace class operator

Suppose I have a trace class operator $A$ of a Hilbert space $H$. Also suppose I have an abstract Wiener space $(H,B)$. Then, $\langle Ax, x \rangle$ is defined almost everywhere in $B$ with respect ...
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2answers
292 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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65 views

Hermitian and self-adjoint operator.

I was trying to understand why this operator is hermitian (I see that) but not self-adjoint: We have the operator $P:\phi(\cdot )\mapsto -i\phi '(\cdot )$, dfined in $C_0^\infty (I)$ (infinitely ...
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156 views

A Banach space that is not a Hilbert space

Can someone give me an example of a Banach space that is not a Hilbert space? I can't think of any because I don't know how to show one space that can not have inner product structure.
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1answer
71 views

right inverse of surjective linear map on hilbert space exists iff kernel is complement subspace

Suppose that $L : H \to H'$ is a surjective continuous linear transformation between Hilbert spaces. Show that there exists a continuous linear transformation $S : H' \to H$ such that $LS = I$. ...
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39 views

Riesz basis $\{|t|^\alpha e^{int}\}_n$

The class of Riesz bases is very large. It is extremely difficult to exhibit at least one bounded basis for a Hilbert space that is not equvalent to an orthonormal basis. We mention without proof the ...
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2answers
64 views

Examples of spectral decompositions

I would like examples of spectral decompositions and how they are obtained for normal compact operators and normal non-compact operators on an infinite dimensional hilbert space. I have googled it, ...
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0answers
34 views

Range of $S$ is orthogonal to the kernel of $L$

Suppose that $L: H \to H' $ is a subjective continuous linear transformation between Hilbert spaces. If $S: H' \to H$ is also a continuous linear transformation such that $ LS = I$. Show that the ...
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1answer
32 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) ...
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105 views

The support of Gaussian measure in Hilbert Space $L^2(S^1)$ with covariance $(1-\Delta)^{-1}$

Let $\mu$ be Gaussian measure defined on Hilbert space $\mathcal{H}=L^2(S^1)$ ($S^1$ - circle) by formula $$ \int e^{(f,g)} d\mu(f) = e^{-\tfrac{1}{2}(g,C g) }. $$ The covariance operator $C$ is ...
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1answer
67 views

The eigenvalues of a compact and self-adjoint operator on Hilbert space

Show that if $K$ is a compact self-adjoint operator on Hilbert space then it has either finitely many eigenvalues or a sequence of eigenvalues $\lambda_n\to 0$ as $n\to \infty$.
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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 ...
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2answers
128 views

Questions about the proof of the Riesz representation theorem

Let $H$ be Hilbert space, $f:H \rightarrow \Bbb F$ linear and bounded map. I'm trying to prove that there exists only one $z_0 \in H$ such that: $ \forall_{x \in H} : f(x)=\langle x,z_0\rangle$ ...
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1answer
85 views

If every closed subspace of a Banach space has a closed orthogonal complement, then it is a Hilbert space.

My professor mentioned this fact in class. FACT: If every closed subspace of a Banach space has a closed orthogonal complement, then it is a Hilbert space. He mentioned that he had never seen the ...
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1answer
70 views

Prove that $A^2$ is an Hilbert Space.

We denote by $A^2$ the space of analytic functions on $B_1=\{z=x+iy\in \mathbb{C}, x,y\in \mathbb{R}||z|<1\}$, such that $$\left(\int\int_{B_1}|f(z)|^2 dx \, dy\right)^{1/2}<+\infty$$ In $A^2$, ...
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1answer
37 views

Inequality on Hilbert spaces in order to prove the nonexpansivity of a mapping.

I have an application $T\colon H\to H$ (where $H$ is a Hilbert space) such that $$(Tx-Ty,x-y)\leq \|x-y\|^2,\forall x,y\in H$$ where $(\cdot,\cdot)$ is the inner product of $H$ and $\|\cdot\|$ its ...
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1answer
34 views

Subspaces of finite dimensional Hilbert spaces

This might be a trivial question but please point out exactly where my reasoning is incorrect. Is every subspace of $\mathbb{R}^n$ closed since $\mathbb{R}^n$ with the dot product is a finite ...
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1answer
172 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 ...
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1answer
47 views

On the isomorphism between bounded sesquilinear forms and bounded operators between two Hilbert spaces

Let $H$ and $K$ be two Hilbert spaces. Let $S(H,K)$ be the vector space of bounded sesquilinear forms $u:H\otimes \overline{K}\to\mathbb{C}$, and let $B(H,K)$ be bounded linear operators from $H$ to ...
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1answer
62 views

Confusion related to hilbert space

I was reading this article related to Hilbert spaces I didn't get why the first function space is not Hilbert space. I mean I can define the same norm $||f|| =\max_{a\leqslant x\leqslant b} ...
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1answer
66 views

Why is $L^2$ function Hilbert space not defined for Riemann Integral

The space of square Lebesgue integrable functions is said to be a Hilbert space. Why is if the integral is Riemann then this is not a Hilbert space? In other words, why not the space of Riemann square ...
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29 views

homomorphism question

Let B(H) be the set of bounded linear operators on a hilbert space H. Let F be a unital commutative subspace of B(H). Give an example of a homomorphism h from F to the complex numbers such that h is ...
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58 views

Surjective homomorphism example

What is an example of a surjective homomorphism $B(H)\to\mathbb C$, where $B(H)$ is the set of bounded linear operators on a Hilbert space $H$, and $\mathbb C$ is the complex numbers.
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35 views

Zero Operators on Complex Hilbert Space

This is a problem from Kreyszig's Introdcutory Functional Analysis with Applications. If for any $x$ in a complex Hilbert Space $<Tx, x> = 0$, show that $T\equiv 0$. Any clue?
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29 views

A couple of proofs on a spectrum

Let $T$ be a normal bounded operator. Let ${\lambda}$ be in $({\sigma}(T))$. Without invoking general algebra theories, show that: a) $p({\lambda},{\lambda}^*)$ is in $({\sigma}(T))$ for all ...
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1answer
44 views

Gelfand transform explicity

Let $T$ be a bounded normal operator. Let $A$ be the algebra generated by $T$ and $T^*$. What is the explicit Gelfand transform $G:A\to C(\sigma(T))$? My book says the image of $T$ is the ...
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1answer
62 views

Expression for orthogonal projection onto Hilbert space (is related to Galerkin method)

Let $H=L^2(\Omega)$ and $V=H^1(\Omega)$. Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily orthogonal). Let $V_m = \text{span}(v_1, ..., v_m)$. Define a projection operator $P_m:H ...
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52 views

Hahn-Banach separation theorem for Hilbert spaces

What is the strongest form of the Hahn-Banach separation theorem for Hilbert spaces? Could you please provide a reference?
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173 views

Does this statement about Hilbert spaces make any sense?

I have found this tweet about git and don't know what to make of it. I think it's written as a joke, but it could have been written in Chinese, and I'd understand ...
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63 views

Difference between unconditional and absolute convergence in Banach spaces

One can show that in any finite-dimensional normed vector space absolute convergence is equivalent to unconditional convergence. It's not hard to show that if we have an orthonormal sequence in ...
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1answer
41 views

Direct product of a family of Hilbert spaces

Let $\{H_i\}_{i\in I}$ be a family of Hilbert spaces, defined $$H = \{f\in \Pi_{i\in I} H_i, \sum_{i\in I}|f(i)|^2<\infty\}$$ and inner product $\langle f,g\rangle : = \sum_{i\in I} \langle ...