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

learn more… | top users | synonyms

2
votes
1answer
59 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
2answers
27 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
0
votes
1answer
26 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.
2
votes
3answers
137 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
84 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, ...
1
vote
3answers
74 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 ...
2
votes
1answer
230 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 ...
0
votes
0answers
72 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 ...
1
vote
1answer
59 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?
1
vote
1answer
68 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 ...
0
votes
1answer
16 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 ...
5
votes
1answer
103 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 ...
2
votes
0answers
68 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 ...
0
votes
2answers
92 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
votes
1answer
30 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 $\langle f,g \rangle = \int^1_0 f'(t)g'(t)dt$. ...
2
votes
2answers
611 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 ...
-1
votes
0answers
83 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 ...
0
votes
3answers
349 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.
1
vote
1answer
99 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 $LS = I$. ...
1
vote
0answers
53 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 ...
4
votes
2answers
72 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, ...
0
votes
1answer
37 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) ...
2
votes
0answers
138 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 ...
0
votes
1answer
110 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 $n\to \infty$.
1
vote
1answer
58 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 ...
2
votes
2answers
149 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$ ...
1
vote
1answer
161 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 ...
3
votes
1answer
76 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 $A^2$, ...
0
votes
1answer
41 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 ...
0
votes
1answer
45 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 ...
5
votes
1answer
237 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
1answer
61 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 ...
0
votes
1answer
73 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} ...
0
votes
1answer
92 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 ...
0
votes
1answer
61 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.
1
vote
1answer
51 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?
1
vote
1answer
50 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 ...
1
vote
1answer
76 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 ...
2
votes
0answers
68 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?
5
votes
2answers
305 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 ...
1
vote
0answers
97 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 ...
4
votes
1answer
46 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 ...
4
votes
0answers
81 views

Reproducing kernel Hilbert sapce

I encountered the following claim (verbatim): Theorem Let $V$ be a subspace of $L^2(\mathbb{R})$ and $\{e_n\}$ be a orthonormal basis of $V$. The $V$ is a reproducing kernel Hilbert space with kernel ...
1
vote
1answer
47 views

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$?

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$? Clearly, every such arrow is a split monomorphism; further, if such an $f$ is ...
0
votes
1answer
35 views

Gelfand Transform in a specific case

What is the gelfand transform of an operator in the algebra generated by a bounded normal operator and it's adjoint? Thanks
2
votes
1answer
53 views

Unbounded extension of bounded operator

Is it possible to construct an unbounded extension of bounded densely defined operator? To be more concrete, let $\mathcal{H}$ be Hilbert space, $\mathcal{D}\subset\mathcal{H}$ - a dense subset, ...
0
votes
2answers
93 views

Parseval's identity, decomposition of inproduct.

Hoi, if $H$ is a seperable real Hilbertspace and $(e_n)$ orthonormal basis, then Parseval's identity $$\sum_n\left\langle x,e_n \right\rangle^2 = \left\|x\right\|^2 = \left\langle x,x ...
1
vote
0answers
51 views

Operators on a Hilbert space question

For a Borel measure $\mu$ define $\langle S_\mu x,y\rangle=\int_H\langle x,z\rangle \langle y,z\rangle \mu(z)$. An exercise in my book that I am reading says that I could find a $\mu$ s.t. $S_\mu$ ...
0
votes
1answer
57 views

Dense subspace in a Hilbert space

Let $H$ be a Hilbert Space and $\{e_n\}_{n\in\mathbb{N}}$ an orthonormal basis. Now let $(x_n)$ be a sequence in $H$ satisfying $$\sum_{n=1}^{\infty}||x_n-e_n||^2<1.$$ Prove that ...
0
votes
1answer
65 views

Common orthogonal basis for $L^2$ and $H^1$

How can we obtain a common orthogonal basis for the space $L^2(U)$ and $H^1(U)$ for some bounded open subset of $\mathbb{R}^n$? That this can be done is mentioned in Evans's Partial Differential ...