Complete normed spaces whose norm comes from an inner product.
4
votes
1answer
283 views
Reproducing kernel Hilbert spaces and the isomorphism theorem
A reproducing kernel Hilbert space is a Hilbert space in which the evaluation functional
$L_x : f \rightarrow f(x)$ is continuous. By continuity, the Riesz representation theorem says that this ...
3
votes
0answers
95 views
Isomorphic Hilbert spaces
As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
3
votes
1answer
71 views
How to show projection of $L^2$ function converges to that $L^2$ function
My teacher said that if $P_n f = \sum_{j=0}^n(f,w_j)w_j$, where $w_j$ is orthonormal basis of $L^2$, then $|P_n f- f|_{L^2} \to 0$ for $f \in L^2$. How do I prove this?
I thought
$$|P_nf - f| = ...
3
votes
2answers
124 views
Find adjoint operator of an operator T
I would like to find the adjoint operator of
$$
T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds.
$$
Here $H^1([0,1])$ is the Sobolev space $W^{1,2}([0,1])$.
I tried to find ...
3
votes
3answers
157 views
Functional analysis-Hilbert spaces
Let $ X$ be an inner product space.
Show that $ X$ is a Hilbert space if and only if for each continuous linear functional $ L$ on $ X$,there exists $ z\in X$ such that $ L(x)=\langle x,z\rangle $ .
...
3
votes
1answer
130 views
Orthonormal basis for Sobolev Spaces
Sobolev spaces of order 2 are known to form a Hilbert space. Consider such a Sobolev space of (order 2) functions on the domain $f:\mathbb{R}\rightarrow \mathbb{R}$. What is an example for the basis ...
3
votes
2answers
156 views
Distance of functions defined on a Hilbert Space
In our Topology class, we touched on Hilbert spaces for a couple of weeks. I've been studying various problems around the topics we covered, and I came across this one on a list of supplemental ...
2
votes
2answers
70 views
Hahn-Banach theorem (second geometric form) exercise #2
Let $X$ be a Hilbert space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that
$$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F),$$
and any kernel of the involved functionals is ...
2
votes
1answer
54 views
Unbounded operator $T $ is bounded below when $\overline T$ is bounded
How to prove the following?
A densely defined symmetric operator $T$ in Hilbert space $H$ has a closure $\overline T$ which is bounded iff both $T,-T$ are bounded below (there exist constants $c,c' ...
2
votes
1answer
109 views
Prove or disprove that the given expression is “always” positive
I have previously asked a question and I tried to solve it by my own and it led to the question below:
Prove or disprove that
...
2
votes
2answers
68 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. ...
2
votes
1answer
86 views
Can this Lemma be extended a little?
Consider this lemma (my question are below):
Lemma Given three pairwise orthogonal subspaces $X$, $Y$, $Z$ of a Hilbert space $H$ that span the whole space, any vector $\nu\in H,\ ||\nu||=1$, can ...
2
votes
1answer
297 views
Bounded linear operator on a Hilbert space
I am having a bit of difficulty with the following homework problem.
Let $\{x_n\}$ be an orthonormal basis in a Hilbert space $V$ over $\mathbb{C}$ and let $\{c_n\}_{n \in \mathbb{N}}$ be a fixed ...
2
votes
2answers
351 views
Hilbert Schmidt operators as an ideal in operators.
Let $H$ be a Hilbert space. For $\{e_n\}$ an orthonormal basis of $H$, we call $T\in B(H)$, a Hilbert Schmidt operator if
$ \|T\|_2^2:=\sum_n \|Te_n\|^2 <\infty.$
I have seen somewhere before ...
2
votes
1answer
134 views
A question on norm of error vector
Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
2
votes
3answers
657 views
Compact operator
If $H$ and $K$ are Hilbert spaces,show that if $T:H\longrightarrow K$ is a compact operator and $\{e_{n}\}$ is any orthonormal sequence in $H$ then $\|Te_{n}\|\to0$.Is the converse true?
thanks.
1
vote
1answer
236 views
$\ell_p$ is Hilbert space if and only if $p=2$
Can anybody please help me to prove this..
Let p greater than or equal to 1,show that the space of all p-summable sequences is an inner product space if and only if p=2
1
vote
1answer
74 views
weak convergence condition
Let $l^{2}=\left\{x=(x^{(1)},x^{(2)},...):\sum_{i=1}^{\infty
}\left\vert x ^{(i)}\right\vert ^{2}<\infty \right\} $. Would you help me
to prove that $({\vert|x_n |\vert})$ is bounded sequence and ...
1
vote
1answer
108 views
Unique extension to a bounded operator
Suppose $\left\{ e_{1},e_{2},\ldots\right\} $ is an orthonormal basis for a Hilbert space $\mathcal{H}$ and for each $n$ there is a vector $Ae_{n}$ in $\mathcal{H}$ such that $\sum\left\Vert ...
1
vote
1answer
41 views
Can this type of series retain the same value?
Let $H$ be a Hilbert space and $\sum_k x_k$ a countable infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation ...
1
vote
1answer
217 views
Numerical range of an operator on Hilbert spaces.
If $H$ is a Hilbert space and $T$ is in $\mathcal{L}(H)$, the numerical range of $T$ is defined by
$$W(T) := \left\{(Tx; x) \mid x \in H,\ \|x\| = 1 \right\}.$$
We have to prove that
The point and ...
1
vote
3answers
465 views
Question on weak convergence ( Example).
Can anybody tell me why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$. I can't see how $\sin(nx)$ can converge?
Explanation with any other example will be nice as well.
0
votes
1answer
103 views
Averaging differential forms
Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1.
I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
0
votes
0answers
64 views
If a sequence is not a frame
A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0<A\leq B<\infty$ such that
$$ A\|f\|^{2} \leq \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq ...
