Complete normed spaces whose norm comes from an inner product.
1
vote
1answer
51 views
Orthonormal Family in a Hilbert Space
If we have an orthonormal family, $\{u_n\}_{i=1}^\infty$ in a Hilbert Space $H$, I need to show that for $x\in H$ we have the following inequality:
$$\left|\left\{n|\langle x, u_n \rangle > ...
1
vote
1answer
94 views
Finding Riesz basis
Let H be a Hilbert space .Is there always a non orthogonal Riesz basis $D$ on it such that following holds?
$$\sup_{g\in D }\sum_{g'\in D,g'\not=g}|\langle g,g'\rangle|<1/3 $$
And is there Riesz ...
1
vote
1answer
53 views
Banach space geometry without bounded operators?
I understand that $B(X)$ can be think of as the collection of symmetries of a Banach space $X$, and that they provide important information concerning the geometric structure of the space. But I am ...
1
vote
1answer
103 views
Show the existence and uniqueness of a closed ball containing a bounded subset of a Hilbert space
The problem:
Assume $A$ is a bounded subset of a Hilbert space $H$. Let $r$ be the infimum of the radii of closed balls containing $A$, so
$r = \inf \{s \geq 0 $ $\vert$ there exists $x \in H$ such ...
1
vote
1answer
73 views
weak convergence condition
Let $l^{2}=\left\{x=(x^{(1)},x^{(2)},...):\sum_{i=1}^{\infty
}\left\vert x ^{(i)}\right\vert ^{2}<\infty \right\} $. Would you help me
to prove that $({\vert|x_n |\vert})$ is bounded sequence and ...
1
vote
1answer
108 views
Unique extension to a bounded operator
Suppose $\left\{ e_{1},e_{2},\ldots\right\} $ is an orthonormal basis for a Hilbert space $\mathcal{H}$ and for each $n$ there is a vector $Ae_{n}$ in $\mathcal{H}$ such that $\sum\left\Vert ...
1
vote
1answer
79 views
Contractibility of the sphere and Stiefel manifolds of a separable Hilbert space
Why are the sphere $$S=\lbrace |x|=1\rbrace$$ and the Stieffel manifolds of orthonormal $n$-frames $$V_n=\lbrace (x_1,\dots,x_n)\in S^n\mathrm{~s.t.~}i\neq j\Rightarrow\langle ...
1
vote
1answer
126 views
Proving Fréchet differentiability
Am learning about Fréchet differentials and was wondering if for a real matrix $X$ and positive semidefinite real matrices $A,B$ the function $f(X)=TrX^TAX-X^TBX$ is twice Fréchet differentiable or ...
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
77 views
Reproducing Kernel Hilbert Space- notation and basics
Am reading about Reproducing Kernel Hilbert Space(RKHS) while reading through Functional Analysis and Hilbert Space material and am unable to get the notation :
$k(·,xi)$ correctly. What does the dot ...
1
vote
1answer
133 views
Representing with Hilbert Schmidt Norm
Am trying to see, if the following Trace function can be expressed using a Hilbert Schmidt Norm: $\operatorname{Tr}(X^TAX)$. Here, $X$ is a matrix whose entries take values that are finite and reals ...
1
vote
1answer
77 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} ...
1
vote
2answers
95 views
Cauchy+pointwise convergence $\Rightarrow$ uniform converges (for an operator in a Hilbert space)
Suppose that the sequence of operators in a Hilbert space $H$, $\left(T_{n}\right)_{n}$,
is Cauchy (with respect to the operator norm) and that there is an
operator $L$, such that ...
1
vote
1answer
123 views
Analysis operator $T_\Phi$ is injective and has a closed range
Definition of the problem
Let $\mathcal{H}$ be a separable Hilbert space on $J\subset\mathbb{N}$
an index set. Let $\Phi:=\left(\varphi_{j}\right)_{j\in J}\subset\mathcal{H}$
be a frame for ...
1
vote
1answer
47 views
An explicit example of an invariant halfspace of the unilateral shift?
In a recent talk, A. Popov stated the following fact
The unilateral shift on $\ell^2$ has invariant halfspaces.
Halfspaces are closed subspaces whose dimension and codimension are both infinite.
...
1
vote
2answers
84 views
closedness of image of closed, unbounded operator
I want to prove the following:
Suppose $\mathcal{H}_1$ and $\mathcal{H_2}$ are Hilbert spaces and let $T: \mathcal{D} \rightarrow \mathcal{H}_2$ be a closed operator, where $\mathcal{D} \subset ...
1
vote
1answer
83 views
Changing of integration and operator
I have a question which maybe looks very simple: Let $T$ be an orthogonal projection on a Hilbert space $H$. If $g(x,u)\in H$, for all $u\in \mathbb R$, and the inner product is defined by
$$\langle ...
1
vote
1answer
95 views
Checking axioms for inner product
I'm going through a question checking that an inner product satisfies the inner product axioms. I have a Hilbert space $H=C[-1,1]$ and for $f,g\in H$ the inner product is defined as
$$\langle ...
1
vote
1answer
150 views
Hilbert Spaces and Closed Subspaces
Let $H$ be a Hilbert Space, and $M$ a closed subspace. Is it true that
$H = M \bigoplus M^{\perp}$
Does this hold if $M$ is not closed? Or only if $H$ is finite/infinite dimensional?
1
vote
1answer
107 views
Hilbert space $H$ is strictly smooth
I am trying to show that every Hilbert space $H$ is strictly smooth with modulus of smoothness $\phi_H(t)=\sqrt{1+t^2} -1 $.
To show this I think I should show $H$ is uniformly smooth first.
...
1
vote
1answer
104 views
Null Space and Range of Particular kind of Operator on Hilbert Space
Let $H$ be the real separable Hilbert space with orthonormal basis $\{e_n\}$ and consider the operator $T:H \times H \to H \times H$ given by
$$T(\sum a_ne_n, \sum b_ne_n) = \sum A_n(a_ne_n, ...
1
vote
2answers
256 views
Question about limits of weakly convergent sequence in $H^1_0(\Omega)$
Let $H = H_{0}^{1}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ whose boundary $\partial\Omega$ is a smooth manifold. We know that the embedding $$H\hookrightarrow L^s(\Omega)$$ is compact for ...
1
vote
1answer
80 views
Riesz sequences in Hilbert spaces
Is it true that if $\{x_{n}\}_{n=1}^{\infty}$ is a finite union of Riesz sequences in a Hilbert space H, then $\{x_{n}\}$ itself will be a Riesz sequence? What about Frames and Bessel seuences, do we ...
1
vote
1answer
138 views
Show $\{u_n\}$ orthonormal, A compact implies $\|Au_n\| \to 0$
I'm having a bit trouble with this homework exercise.
Let $\mathcal{H}$ be a Hilbert space and $\{u_n\}_{n=1}^\infty$ an
orthonormal sequence in $\mathcal{H}$. Let $A$ be a compact operator
on ...
1
vote
0answers
15 views
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
1
vote
0answers
20 views
closest point property of subset of Hilbert space - what are the conditions for existence of inf?
I'm proving the closest point property of a subset of a Hilbert space, ie:
$$H$$
is a Hilbert space with a norm generated by the inner product and so on.
$$h\in H$$
is a point in H
$$M\subset H$$
M ...
1
vote
1answer
28 views
Weak convergence in Hilbert space L2 implies convergence in distribution?
Does weak convergence in $L^2$ (for $X_n, X \in L^2$ we say that $X_n$ converges weakly to $X$ ($X_n \rightarrow^w X$) if for every $Y\in L^2$ we have $\mathbb{E}X_nY \rightarrow \mathbb{E}XY$) ...
1
vote
1answer
53 views
Computing an explicit solution to an integral equation via the Neumann Series.
I am hoping that someone can provide guidance for solving the integral equation
$$u=f+\lambda Au$$
where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
1
vote
0answers
32 views
Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit
$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
1
vote
0answers
32 views
Unbounded self- adjoint and von Neumann algebra
I am reading Conway's Functional Analysis. Here is one exercise problem.I don't know how to show the following fact. For unbounded self-adjoint $T$ in Hilbert space $H$
1) $T$ commutes with its Borel ...
1
vote
1answer
47 views
Orthogonal family in Hilbert Space
Let $(x_k)_1^\infty$ be an orthogonal family of points in X a Hilbert space. Then $\sum_{i=1}^\infty x_i$ converges if and only if $\sum_{i=1}^\infty ||x_k||^2$ converges. Also need to show that ...
1
vote
0answers
15 views
Weighted inner product space and representation of dual space
Let $H$ be a Hilbert space and define $H_c$ to be the weighted Hilbert space with inner product
$$(u,v)_{H_c} = c(u,v)_H$$
where $c$ is a positive constant.
Then is it true that
$$c\langle f, u ...
1
vote
0answers
38 views
Countable orthonormal basis of product of separable Hilbert spaces
If I have 2 separable Hilbert spaces $X$ and $Y$ which have (different) orthonormal bases $x_i$ and $y_i$, then clearly $x_i \times y_j$ is a basis for $X \times Y$ (which is also a separable space).
...
1
vote
1answer
73 views
Riesz Representation theorem-pde
Consider $\sum_{i,j=1}^n \displaystyle\int_{\mathbb{R}^n} \dfrac{\partial^2 u}{\partial^2 x_i} \overline{\dfrac{\partial^2 v}{\partial^2 x_j} } dx + \lambda \displaystyle\int_{\mathbb{R}^n} u ...
1
vote
1answer
22 views
Is the Strong Limit of a Linear Operator in a Hilbert Space the Same as the Norm Limit?
If $H$ is a Hilbert Space, and I have an operator $F:H \rightarrow H$ which is the limit of a sequence of operators $F_n$ with respect to the operator norm; and this same sequence of operators ...
1
vote
1answer
38 views
Weak convergence-exercice
Let $\Omega$ be an open set in $\mathbb{R}^n$ and let $(u_n)$ be a bounded sequence in $H^1_0(\Omega).$
Who's the theorem say that we can extract a subsequence denoted $u_{n}$ as $u_n$ weakly ...
1
vote
0answers
30 views
How can projection operators be limits of powers of unitary operators?
Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact ...
1
vote
1answer
57 views
Calculating the Norm of an operator in $L^2(0,1)$
If I have the following operator for $H=L^2(0,1)$:
$$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this:
I know that in $L^2(0,1)$ we have that ...
1
vote
0answers
30 views
Closed unit ball in infinite dimensional normed linear space
I have to prove that in any infinite dimension normed linear space we have that the closed unit ball is not compact.
I know that I have to construct a sequence such that $||x_n||=1$ and ...
1
vote
0answers
38 views
Orthogonal Projection on hilbert spaces
I found this exercise on a book, I guess it's not hard but don't know what to do.
Let $H$ be a Hilbert space and let $P:H \rightarrow H$ be linear. If $P$ is a projection, i.e $P^2 =P$, and ...
1
vote
0answers
26 views
Prove that $S$is a closed subspace of $H^2$ invariant under multiplication by $z$. Find the inner function $F$ such that $S=FH^2$
Let ${\alpha_n}$ be a sequence of points in the open unit disc such that $\sum(1-|\alpha_n|)<\infty$. Let $S$ be the set of all functions $f$ in $H^2$ spaces such that $f(\alpha_n)=f'(\alpha_n)=0$ ...
1
vote
0answers
51 views
Confused about Bessel's inequality
I know that if $H$ is a Hilbert space and $(e_{j})_{j\in\mathbb{N}}$ is an orthonormal system in $H$ and $f\in H$. Then one has Bessel's inequality
$$\sum_{j=1}^{\infty}|\langle f,e_{j}\rangle ...
1
vote
1answer
44 views
Particular series on Hilbert Space
Let $(H, \langle\cdot,\cdot\rangle)$ a Hilbert space and consider a sequence $\{x_n\}_{n\in\mathbb{N}}$ of $H$ such that:
$$\langle x_n,x_m\rangle\ =\ \delta_{mn}\ =\ \left\{\begin{array}{ll}1, & ...
1
vote
1answer
35 views
Norm of oblique projector and angle between subspaces
Take $V$ and $W$ closed subspaces of $H$ a Hilbert space with $V\oplus W=H$ (we'll assume this holds in the sequel, it may not be required everywhere but in the context of interest, it is always ...
1
vote
1answer
85 views
Hilbert spaces and orthogonality sets
I need to prove if $X$ is a Hilbert space and $M$ and $N$ it's closed:
$$
(M+N)^\perp=M^\perp\cap N^\perp
$$
thanks
1
vote
1answer
83 views
Orthonormal basis in Hilbert spaces
I have a general question but I'm going got ask it in a very restrictive setup.
It is known that an equivalent condition for a system $\left\{e^{i\lambda t}\right\} _{\lambda\in\Lambda}$ being an ONB ...
1
vote
1answer
68 views
Infimum of a Hilbert space inner product
This is exercise 5.11 in Brezis's Functional Analysis, Sobolev Spaces, and PDEs.
Let $H$ be a Hilbert space, and let $M \subset H$ be a nonzero closed linear subspace. Let $f \in H$, $f \notin ...
1
vote
0answers
37 views
Hilbert basis of $L^2([-1,1])$?
Could you please specify hilbert basis of $L^2([-1,1])$? How will be the representation of a function f $\in L^2([-1,1])$ by means of its Fourier series?
My solution:
$E_k=1/\sqrt2 e^{kit\pi}, k\in ...
1
vote
1answer
31 views
Kernel inclusion implies factorization
I have a question whether a certain fact is true for arbitrary operators on a Hilbert space. Namely, consider Hilbert spaces $H,K$, an operator $A\in B(H)$ and another $B\in B(H,K)$. Moreover, assume ...
1
vote
1answer
24 views
Functional to inner product in Hilbert triple
If $V \subset H \subset V^*$ is a Hilbert triple, and $f \in V^*$, I cannot represent $f(v) = (e,v)_V$ because we don't identify $V$ with $V^*$. But is it true that
$f(v) = (e,v)_H$ for some $e$?
