Complete normed spaces whose norm comes from an inner product.
1
vote
0answers
17 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.
4
votes
0answers
36 views
Concerning unbounded linear operators on a Hilbert space
Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
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 ...
2
votes
1answer
16 views
Example for a sequence of operators converging pointwise, but not with respect to the operator norm
I am trying to understand the following example.
Define $$T_n: l^2 \rightarrow l^2$$
$$T_n(x)=(0, ..., 0, x_{n+1}, ...).$$
It's rather clear that $T_n(x)$ converges for $0$ for every $x \in l^2$. ...
5
votes
2answers
60 views
Hilbert space with all subspaces closed
Does there exist an infinite-dimensional Hilbert space with all subspaces closed?
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 $\mathbb{E}X_nY \rightarrow \mathbb{E}XY$) ...
0
votes
1answer
27 views
Can a Accumulation Point be an Eigenvalue?
I have a discrete (separable) infinite dimensional Hilbert Space with a compact operator defined on it. So 0 is an accumulation point (some theorem says so). Can 0 also be an eigenvalue? And how would ...
2
votes
0answers
49 views
How to prove $\mathcal{L}^2[(0,1)]$ is a Hilbert Space
Let $\mathcal{L}^2[(0,1)]$
denote the set of $\mathbb{C}$-valued square integrable functions on the interval [0,1].
Prove that $\mathcal{L}^2[(0,1)]$ forms a Hilbert Space.
I believe that I can ...
1
vote
1answer
33 views
What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.
I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
0
votes
1answer
29 views
Prove a non-empty subset is closed in an inner product space
I hope someone would be able to help me with the finer details of this proof.
Problem:
M is a non-empty set in an Inner Product Space (IPS) X.
I need to show that the annihilator of M which is ...
0
votes
1answer
38 views
Need explanation of problem in Temam (convergence, weak derivatives)
Let $V \subset H \subset V$ be Hilbert triple. We have $u_m$ is infinite differentiable from $[0,T]$ to $V$.
Suppose $u_m \to u$ in $L^2(0,T;V)$ and $u_m' \to u'$ in $L^2(0,T;V^*)$
Suppose that it ...
2
votes
1answer
23 views
If $u_m \to u$ and $v_m \to v$, does $b(u_m,v_m) \to b(u,v)$?
In a Hilbert space $H$, if $u_m \to u$ and $v_m \to v$, does $b(u_m,v_m) \to b(u,v)$ if $b$ is a bounded bilinear form on $H$?
2
votes
1answer
45 views
Spectral Theorem for bounded compact, self-adjoint operators as corollary of Hilbert-Schmidt theorem
I'm following Debnath and Mikusinksi's "Introduction to Hilbert Spaces with Applications" and am trying to understand how the spectral theorem for compact self-adjoint operators is a corollary of the ...
2
votes
1answer
29 views
Is every symmetric bilinear form on a Hilbert space a weighted inner product?
Is every symmetric bilinear form on a Hilbert space a weighted inner product?
i.e. can I write that $b(u,v) = (wu,v)_H$ for all $u, v \in H$?
I am not sure about this. Maybe something to do with Riesz ...
0
votes
1answer
52 views
self-adjoint operator proof
Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. A number $\lambda \in C$ is called an approximate eigenvalue of T if there is a sequence ${X_n} \subset D(T)$, with ...
2
votes
0answers
23 views
Orthogonal projections for minimization problem
I have trouble to understand the existence of a minimization problem in a Hilbert space. Let $(\Omega,\mathcal{F}_T,P)$ be a filtred probability space with filtration $(\mathcal{F}_t),0\le t\le T$. We ...
2
votes
1answer
43 views
Spectrum proofs
Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. Show that if $\lambda$ is a point in the residual spectrum of $T$, then $\bar{\lambda}$ is in the point spectrum of the ...
0
votes
1answer
62 views
proof related to Hilbert Spaces
Let $T$ be a bounded linear compact operator on a Hilbert space $H$ over $C$, $A$ is a positive self-adjoint operator on $H$. How to show that $T=UA$ where $U^{+}U=I$ on the range $R(A)$ of $A$
2
votes
1answer
53 views
Limit of a sequence in the space $\ell_2$
I have difficulties in the following problem.
Let $H=\ell_2$ be the space of square-summable sequences. Let $\alpha\in (0,1)$ and $\{u^k\}\subset H$ be such that
$$
u^{k+1}=(1-\alpha)u^k+\alpha ...
1
vote
1answer
51 views
exercise: limit orthonormal sequence, “Banach Space Theory”
I have an exercise from "Banach Space Theory":
Suppose $\{x^k\}_{k=1}^\infty$ is an orthonormal sequence in $l_2$, where $x^k:=(x_i^k)$. Show that $\lim_{k\rightarrow \infty} x_i^k =0 \ \forall_{i\in ...
1
vote
1answer
54 views
Computing an explicit solution to an integral equation via the Neumann Series.
I am hoping that someone can provide guidance for solving the integral equation
$$u=f+\lambda Au$$
where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
1
vote
0answers
32 views
Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit
$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
0
votes
1answer
25 views
Self-adjoint operator on a Hilbert space.
Let $T$ be a self-adjoint operator on a Hilbert space $H$. If for all $x\in H$, $\langle Tx,x\rangle=0$, is $T=0$?
0
votes
1answer
23 views
Is my proof of characterisation of self-adjoint operators on complex Hilbert spaces okay?
I wish to show the following theorem:
Let $T:H\to H$ be
a bounded linear operator on a complex Hilbert
space $H$. Then if $\left\langle Tx,x\right\rangle \in\mathbb{R}$
for all $x\in H$, then $T$ is ...
1
vote
1answer
28 views
Orthogonal Projectors
Please, I need help with this proble.
Let $(H,\langle\cdot,\cdot\rangle)$ be a Hilbert space and let
$V_1,V_2,\ldots,V_N$ closed subspaces, mutually orthogonal of $H$,
that is, $v_i\perp v_j$ ...
2
votes
2answers
66 views
Hahn-Banach theorem (second geometric form) exercise #2
Let $X$ be a Hilbert space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that
$$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F),$$
and any kernel of the involved functionals is ...
2
votes
1answer
31 views
Two isomorphic inner product spaces, one is complete, is the other also complete?
If you two have inner product spaces and one is complete, and there is an isomorphism between the two spaces, is the other space also complete?
Or do we absolutely require equivalence of norms?
0
votes
0answers
49 views
existence of solution - exercice
let the problem $$-u'' + a(x) u = f , x \in \Omega = ]0,1[, u'(0) = u(0); u(1) = -1$$
where $f \in L^2(\Omega) , a(x) \geq a_0 > 0, a \in L^{\infty}(\Omega)$
1- Prove that the variational ...
4
votes
1answer
76 views
Hahn-Banach theorem (second geometric form) exercise
Let $X$ be a vector normed space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that
$$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F).$$
Apply the Hahn-Banach theorem (second ...
-1
votes
1answer
27 views
if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then$ f\in H^k (\Omega')$
prove that if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then $f\in H^k (\Omega')$
1
vote
0answers
33 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
27 views
Riesz representation theorem on Hilbert space with equivalent norms
If we have a Hilbert space that has two equivalent norms (and inner products), are the Riesz maps (from Riesz representation theorem) associated with each inner product the same?
0
votes
1answer
19 views
Equivalent norms and density/separability
$V \subset H$ are Hilbert spaces with inner products $(\cdot,\cdot)_V$ and $(\cdot,\cdot)_H$. Suppose $V$ is dense in $H$ and both spaces are separable. If $(\cdot,\cdot)_{V_2}$ and ...
1
vote
2answers
55 views
Sequence of operators in a Hilbert space
The question is:
Let $H$ be a Hilbert space and $\{T_n\}$ be a sequence in $B(H)$ such that $\lim_{n\rightarrow\infty}\langle x, T_n y \rangle = 0$ for all $x, y \in H$. Prove or disprove $\sup_n ...
2
votes
2answers
50 views
little question about linear operators
Let H be a complex Hilbert Space. Let $P \in L(H)$ be an idempotent operator ($P^{2} = P$). Also, let $\parallel P\parallel = 1$. I want to prove that $P$ is an orthogonal operator. I defined $M = ...
0
votes
1answer
60 views
How can I able to show that $(S ^{\perp})^{\perp}$ is a finite dimensional vector space.
Let $H$ be a Hilbert space and $S\subseteq H$ be a finite subset. How can I able to show that $(S ^{\perp})^{\perp}$
is a finite dimensional vector space.
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
1answer
33 views
some inclusions regarding linear operators
Let $H$ be a Hilbert Space and $T:H\rightarrow H$ a linear operator.
Let $T^*$ be the adjoint operator of $T$ and let $\operatorname{Cl}(X)$ be the topological closure of the set X and $X^{\perp}$ ...
0
votes
1answer
21 views
Does this dual space functional pairing = 0 imply functional = 0?
If $V$ is a Hilbert space, is it true that if $\phi_1, \phi_2 \in C_c^\infty(0,T)$,
$$\int_0^T \langle \phi_1(t)g +\phi_2(t) f, v \rangle_{V', V} = 0$$ for all $v \in V$, then $\phi_1g + \phi_2f ...
1
vote
0answers
15 views
Weighted inner product space and representation of dual space
Let $H$ be a Hilbert space and define $H_c$ to be the weighted Hilbert space with inner product
$$(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 ...
6
votes
1answer
50 views
Approximating a Hilbert-Schmidt operator
Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$
where ...
1
vote
1answer
30 views
Why is $\langle f, u \rangle_{H^{-1}, H^1} = (f,u)_{L^2}$ when $f\in L^2 \cap H^1$ and not $\langle f, u \rangle_{H^{-1}, H^1}=(f,u)_{H^1}$?
More generally, if $V \subset H \subset V'$ are Hilbert spaces, why is $$\langle f, u \rangle_{V',V} = (f,u)_{H}$$ when $f\in H \cap V$ and not $$\langle f, u \rangle_{V',V}=(f,u)_{V}?$$
Is this what ...
2
votes
2answers
83 views
True or False; Functional Analysis
Given $T: V \to W$ with $V,W$ being Hilbert Spaces. We always have $\| T^ *\| = \| T \|$.
I think it is true because of Riesz' Theorem, but I am not sure if a proof is necessary.
EDIT: In case ...
0
votes
0answers
8 views
Weak limits and structure of a generated semigroup
I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following:
Theorem. (Jacobs–Glicksberg–de Leeuw ...
1
vote
1answer
66 views
Bounded linear operator in weak topology
Let $B$ be a bounded linear operator on $H$. Prove $B\colon (H,w)\to (H,w)$ is continuous. $(H,w)$ is a Hilbert space with its weak topology.
3
votes
0answers
63 views
Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.
For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$
Here is a quick one. If someone could improve this it would be great
Proof
By Cauchy Schwarz, $\langle x,z \rangle ...
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 ...
