Complete normed spaces whose norm comes from an inner product.
3
votes
0answers
45 views
Existence of an ergodic-looking limit in a Hilbert space
This is part of a problem from Reed & Simon's Functional Analysis -- I'll write the problem first.
Let $V$ be a linear transform on the Hilbert space $H$, such that its powers are uniformly ...
3
votes
0answers
332 views
Question about example of non-separable Hilbert space
I have come across the following example of a non-separable Hilbert space:
Why do I need the discrete topology on $I$? Or more generally: why do I need a topology? If we talk about $L^p$ spaces in ...
3
votes
1answer
101 views
What is $\mathcal{C}(S^{1})$? (Where $S^1$ denotes unit circle)
What is $\mathcal{C}(S^{1})$ (Continuous function on a unit circle)? (Where $S^1$ denotes unit circle)
I saw this in a proof of showing Fourier Basis $S:=\{1,\sqrt{2}\cos{nx},\sqrt{2}\sin{nx}\}$ is ...
3
votes
0answers
125 views
Is there a deeper connection between the two Riesz's Representation Theorems?
I have been reading Kreyszig's Functional Analysis when I encountered two versions of Riesz's Representation Theorems:
(1) Every bounded linear functional $f$ on a Hilbert space $H$ can be
...
3
votes
2answers
87 views
Limit of Inner Products in Hilbert Space
Let $H$ be an infinite dimensional Hilbert space. Then there exists an orthonormal basis $\{e_{i}\}_{i = 1}^{\infty}$. Suppose we know that $\lim_{k \rightarrow \infty}(f_{k}, e_{j}) = (f, e_{j})$ for ...
3
votes
0answers
244 views
How does the parallelogram law imply the existence of an inner product for a given norm? [duplicate]
Possible Duplicate:
Norms Induced by Inner Products
I am trying to prove to that if a norm of a vector space satisfies the parallelogram law ($\| \vec x + \vec y \|^2 + \| \vec x - \vec ...
3
votes
0answers
83 views
Counting balls in Hilbert spaces
Let $W$ be a real Hilbert space of dimension $n$ and $V$ a Hilbert subspace of dimension $m$. Assume that $f_1,\cdots,f_k$ are points in $W$ such that the following holds:
there exists $\sigma>0$ ...
2
votes
3answers
813 views
Canonical examples of inner product spaces that are not Hilbert spaces?
That is, what are some good examples of vector spaces which are inner product spaces but in which not every Cauchy sequence converges?
2
votes
2answers
789 views
A proof of the Riesz representation theorem
I'm having trouble filling the steps in this guided proof of Riesz's representation theorem. (I already have a proof I can understand, but I'd like to understand this one too.)
Let $H$ be a Hilbert ...
2
votes
3answers
381 views
Orthonormal basis in $L^2(\Omega)$
In the one dimension case, where $\Omega\subseteq{\bf R}$ is a bounded domain, for example $\Omega=[0,2\pi]$, one can find the orthonormal basis for $L^2(\Omega)$:
$$\{e_n\}_{n\in {\bf Z}}$$
where ...
2
votes
3answers
298 views
$C[0,1]$ is NOT a Banach Space w.r.t $\|\cdot\|_2$
I'm trying to find a cauchy sequence in $C[0,1]$ that converges under $\|\cdot\|_2$ to a limit which isn't continuous.
Any ideas?
2
votes
1answer
105 views
Non-linear functional on $L^2$
Let $a,b,g,h$ be real numbers. How to prove that the functional $F\colon L^2 [a,b]\to \mathbb{R}$, given by
$F(u)=\int_a^b (u^2(x)-gu(x)-h)\,dx$
is continuous?
Thank you
2
votes
2answers
92 views
elliptic pdes and associated bilinear forms for Lax-Milgram
I have a simple question on elliptic pdes, actually I can not understand clearly from definitions. Thats why I want to try think on an example.
Let us have an elliptic pde $$-A \Delta ...
2
votes
2answers
129 views
$f'$ is in $L^2[0,1]$
Let $f$ is absolutely continuous function on $[0,1]$, $f(0)=0$ and $f' \in L^2[0,1]$. Would you help me to prove that there is constant $c$ such that
$$|f(t)| \leq c \left( \int_0^1 |f'(t)|^2 dt ...
2
votes
2answers
126 views
I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed.
Let $x_n$ be a sequence in a Hilbert space such that
$\left\Vert x_n \right\Vert=1$ and $ \langle x_n,\ x_m \rangle =0 $, for all $n \neq m$.
Let $ K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\} $.
...
2
votes
2answers
201 views
Approximating $|x|$ by a linear combination of $1, \cos x, \sin x, \cos 2x, \sin 2x$
Let $\phi(x) = |x|$ for $x \in (-\pi, \pi)$. Suppose we approximate $\phi(x)$ by a linear combination of the functions $\{1, \cos x, \sin x, \cos 2x, \sin 2x\}$. What linear combination of the form:
...
2
votes
2answers
215 views
A non complete orthogonal system whose perpendicular space is trivial
Let $Y$ be an inner-product space, and let $A$ be an orthonormal system.
We're trying to find a case to demonstrate the fact that even if for any given $x$ in $Y$ there's some $u$ in $A$ such that ...
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 = ...
2
votes
3answers
241 views
Compact operators and uniform convergence
Suppose $T: H \rightarrow H$ is a compact operator, $H$ is a Hilbert space, and let $(A_n)$ be a sequence of bounded linear operators on $H$ converging strongly to $A$. Show that $A_nT$ converges in ...
2
votes
2answers
134 views
Is $L^2(D)$ separable?
Let $D$ be a bounded connected open subset of $R^n$ and $μ$ is a finite measure on $D$, say the Lebesgue measure. Is $L_2(μ)$ separable? Is a bounded sequence $\{f_k\}$ of $L^2(μ)$ pre-compact?
2
votes
1answer
183 views
How to characterize self-adjoint operators in terms of orthogonal diagonalizability
Have a look at the following excerpt of Tosio Kato (taken from Zeidler Applied functional analysis vol. I):
The fundamental quality required of operators representing physical quantities in ...
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 ...
2
votes
3answers
56 views
Solving for positive semidefiniteness
Given a real matrix M, is there a matrix function f(M) such that $f(M)-M$ is guaranteed to be positive semidefinite, other than the idea of multiplying $M$ with its transpose and apart from the ...
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
80 views
Is it ok to switch the limits in $L_2$?
Let $(X,B,\mu)$ be a probability space and let $U$ be a unitary operator on $L_2(X,B,\mu)$.
Suppose that $g_n$ is a convergent sequence in $L_2(X,B,\mu)$, $g_n\rightarrow g$. Suppose also that there ...
2
votes
2answers
94 views
Completion of pre-Hilbert space in H. Brezis' Functional Analysis
I'm trying to solve the problem 5.12 of Harim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations; but I'm stucked understanding the statement which comes as follows:
...
2
votes
2answers
92 views
Perturbation theorem of Weyl
Does anyone know where to find something about the perturbation theorem of Weyl, preferably
on the internet. The theorem I'm talking about states:
let $A$ be a self-adjoint operator on a Hilbert ...
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
3answers
311 views
Separability of the space of bounded operators on a Hilbert space
Let $H$ be a (separable) infinite dimensional Hilbert space, and $B(H)$ the space of bounded operators on $H$. Is $B(H)$ separable in the operator norm topology? What about in the strong and weak ...
2
votes
1answer
172 views
The commutator subgroup of the group of bounded invertible linear operators
I am curious to know what the commutator subgroup of the group of (bounded) invertible linear operators on a complex Hilbert space is?
Note that by "commutator subgroup" I mean the subgroup ...
2
votes
1answer
425 views
Riesz Lemma to the Riesz Representation Theorem
Let $H$ be a Hilbert Space and let $H^*$ be the dual space of $H$.
The Riesz Lemma states that for each $T\in H^*$, there is a unique $y_T\in H$ such that $T(x)=(y_T,x)$ $\forall x\in H$. Also, ...
2
votes
2answers
385 views
Hilbert spaces, square integrability etc
(Someone may please change the title if they can think of a better one)
We have a Hilbert Space $\mathcal{H}$ that consists of all functions $\psi(x)$ such that
$\int_{-\infty}^{\infty} |\psi(x)|^2 ...
2
votes
1answer
65 views
Bounded sequence in Hilbert space contains weak convergent subsequence
In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence.
Is there any short proof? Thanks a lot.
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 ...
2
votes
1answer
69 views
Matrix Representation of Trace Class Operators
Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e:
$A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
2
votes
1answer
26 views
Is the adjoint of a quasinormal operator quasinormal as well?
I am trying to make sense of the various properties of operators on Hilbert spaces that generalise the notion of normality. It is known that for a (bounded) operator $A$ there are the following ...
2
votes
1answer
83 views
Estimate on the norm of a self-adjoint operator
EDIT: thks to Martin's comment I realize the previous version was wrong. Here is the correct version of what I need to show:
I am trying to show that if $A$ is a self - adjoint operator in a Hilbert ...
2
votes
1answer
56 views
Finding the minimizing vector of a $l_{2}$ sequence
I am working on a problem sheet and this question has me stuck. A little guidance will be appreciated.
Let $X = l_{2}$. Let $x \in X$ be given by
$x = \{\frac{1}{2^{i}} \}^{\infty}_{i=1}$
Let
$M ...
2
votes
2answers
201 views
Every Hilbert space has an orthonomal basis - using Zorn's Lemma
The problem is to prove that every Hilbert space has a orthonormal basis. We are given Zorn's Lemma, which is taken as an axiom of set theory:
Lemma If X is a nonempty partially ordered set with the ...
2
votes
2answers
85 views
Theorem about orthogonal system in inner product space.
It is known that "If $\{x_n\}$ is a sequence in a real Hilbert space $H$
satisfying
$$
\langle x_n, x_m\rangle =0 \quad\forall n\ne m,
$$ then $\displaystyle\sum_{n=1}^{\infty}x_n$ is convergent if ...
2
votes
2answers
117 views
proving “$C^1([−1,1])$ is dense in the given space with given norm”
Define $$E = \left \{ f \in W^{1,2} (-1,1) \; | \; \| f \|_E := \left( \int_{-1}^1 (1-x^2 ) | f' (x) |^2 dx + \int_{-1}^1 | f(x) |^2 dx \right)^{\frac{1}{2}} < \infty \right \}.$$ Then how can I ...
2
votes
2answers
88 views
Minimizing a functional on $L^2$
Let
$$
\mathcal{M} := \left\{f \in L^2([0,\pi]): \int_0^\pi f(x)\cos x dx = \int_0^\pi f(x)\sin x dx = 1\right\}.
$$
Solve this problem:
$$ \tag{P}
\min_{\mathcal M} \int_0^\pi ...
2
votes
1answer
174 views
Projections on Hilbert space
My question is:
Let $H$ be a Hilbert space and $T \in B(H)$. Prove that $T$ is a
projection if and only if $T$ is the identity on the orthogonal
complement of its kernel.
Thanks
2
votes
1answer
149 views
Why are only Sobolev spaces with certain exponents Hilbert Space?
I would like to know why $W^{k,2} (\Omega) $ is a Hilbert space , why is it impossible to define inner product in other Sobolev spaces, ie exponent $\ge2$ .
Here $||u||_{W^{k,2} (\Omega)} $ = ...
2
votes
1answer
119 views
how to show $f$ attains a minimum?
Let $H$ be a Hilbert space and let $f\colon H\rightarrow \mathbb{R}$ be a continuous convex function such that $f(x_n)\rightarrow\infty$ whenever $\lVert x_n\rVert\to\infty$. We need to show that $f$ ...
2
votes
2answers
392 views
$C[0,1]$ is not Hilbert space
Prove that the space $C[0,1]$ of continuous functions from $[0,1]$ to $\mathbb{R}$ with the inner product $ \langle f,g \rangle =\int_{0}^{1} f(t)g(t)dt \quad $ is not Hilbert space.
I know that I ...
2
votes
1answer
346 views
Dual of $C[0,1]$, Hilbert space and Riesz representation.
Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. I need help proving the following claim:
...
2
votes
1answer
91 views
Hilbert sum of $L_2(X_\nu,\mu_\nu)$ spaces.
Let $\{(X_\nu,\mu_\nu):\nu\in\Lambda\}$ be a family of measurable spaces. Is it true that
$\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}$ isometrically isomorphic to ...
2
votes
1answer
99 views
A question about projection in Hilbert space .
Let $a$ be a non-zero element of an Hilbert space $H$. I try to prove that for every $x\in H$,
$$
d(x, \{a\}^{\perp})=\frac{\left|\langle x,a\rangle \right|}{\left\|a\right\|}.
$$
So $d(x, ...
2
votes
1answer
98 views
Hilbert space on a finite set
If X is a finite set, what does the Hilbert space $L^2(X)$ means? - saw this notion on The Princeton Companion to Mathematics.
