Complete normed spaces whose norm comes from an inner product.
4
votes
1answer
194 views
How to find an orthonormal basis for $L^2(\mathbb{R},\mathbb{C})$?
Consider the Hilbert space
$X:=L^2(\mathbb{R},\mathbb{C})$
Now consider the operator that takes the second derivative, i.e.
$A := \partial_{x}^2$, i.e. $A: H^2(\mathbb{R},\mathbb{C}) ...
4
votes
1answer
77 views
Show that if the Riesz map is surjective on $H$, then $H$ is a Hilbert space
Let $H$ be a vector space equipped with an inner product $(\cdot, \cdot)$ and $f:H\to H',\ f(x)=(\cdot,x)$ surjective. Now, why $H$ is a Hilbert space?
The other direction is clear by Riesz' ...
4
votes
2answers
81 views
Orthogonality checking in Kreyszig exercise
Let $H$ be inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $\lVert \cdot\rVert$. Let $x,y \in H$. Would you help me to prove that $\langle x,y\rangle=0$ if and only if ...
4
votes
1answer
63 views
Invertible operator not preserving Hilbert dimension
It is known that for a bijective linear operator $T:X\to Y$ the algebraic dimensions of the linear spaces $X$ and $Y$ coincide.
I am asking for an example of an invertible (bounded) linear operator ...
4
votes
1answer
236 views
Dense subset of a Hilbert space
Let $A$ be a dense subspace of a Hilbert space $H$.
Denote $\ell^2$ the Hilbert space of (complex valued) square-summable sequences
and denote $\ell^2(H)$ the Hilbert space of $H$-valued ...
4
votes
1answer
1k views
Why are $l^{\infty}$ and $L^{\infty}$ non separable spaces?
$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces.
What on earth has changed when the value of p turns from a finite number to ${\infty}$?
Our teacher gave us some hints that there ...
4
votes
1answer
111 views
The union of cyclic subspace is also a cyclic space
Given a separable Hilbert space $H$, $U$ is a unitary operator. A cyclic subspace, denoted as $Z(x)$ for some $x\in H$, is defined as the closure of linear span of $U^nx$, where $n\in \Bbb Z$ is any ...
4
votes
1answer
224 views
Sum of Closed Operators Closable?
Let $A$ and $B$ be closed operators on a (separable complex) Hilbert space with dense domains $D(A)$ and $D(B)$ respecitvely. Then, we may define the operator $A+B$ on $D(A)\cap D(B)$. In general, ...
4
votes
3answers
224 views
Convergence of a sequence of periodic functions
Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question.
Let the ...
4
votes
1answer
72 views
Multiplication operator and trace class
Suppose we work in $H=l^2(\Bbb{N})$ and suppose the multiplication operator $T_f$ such that $T_f\psi=f\psi$ and $f:\Bbb{N}\rightarrow \Bbb{C}$. We denote by $B_1(H)$ the trace class of operators.
...
4
votes
1answer
42 views
Help with proof of closedness of a set
Let $u_n$ be a sequence in Hilbert space such that $\|u_n\|=1$ for all $n$, and $\langle u_n|u_m\rangle=0$ whenever $n\neq m$.
Why is the following set closed: $\{0\}\cup \{u_n \mid n\geq 1\}$?
Thanks ...
4
votes
1answer
85 views
Showing a representation is irreducible by showing that a degenerate subspace has codimension one.
Throughout $\phi$ be a continuous character from a locally compact abelian group $G$ to the circle.
I'm trying to understand this implication. Basically we want to show that a certain representation ...
4
votes
1answer
218 views
Showing the sum of orthogonal projections with orthogonal ranges is also an orthogonal projection
Show that if $P$ and $Q$ are two orthogonal projections with orthogonal ranges, then $P+Q$ is also an orthogonal projection.
First I need to show $(P+Q)^\ast = P+Q$. I am thinking that since
...
4
votes
2answers
470 views
How to show a compact, closed-range operator on an infinite-dimensional Hilbert space has finite rank, without using the open-mapping theorem?
If $H$ is an $\infty$-dimensional Hilbert space and $T:H\to{H}$ is a compact operator with closed range, how do I show that $T$ has finite rank, without using the open-mapping theorem? (The ...
4
votes
1answer
257 views
Weak convergence
Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $V,W$ be two closed subspaces. For $x_0\in H$ we may define the sequence of projections $$x_{2n+1}=P_W(x_{2n}), \qquad ...
4
votes
1answer
118 views
An interesting condition for the completeness of an orthonormal system in $ L^2([0,1]) $
Let $\{u_n\}$ be an orthonormal system in $L^2([0,1])$, prove that $\{u_n\}$ is complete iff
$$ \sum_{n=1}^\infty \intop_0^1 \left|\intop_0^x u_n(t)\;dt\right|^2 dx = 1/2.$$
It should be noted that ...
4
votes
2answers
334 views
Orthonormal basis in Hilbert space
I an reading a book about functional analysis and there is one thing i really don't understand. Let $\mathcal{H}$ be a Hilbert space. And $U \subset \mathcal{H}$ a closed subspace. Is it possible to ...
4
votes
3answers
1k views
How to prove that square-summable sequences form a Hilbert space?
Let $l^2$ be the set of sequences $x = (x_n)_{n\in\mathbb{N}}$ ($x_n \in \mathbb{C}$) such that $\sum_{k\in\mathbb{N}} \left|x_k\right|^2 < \infty$, how can I prove that $l^2$ is a Hilbert space ...
4
votes
1answer
62 views
Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?
Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
4
votes
1answer
115 views
Convergence of a series involving inner products
Let $\{A_{j}\}$ be a sequence of bounded operators on a Hilbert space satisfying $\|A_{j}^{\ast}A_{k}\| \leq C_{j - k}$ and $\|A_{j}A_{k}^{\ast}\| \leq C_{j - k}$ where $\sum C_{i} < \infty$. Fix an ...
4
votes
0answers
37 views
Concerning unbounded linear operators on a Hilbert space
Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
4
votes
1answer
129 views
Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces
I am trying to prove the existence of a weak solution of the problem:
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
on the bounded open set $U\subset\mathbb{R}^n$ ...
4
votes
0answers
119 views
Inverse of Identity plus Volterra operator
consider the following operator or $L_2(0,1)$,
$(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial.
I am trying to construct the inverse of this ...
4
votes
1answer
137 views
Brownian Motion Covariance: max instead of min
It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion.
Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
4
votes
0answers
86 views
What is the dual space of $C([0,T];X)$ ($X$ Hilbert space)?
What is the dual space of $C([0,T];X)$, where $X$ is a Hilbert space? Is it $\operatorname{BV}([0,T]; X^*)$?
As we know, for $C([0,T])$, the dual space is $\operatorname{BV}([0,T])$, but when it is ...
4
votes
1answer
119 views
Canonical field of Hilbert spaces in Dixmier; Plancherel Theorem
I'm working through the proof of Plancherel's Theorem in $C^{*}$-algebras by Dixmier, section 18.8. For the most part, I'm happy with it, although I have one problem. From Dixmier, I have the proof ...
4
votes
0answers
194 views
Spectral theorem for unitary operators
I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
4
votes
0answers
183 views
Sum of operator and adjoint is self-adjoint
In abstract Hodge theory there is the following lemma:
Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
4
votes
0answers
63 views
Relations between spectrum and quadratic forms in the unbounded case
Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
4
votes
0answers
136 views
Set of all compact operators $K(H)$ is the unique ideal in $B(H)$?
I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
4
votes
0answers
133 views
When functions, analytically continued, carry over certain properties
Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
3
votes
4answers
178 views
Orthogonal complement of a Hilbert Space
I have this problem:
Let $S$ be a subset of a Hilbert $H$ and let $M$ be the closed subspace generated by $S$. Show that
$M^{\perp} = S^{\perp}$
$M = (S^{\perp})^{\perp}$
if $V$ is a subspace of ...
3
votes
2answers
169 views
Boundedness of operator on Hilbert space
I have the following question: let $\mathcal{H}$ be a Hilbert space and $\{\varphi_{i}\}_{i \in \mathbb{N}}$ be an orthonormal basis. Furthermore let $T: \mathcal{H} \rightarrow \mathcal{H}$ be an ...
3
votes
2answers
272 views
A counterexample to theorem about orthogonal projection
Can someone give me an example of noncomplete inner product space $H$, its closed linear subspace of $H_0$ and element $x\in H$ such that there is no orthogonal projection of $x$ on $H_0$. In other ...
3
votes
2answers
128 views
A continuity condition for a bilinear form on a Hilbert space
Let $H$ be a real Hilbert space, and let $B : H \times H \to \mathbb{R}$ be bilinear and symmetric. Suppose there is a constant $C$ such that for all $x \in H$, $|B(x,x)| \le C \|x\|^2$. Must $B$ be ...
3
votes
2answers
278 views
Paradox or Error: On the inclusion of dense subspaces into Hilbert spaces
the following observations are very simple, but I suppose they contain an error, which I haven't been able to find it so far. Maybe somebody can help how to fix it:
Let $H$ be a Hilbert space, $U$ be ...
3
votes
5answers
396 views
Subspaces of Hilbert Spaces of finite dimension
Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ ...
3
votes
1answer
93 views
Dense subspace of $\ell^2$
Is the set \begin{align} A=\left\{a=(a_1,a_2,\dots)\in\ell^2 \ \ \lvert \ \ \sum_{k=1}^\infty \frac{a_n}{n}=0 \right\}\subset\ell^2
\end{align} dense in $\ell^2$
Is the following argument ...
3
votes
1answer
242 views
$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$
My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$
Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
3
votes
3answers
156 views
Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional
Suppose that we have a Hilbert Space $H$ and $M$ is a closed subspace of $H$. Let $T\colon H\rightarrow M$ be the orthogonal projection onto $M$.
I have to show that $T$ is compact iff $M$ is finite ...
3
votes
1answer
305 views
Hilbert-Schmidt Operator
We have just covered Hilbert-Schmidt operators in class (which I missed) and I am having a hard time getting my head around them. I know the definition:
If $H$ is a Hilbert space and ...
3
votes
1answer
131 views
A problem on Fourier transforms and orthogonality
Let $f$ be a square integrable function, strictly positive almost everywhere. Consider the family of functions $f_a=f(x+a)$, where $a$ is any real number. I want to prove that if a function is ...
3
votes
1answer
138 views
Compact operators between Hilbert spaces
I have the suspect that the following statement is true, but I don't how to prove it. Any suggestion? Thanks to all!
Let $X$, $Y$ be Hilbert spaces and let $T \colon X \to Y$ be a linear continuous ...
3
votes
2answers
198 views
The image of orthonormal basis under compact operator
I need a help to prove that statement: if $\{e_n\}$ an orthonormal basis in Hilbert space $H$ and $A$ is a compact operator from $H$ to $H$, then $Ae_n\rightarrow 0$. Thx for any help.
3
votes
3answers
87 views
Maximal Value of Integral
Calculate the maximal value of $\int_{-1}^1g(x)x^3 \, \mathrm{d}x$, where $g$ is subject to the conditions
$\int_{-1}^1g(x)\, \mathrm{d}x = 0;\;\;\;\;\;\;\;$ $\int_{-1}^1g(x)x^2\, \mathrm{d}x = ...
3
votes
3answers
78 views
$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ for functions in $C([0,1])$?
Why does the following hold for continuous functions on $[0,1]$?
$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$
3
votes
2answers
154 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 ...
3
votes
2answers
67 views
Show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ is not closed in $\ell^2$
How to show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ where $e_n=(\delta_{nk})_{k\in\mathbb N}$ is not closed in $\ell^2$?
3
votes
6answers
120 views
Is $\operatorname{range} =\ker^\perp$ only true for projection?
Let $P$ be a linear operator on a Hilbert space $H$. If $\operatorname{range} P=(\ker P)^\perp$, is $P$ necessarily a projection, i.e., $P^2=P$?
3
votes
2answers
134 views
Minimization problem in Sobolev spaces
This is a homework problem and I don't know how to solve it:
Consider the following minimization problem on the real-valued sobolev space $H^{1,2}(\Omega)$ with dimension $n=1$ and $\Omega=(0,1)$:
...
