Questions tagged [hilbert-spaces]
For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.
0
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0answers
29 views
A Hilbert space is separable if and only if it admits a countable orthonormal basis [duplicate]
According to Wikipedia:
A Hilbert space is separable if and only if it admits a countable orthonormal basis
While this statement seems very reasonable, it is not clear to me how one would go about ...
1
vote
1answer
42 views
Polynomial ring as a (complete) metric space
Let $k$ be a field of characteristic zero (for example $\mathbb{R}$ or $\mathbb{C}$), and let $R=k[x_1,\ldots,x_n]$ be the $k$-algebra of polynomials in $n$ variables $x_1,\ldots,x_n$, $n \geq 1$.
...
1
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0answers
22 views
Isomorphisms and Implicit Function Theorem in the context of Hilbert spaces
Let X be a Hilbert space with inner product denoted $(\cdot,\cdot)$ and norm $\|\cdot\|$ induced by it. Suppose $F\,\colon\,X \times \mathbb{R} \to X$ is continuously Frechet differentiable with ...
1
vote
1answer
22 views
A condition to ensure that adjoint and inverse commute
Let $A$ be a densely defined operator on a Hilbert space $H$. Let us assume that $A$ is injective and $Ran(A)$ is dense. Then I want to show that
$$
(A^*)^{-1} = (A^{-1})^*.
$$
So far, I was able to ...
0
votes
1answer
37 views
A question about self-adjoint operators
Let $A$ be a densely defined symmetric operator on some Hilbert space $\mathcal{H}$. Let's say that we know that there exists $(A + 1)^{-1}$ and that this is a self-adjoint operator.
How can I prove ...
3
votes
1answer
46 views
Show that finitely generated cone in Hilbert space is closed.
Let $X$ be a real Hilbert space. Let $\{c_i\}_{i=1}^m\subset X$. Denote
$$
Y=\left\{\sum\limits_{i=1}^m\lambda_ic_i\Big|\lambda_i\in[0,+\infty),\quad 1\leq i\leq m\right\}.
$$
Show that $Y$ is a ...
2
votes
4answers
52 views
Hilbert space and orthonormal basis.
Let $H$ be a Hilbert space and let ${e_n} ,\ n=1,2,3,\ldots$ be an orthonormal basis of $H$. Suppose $T$ is a bounded linear oprator on $H$. Then which of the following can not be true?
$$(a)\quad T(...
1
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0answers
39 views
Lagrange multiplier method on Banach spaces (in Dirac notation)
I want to prove Cauchy–Schwarz' inequality, in Dirac notation, $\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2$, using ...
4
votes
3answers
52 views
Prove that a subset in Hilbert space is open
I'm a undergraduate who enjoys math and follows his first course on real Analysis. I follow this course for fun and i'm stuck on the following problem:
Let $H^{\infty}$ be the Hilbert cube (the ...
3
votes
1answer
45 views
Reference Request: KKT in Hilbert Space
Are there analogues of Slater's condition and the KKT conditions in separable Hilbert spaces? Does the infinite dimensionality pose a problem?
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2answers
41 views
Show Cauchy-Schwarz inequality from triangle inequality for $\mathbb{C}$-vector space
In many textbooks the proof of Cauchy-Schwarz inequality needs to introduce a parameter $\lambda$ to take $\langle x+\lambda y,x+\lambda y \rangle$ as the first step. I am trying to proof the Cauchy-...
1
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0answers
36 views
GNS representation and states of $C^*$-Algebras
Let $\varphi$ be a state of the $C^*$-algebra $A$, $B\subset A$ a hereditary subalgebra and $K_\varphi:=\{x\in B : 0\le x \le 1, \varphi(x)=1\}$. Let $\pi_\varphi:A\rightarrow \mathcal{B}(H_\varphi)$ ...
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votes
2answers
38 views
$T$ is an isometry if and only if $\langle Tx, Ty \rangle = \langle x, y \rangle$
I want to prove:
A linear mapping $T:X \to Y$ between two pre-Hilbert spaces is an
isometry if and only if the inner products $\langle Tx, Ty \rangle = \langle x, y \rangle$ for all $x, y \in ...
2
votes
1answer
34 views
Spectrum included in spectral measure support
Let $\mathcal{H}$ be an Hilbert space and $A=\displaystyle{\int} \lambda \,dE(\lambda)$ a spectral operator on $\mathcal{H}$ associated to the spectral measure $E$.
I am interested in the very basic ...
1
vote
2answers
41 views
A linear bounded operator has to be continuous?
I just found on the wiki https://en.wikipedia.org/wiki/Continuous_linear_operator stating that
An operator between two normed spaces is a bounded linear operator if
and only if it is a continuous ...
5
votes
0answers
53 views
Show the average of first n terms of a power series converges in Hilbert space
Let $X$ be a Hilbert space. Let $A:X\mapsto X~$ be a continuous linear operator such that $||A||=1$.
For any $x\in X$, define a sequence $\{y_n\}_{n=1}^{\infty}$ as
\begin{equation*}
y_n=\frac{1}{n}\...
4
votes
1answer
53 views
Why a countable set of numbers can characterize a function in Hilbert space?
I'm a new graduate student major in physics; the only mathematical course I was formally trained is Real Analysis, so my following statement of my question may seem a little unprofessional, and I hope ...
-1
votes
1answer
26 views
Image of a dense set [closed]
Is the image of a dense set under an isometric operator is again dense set?
i.e.,
Given two Hilbert spaces $X, Y$.
If $T:X\to Y$ is an isometric operator and $S$ is a dense span subset of $X$, is ...
2
votes
1answer
41 views
Hilbert-Schmidt operator defined by non-orthogonal basis
I have the following operator on $L^2(0,1)$
$$ Tf = \sum_{n \geq 0} 2^{-n}\langle f,v_n\rangle v_n$$
Where $v_n(t) = t^n$.
I was able to prove that it is Hilbert-Schmidt, but now I need to ...
0
votes
0answers
9 views
Proving a proposition about special functions in finite dimensional Hilbert space
We call frame function any mapping $f : \{ x \in H, \| x \|=1 \} =:\mathbb{S}(H) \mapsto \mathbb{R} \cup \{ \pm \infty \}$ (here $H$ denotes a finite dimensional Hilbert space) satisfying the ...
1
vote
1answer
22 views
Projection onto the span of linearly independent vectors in Hilbert spaces
Let $H$ be a Hilbert space and $S\subseteq H$ a closed subspace. Moreover, let $\{s_{n}\}_{n=1}^{\infty}\subseteq S$ a complete and linear independent sequence in $S$, i.e.
$S=\overline{\text{Span}\...
1
vote
1answer
38 views
Find orthogonal basis for $V^{\perp}$ and find orthogonal projection on $V$.
I have a task in the functional analysis exam that I could not do.
Problem: In Hilbert space $L_2 [0,1]$ (with Lebesgue measure) let:
$$V = \left\{ f \in L_2 [0,1]: \int_{0}^{1} t \cdot \overline{f(...
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votes
1answer
23 views
Measurable function on Hilbert space
Every continuous function defined on $\mathbb{R}$ is Borel measurable. Is it also true in Hilbert spaces ? any references are there?
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0answers
40 views
The Hardy space $\mathbb{H}^{n\times m}_{2}$ in the upper half plane
Given $f\in \mathbb{H}^{1\times m}_{2}(U) $ and $g\in\mathbb{H}^{m\times m}_{2}(U)$.
Is it true that $fg\in \mathbb{H}^{1\times m}_{2}(U)$ ??
If not, what conditions needed so that it would be ...
0
votes
0answers
18 views
How can we have a complex basis for a real-valued space?
I have started studied functional analysis few times ago and now I am stuck in a doubt which I think is really trivial, but I cannot still find an answer.
Consider the Hilbert Space $X = L^2[-\pi,\pi]...
1
vote
2answers
30 views
Showing that the space $L^2(\Omega \rightarrow H)$ of Square-integrable Hilbert-valued functions is a Banach Space
I am trying to fill in the details of Nate Eldredge's answer to "Is this set of random variables a Hilbert space?". In particular, let $(\Omega,\mathcal{F}, \mu)$ be a measure space and $H$ be a ...
1
vote
0answers
108 views
Is there a Hilbert space containing exactly 128 elements? [closed]
We can construct a Metric Space having exactly 128 elements. In fact we can define a discrete metric space on finite set containing 128 elements.
The questions are "Can we construct a Vector Space/...
2
votes
1answer
50 views
A question about a theorem in *Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators*
I'm studying the paper Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators by Changsoo Bahn and Chul Ki Ko and Yong Moon Park(https://arxiv.org/abs/math-ph/0505026v1)....
1
vote
1answer
14 views
Canonical isomorphism between finite-dimensional Hilbert spaces
I want to ask is there a canonical isomorphism between two same finite dimensionality Hilbert spaces.
My intuition is telling me no, there is an isomorphism, but not canonical. The reason is because ...
1
vote
3answers
41 views
Orthonormal basis in every Hilbert space?
Is it true that in every Hilbert space (not separable) there exists an orthonormal basis (complete and countable orthogonal system)? So equivalently asked can we have a Hilbert space in which it is ...
0
votes
1answer
51 views
Non-orthogonal extension of an inner product
Let $W$ be a real linear vector space of possibly infinite dimension and let $<\cdot, \cdot>_W$ be an inner product on $W$. Further let $V$ be a superspace of $W$, such that holds $V = W \...
2
votes
0answers
44 views
Is the simplicial complex with vertices the critical points of a $C^2$-function on a Hilbert manifold $X$, homotopy equivalent to $X$?
Let $X\subset\mathbb{R}^{\infty}$ be a paracompact connected Hilbert manifold. Moreover, let $f:X\to\mathbb{R}$ be a $C^2$-function.
Then is the simplicial complex $\Delta:=\{p\in X:df(p)=0\}$, ...
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0answers
27 views
How do we compute homology from the Čech complex $\text{Čech}(r)$ with $B^{\infty}$ balls covering the Hilbert manifold $M$?
Suppose a Hilbert manifold $M$ is covered by the union of $\infty$-balls (in the sense of Baire spaces), namely $M=\bigcup_{\alpha\in A}B^{\infty}$, without knowledge of intersections. The only ...
2
votes
2answers
33 views
Convergent sequence in Hilbert space
Let $H$ be a Hilbert space with orthonormal basis $(e_n)_n$ and $(a_n)_n \in \ell^2(\mathbb{N})$, so $$\sum_{n=0}^{\infty} |a_n|^2 < \infty$$
Now, define the sequence $(\xi_n)_n$ in $H$ as $$\xi_n= ...
0
votes
1answer
25 views
The average effect of a self-adoint projection
First, here is a brief non-rigorous version of the question:
Staring with two arbitrary rays (one-dimensional subspaces)
in a finite-dimensional
Hilbert space, what is the average effect of a
...
2
votes
2answers
73 views
$\lim_{n \to \infty} \langle\bar A_nx,y\rangle = \langle Ax,y\rangle$
Let $A$ be an operator on a separable Hilbert Space $H$. So we can consider $A=(a_{ij})_{1\leq i,j\leq \infty}$.
Let $\tilde A_n = $$
\begin{pmatrix}
A_n & 0 \\
0 & -I
\end{...
1
vote
2answers
29 views
An application of Riesz representation theorem $\lim_{n\rightarrow \infty} \left< x_n,e_k\right>=\left< x,e_k \right>$.
Suppose that $\mathcal{H}$ be a real, separable Hilbert space and $\{e_k \}$ be a countable orthonormal basis. And let $x\in \mathcal{H}$ and $\{x_n\}\subset \mathcal{H}$ is a bounded sequence. Than ...
3
votes
1answer
38 views
Semi-inner product structure in complex Hilbert spaces
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $(F,\langle\cdot,\cdot\rangle)$. Any $M\in \mathcal{B}(F)^+$ (i.e. $\langle Mx\;, \;x\rangle\geq 0$ for all $...
1
vote
1answer
39 views
One-Dimensional Subsets of Hilbert Space vs. Euclidean Space
Motivated by a question I received (though a bit different), is there a one-dimensional subset of Hilbert Space that cannot be embedded in $\mathbb{R}^3$?
Here, dimension refers to covering or ...
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1answer
21 views
If f is bounded ($\vert f(x)\lvert \leq M$), and $f\in L^1(R)$, then
I’m doing this problem:
If f is bounded ($\vert f(x)\lvert \leq M$), and $f\in L^1(R)$, then:
$f\in L^2(R)$ with $\lvert\lvert f\lvert\lvert_{L^2(R)}\leq M^{1/2}\lvert\lvert f\lvert\lvert_{L^1(R)}^{1/...
4
votes
1answer
45 views
Is the Banach space of continuous functions $I \to \ell^2$ separable?
Inspired by this question.
Consider the vector space $V$ of all continuous functions $I \to \ell^2$ for $I=[0,1]$ the closed unit interval and $\ell^2$ the Hilbert space of all square-summable ...
2
votes
1answer
35 views
Why do we need to use dominated convergence theorem?
I was thinking of this problem:
If $f_n\rightarrow f$ pointwise a.e. and $\lvert f_n\lvert \leq g$ for some $g\in L^p$, then prove that $f_n \rightarrow f$ in $L^p$.
To use the dominated convergence ...
6
votes
1answer
71 views
Proof explanation related to the operator matrices
Let $F$ be a complex Hilbert space. Let $A,B,C,D\in \mathcal{B}(F)$. Consider the operator matrix $T$ such that
\begin{equation*}
T=\begin{pmatrix}A & B \\
C & D
\end{pmatrix}\in \mathcal{B}(F\...
2
votes
1answer
50 views
Suppose $f_n\rightarrow f$ Point wise a.e.
Suppose $f_n\rightarrow f$ pointwise a.e. Show that $f_n\rightarrow f$ in $L^1$ if and only if $\lvert\lvert f_n\lvert\lvert_1 \rightarrow \lvert \lvert f\lvert\lvert_1$.
What happens if it’s all in ...
1
vote
1answer
43 views
Let $f\in L^p (R)$ for $1\leq p<\infty$ and let $f_t(x)=f(x-t)$.
Let $f\in L^p (R)$ for $1\leq p<\infty$ and let $f_t(x)=f(x-t)$.
Prove each $f_t\in L^p(R)$.
Prove $\lvert \lvert f_t-f\lvert\lvert_p\rightarrow0$ as $t\rightarrow0$. What happens as $t\rightarrow ...
1
vote
1answer
13 views
Suppose $\mu$ and $\nu$ are finite positive measures on a measuable space
The problem is as follow:
Suppose $\mu$ and $\nu$ are finite positive measures on a measuable space (X, M). Show there is $f\in L^1(X,\mu)$ so that $\int f \ d\mu=\int (1-f) \ d\nu$.
I know that $f\...
1
vote
1answer
30 views
For an orthonormal set in $L^{2}[0,1]$, show that $ \int_{A} \sum_{k=1}^{\infty} |f_{k}(x)|^{2} \geq 1$ for a non-zero measurable subset $A$.
I was stuck on the following question
If ${f_1, f_2, . . . }$ is a complete orthonormal set in $L^2
[0, 1]$ and $A$ is an arbitrary subset of
positive Lebesgue measure in [$0,1]$ show that
$$ \...
1
vote
1answer
58 views
Is there an orthonormal basis for $L^2([0,1])$ consisting of only even-degree polynomials?
Problem:
Does $L^2([0,1])$ equipped with the inner product $$\langle f,g \rangle = \int_0^1 \overline{f(x)}g(x) \, dx$$ have an orthonormal basis of the form $\left\{f_n(x):=\sum\limits_{k=0}^n a_k ...
0
votes
0answers
60 views
Dirac delta function like a “function” of an operator
Let $\mathcal{S}\subset \mathcal{H}=L^2(\mathbb{R}^n)\subset \mathcal{S}^*$ be the Schwartz space, the Hilbert space and the space of tempered distributions respectively. Consider an algebra $\mathcal{...
0
votes
0answers
18 views
Composition of absolutely continuous function
if I have a function $x:[0,\infty)\rightarrow H$ (H is an hilbert space) that is absolutely continuous (AC), why should the function $\|x(t)-y\|^2$ be AC, for $y\in H$?
Yet again, why should $\|x_{\...
