Questions tagged [hilbert-spaces]
For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.
0
votes
5answers
32 views
Closure of the orthogonal complement.
Let H be a Hilbert space and let $A \subset H$. Let the orthogonal complement of A be:
$A^\perp$ = {$x \in H : x \perp A$}.
How do I show that $A^\perp$ is a vector space and that it is closed? I ...
1
vote
1answer
16 views
Reducing subspaces of unilateral shift on $l^2$
Show that there is no non-trivial reducing subspace of the unilateral shift $T:l^2\to l^2$ given by $T((x_1,x_2,...))=(0,x_1,x_2,...)$.
So suppose there is a non-trivial reducing subspace $M$ then we ...
2
votes
0answers
21 views
Continuous curve on a Hilbert Space where any three points form a right angled triangle
Give an example of a continuous curve $\gamma$ in the Hilbert space (i.e., an image of a continuous map $\phi : [0,1] \rightarrow H)$ with the following “miraculous” property: for every three points $...
0
votes
0answers
29 views
Uniqueness of Reproducing Kernel Hilbert Space
Digging in the definition of Reproducing Kernel Hilbert Spaces (RKHS) I came across the following example taken from pages 49-51 of [1]:
Given the kernel $k(x,y) = \langle x,y\rangle^2$, with $x,y\in ...
2
votes
1answer
34 views
Inequality on Hilbert Space
The problem is a sort of Cauchy-Schwarz inequality:
Let $(x_n)_{n\in\mathbb{N}}$ be a sequence in a Hilbert Space $H$, and $(c_n)_{n\in\mathbb{N}}\in l^2(\mathbb{N}).$ Let also $F$ be a finite ...
0
votes
1answer
12 views
The difference between closed linear span and linear span in Hilbert spaces.
$H$ is a Hilbert space and $M$ is an orthonormal set(not necessarily finite). What is the definition of:
1)closed linear span of $M$
2)the linear span of $M$
3)the closure of a linear span
I was ...
-1
votes
0answers
28 views
$H$ is a Hilbert space. Let $S$ denote the subspace spanned by $\{e_n \}_{n=1}^{\infty}$ . Prove that $S$ is a closed subspace.
$H$ is a Hilbert space and $\{e_n \}_{n=1}^{\infty}$ is a sequence of elements in $H$ with $\vert \vert e_n \vert \vert =1$ for all $n$ and $(e_n,e_m)=0$ whenever $n \neq m$ . Let $S$ denote the ...
2
votes
0answers
17 views
Topologies on isometry groups
Let $X$ be a (complete) Hilbert space, not necessarily separable. What are the known interesting topologies on the group $Isom(X)$ of its isometries? Are there interesting metrics on this group?
0
votes
1answer
27 views
Definition of closed subspace in Hilbert space
There's some definition of concept in stein's Real analysis Page-175 made me so confused.
(0)$H$ is a Hilbert space. $T$ is a bounded symmetric linear operator on $H$
(a) a linear subspace $S$ ...
2
votes
2answers
24 views
Completeness of variation on $L_2$-space
I'm trying to show completeness of the space of measurable functions $f:[0,1]\rightarrow \mathbb C$ with the condition $\int|f|^2kd\mu<\infty$. $k$ is a measurable function, $k:[0,1]\rightarrow[0,\...
2
votes
2answers
24 views
If $A+B\ge C$, can we find positive operators $A_1\le A$ and $B_1\le B$ such that $A_1+B_1=C$?
Let $A$ and $B$ be two positive operators on a Hilbert space. $C$ is a positive operator with $A+B\ge C$. Can we find positive operators $A_1\le A$ and $B_1\le B$ such that $A_1+B_1=C$?
0
votes
0answers
16 views
Spectral theorem of infinite-dimensional case
Spectral theorem (infinite-dimensional):Suppose $T$ is a compact symmetric operator on a Hilbert space $H$.Then there exists an orthonormal basis $\{\phi_k\}_{k=1}^{\infty}\,\,\,$ of $H$ that consists ...
1
vote
1answer
23 views
An example of the Pseudo-inverse of an operator
Let $E$ an infinite dimensional complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.
Definition: Let $T \in \mathcal{L}(E)$. The Moore-Penrose inverse of ...
1
vote
2answers
30 views
Show that $||e^{-\lambda k}x-x||\to0$ as $\lambda\to0$ when $x = \{x_k\}_k \in \ell^2$
Let be $x=\{x_k\}_k\in \ell^2(\mathbb{R})$ , consider $\lambda>0$ and show that $||e^{-\lambda k}x-x||\to0$ as $\lambda\to0$.
I tried to solve this exercise by this way:
$$||e^{-\lambda k}x-x||^2=...
2
votes
0answers
24 views
Understanding the product of Normal Random Variable and Eigenfunction
Consider a symmetric function (Mercer Kernel) $K : T \times T \rightarrow \mathbb{R}$ and define an operator $H_k: L^2(T,\nu) \rightarrow L^2(T,\nu)$ where $H_kf(x) = \int_X K(x,y)f(y)d\nu(y)$.
...
0
votes
2answers
32 views
What is an example of a Hilbert Space that is not ${\mathbb R}^n$, $\mathbb{C}^n$ or $L^2$?
What is an example of a Hilbert Space that is not any subset of ${\mathbb R}^n$, $\mathbb{C}^n$ or $L^2$ (n-dimensional reals, n-dimensional complex numbers, or Lebesgue integrable functions)?
I'm ...
2
votes
3answers
44 views
Orthonormal basis: Countable $\infty$ vs. Uncountable $\infty$
My doubt is the following, when you create an orthonormal basis for a space, the number of coefficients in each vector, and the number of vectors is equal to the dimension of the space (at least in ...
0
votes
1answer
15 views
How to recognise an orthonormal basis in a complex inner product space?
All the N-tuples of complex numbers $\omega=(w_1,...,w_N)$, all the complex scalars define a inner-product space.
The function $\langle \omega, \chi \rangle=\frac{1}N\sum_{i=1}^N w_ix_i^*$ is an inner ...
1
vote
0answers
27 views
Two independent sequences in $l_\infty$ that are bounded by $1$, and their sum is bounded by $2$
Prove that there are $2$ independent vectors $x,y \in(l_\infty)$ such that $||x||=||y||=1$ and $||x+y||=2$
(Here $l_\infty = \{(a_n)|a_n\in \mathbb{C}, sup|a_n|<\infty\}$ and $||x||=sup|x_n|$)....
0
votes
0answers
14 views
Variational Inequality only for Real Part
In Functional Analysis, we have proven what we have called Variational Inequality today.
Let $X$ be a Hilbert space and $\emptyset \neq K \subseteq X$ closed
and convex. Then for every $x \in X$ ...
0
votes
1answer
21 views
Square summable coefficients in infinite linear combination.
I understand the reasoning behind all this paragraph, what I would like to know is if the condition $\sum_{k=1}^\infty|\beta_k|^2<\infty $ is a necessary condition for all infinite linear ...
1
vote
2answers
22 views
Convergence of the inner product in Hilbert Space
I'm starting to study Hilbert Spaces for the very first time in my life and I had difficulty to understand one very simple proof:
Let $\{x_n:n=1,2,...\}$ be a sequence of vectors in the space; the ...
0
votes
0answers
11 views
What is cylindrical Brownian Motion / Wiener Process
I have been given some reading on the Krylov-Bogoliubov Method for constructing invariant measures.
An SDE in Hilbert space H is introduced as
$$d(X)=b(X)dt + \sigma(X)dW $$
Where W is the ...
-2
votes
0answers
27 views
Left multiplication is continuous in weak operator topology [closed]
Suppose $H$ is $l^{2}$ space and $B(H)$ is the space of all continuous linear maps. Fix $A \in B(H)$, define $L_{A}:B(H)\rightarrow B(H)$ by $L_{A}(T)=AT$ for all $T\in B(H)$, where $AT(x)=A(T(x))$ ...
-1
votes
1answer
27 views
Strong open neighborhood contains an infinite dimensional subspace
Suppose $H$ is $l^{2}$ space, let $B(H)$ denote the space of all continuous linear maps. How to prove that every open neighborhood of $0\in B(H)$, as in strong operator topology, contain an infinite ...
1
vote
1answer
36 views
Show that $P_Eh=h-\langle h,e_1\rangle e_1-\langle h,e_2\rangle e_2$
I have that $E=\{e_1,e_2\}^\perp$, and that $(e_n)_{n=1}^\infty$ is an orthonormal basis for the Hilbert space. Furthermore, I have that $P_Eh$ is called orthogonal projection of $h$ onto $E$.
I have ...
0
votes
2answers
16 views
Self-adjoint operator or not?
let be the operator on $L^2(\mathbb{R})$ define by
$$ T[f](x)=f(-x) $$ and I tried to determinate the adjoint with a change of variables:
$$
(T^*g,f)=(g,Tf)=\int_{\mathbb{R}}g(x)f(-x)\,dx=[y=-x; dx=-...
1
vote
1answer
19 views
Spectrum and Convergence of an operator
can you help me to solve this exercice? The first point is ok but I have problems with the others.
Let $S_\varepsilon$ an continuous operator on $L^2(\mathbb{R})$ define as $$ S_\varepsilon[f](x)= \...
3
votes
1answer
14 views
Fourier Series in any interval [a,b]
Let $e_n=e^{inx}$ and Fourier coefficients are $\alpha_n= \frac{1}{2\pi} \int_{-\pi}^{\pi}f(x)\overline{e^{inx}}$ where $\overline{e^{inx}}$ is complex conjugate and $f \in L^2[-\pi,\pi]$.
I’m ...
3
votes
2answers
33 views
Limit of an operator and eigenvalues
can you help me how to solve this exercice? Define in $L^2(-1,1)$ $$ T_k[f](x):= \int_{-1}^1 \frac{(-1)^k}{(2k+1)!}(xy)^{2k+1} f(y)\,\,dy $$
i) show that $||T_k||\to 0$;
ii) find eigenvalues of $T_0$....
0
votes
1answer
17 views
Kernel of an operator
can you help me to solve this exercice?
Show that the kernel of the operator $M:L^2((-1,1))\to L^2((-1,1))$ has infinite dimension.
The operator is: $$ M[f](x):= \int_{(-1,1)} \sin(xy)f(y)\,\,\,dy. $$
...
0
votes
0answers
22 views
Proving that an operator on a Hilbert space is compact
Let $V$ be the usual complex vector space on the set of analytic, complex-valued functions defined on the interval $[0,L]$ such that $f(0)=f(L)$ for all $f\in V$. Equip $V$ with the inner product $\...
0
votes
0answers
15 views
Self-Adjoint Integral Operator
I'm dealing with a new problem of functional Analysis:
Let $f(t) = (1-\frac{e^{it}}{2})^{-1} \in L^2(-\pi,\pi)$ and let $T$ be the integral operator with integral kernel $K(x,y) = f(x-y)$.
a)Show ...
0
votes
1answer
18 views
Two subspaces of a Hilbert space are equal if their orthogonal complements are equal
I have the question: "If $F_1$ and $F_2$ are subspaces of the hilbert space $H$, and they satisfy $F_1^\bot=F_2^\bot$, is it true that $F_1=F_2$?"
I would say that is it true, as one have that if $X$ ...
1
vote
1answer
33 views
Disjoint isometric copy of countable-dimensional vector subspace of Hilbert space
Let $H$ be a complex separable Hilbert space. Let $A \subset H$ be a vector subspace of countable dimension. Does there exist unitary operator $U$, such that $A \cap U(A) = \{ 0 \} $ ?
0
votes
1answer
43 views
$P$, $Q$ are projections on a Hilbert space such that $|P-Q|<1$ then $\dim(\operatorname{Range}(P))=\dim(\operatorname{Range}(Q))$
$P,Q$ are projections on a Hilbert space such that $|P-Q|<1$ then $\dim(\operatorname{Range}(P))=\dim(\operatorname{Range}(Q))$.
Is there a intuition for this to happen, also I am not being able ...
13
votes
1answer
258 views
Is the following set dense in $L^2$?
Lately I was talking to a friend of mine and we came up with the following question
Denote by $\mathcal{P}$ the set of all real valued polynomial functions. Is the set
$$ \{ p(x) e^{- \alpha \...
1
vote
0answers
10 views
Prooving independece of random variables defined on Hilbert space
I have a problem with one proof concerning my thesis on Gaussian measures on Hilbert spaces.
So we have separable Hilbert space $H$, Borel $\sigma$-algebra $\mathscr{B}(H)$ and Gaussian measure $\mu$,...
1
vote
1answer
49 views
The collection of (unitary representations on) Hilbert spaces is a set
Let $G$ be a locally compact group. I know that the collection of all unitary representations of $G$ is not a set, since there are unitary representations on inner product spaces with bases of any ...
2
votes
0answers
73 views
What is the orientation of a hypercube in $N$-space needed for a certain projection?
There exists a parallel projection of an $N$ dimensional cube from $N$ dimensional (Euclidean) Hilbert space to the $x-y$ plane whose projected perimeter approaches a circle as $N$ increases. The ...
1
vote
1answer
34 views
Determining Adjoint Operator
I was dealing with this exercise of my functional analysis course :
Let $\mathcal{H} = l^2(\mathbb{Z})$, $\hspace{2mm}$ $U:\mathcal{H} \rightarrow \mathcal{H}$ such that $(Ux)_k = x_{k+1}$
a)Find $U^...
2
votes
1answer
13 views
Orthogonality in subspaces of Hilbert Spaces
Let $X$ and $Y$ are subspaces of the Hilbert space $H$
If $X+Y=\{x+y : x\in X , y\in Y\}$
Show that $(X+Y)^{\perp}=X^{\perp} \cap Y^{\perp}$
My tend:
I have tried to prove coverings from bothside....
1
vote
1answer
35 views
Question about Bounded Hilbert Operator and Riesz Theorem
I was dealing with this:
Let
\begin{equation}
[ \cdot, \cdot ] : H \times H \rightarrow \mathbb{C}
\end{equation}
such that $\forall x,y,z \in H$, $\lambda, \mu \in \mathbb{C}$,
$$ \langle x, \...
2
votes
1answer
25 views
Quantum mechanics: total orthonormal sets & position/momentum space
Consider the position and momentum vector sets
$$X= \text{{|x> | x $\in$ $R^3$}}$$
$$P=\text{{|p> | p $\in R^3$}}$$
By the assumption of quantum mechanics, both $X$ and $P$ are total ...
2
votes
0answers
35 views
Minimal rotational degrees of freedom needed in N-space
How many rotational matrices are needed in N-space to achieve an arbitrary attitude when allowed to be successively applied?
Each N by N rotation matrix will allow only a rotation in a plane, so only ...
2
votes
1answer
29 views
Showing that an operator is positive
Let $\mathcal{H}$ be a Hilbert space, and $T$ an operator. The task is to prove that the corresponding self adjoint operator $
\begin{bmatrix}
1 & T \\
T^{*} & 1\\
\end{...
0
votes
1answer
27 views
Hilbert Space: infinite or finite? - All real inner product spaces are Hilbert spaces?
I am confused about Hilbert spaces. I want to understand them better.
If I understand correctly a Hilbert space is an inner product space that has either finite or infinite dimension over real or ...
2
votes
1answer
55 views
Several questions on the mini-max theorem for self-adjoint operators
I am reading the proof of mini-max theorem for bounded self-adjoint operators following "Unbounded Self-adjoint Operators on Hilbert Space" by Konrad Schmüdgen and it seems a totally mess.
Given a ...
4
votes
0answers
35 views
Convergence of (unbounded) self-adjoint operators
I'm learning about the dynamical convergence (i.e, convergence of the unitary group associated with each operator) and resolvent convergence of (unbounded) self-adjoint densely defined operators. I ...
1
vote
1answer
115 views
Why does this map define an $n$-dimensional manifold?
Let $0<\xi_1 <\xi_2<\ldots <\xi_n<1$ be $n$ variables, $\xi=(\xi_1,\ldots,\xi_n)$; and $\Gamma=\{\xi|0<\xi_1<\ldots <\xi_n<1\}$.
Let $U(\cdot,\cdot)\colon\Gamma\times [0,1]\...
