For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

learn more… | top users | synonyms

5
votes
1answer
138 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
0
votes
1answer
145 views

Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
1
vote
1answer
155 views

Weak convergence equal to coordinate-wise convergence

Show for the Hilbert Space $\ell^2(\mathbb{N})$ weak convergence of a bounded sequence $(x_k)_{k \in \mathbb{N}}$ is equal to coordinate-wise convergence. A sequence $(x_k)$ is weak ...
1
vote
0answers
40 views

Continuity of certain projections in the weak topology.

I'd like to prove that: Given a Hilbert space H and S a closed subespace, $S \subseteq H$, the projection $P_{S}:H \to S$ is continuous in the weak topology. I have tried the following. ...
1
vote
0answers
34 views

$C_0$-Semigoups - References needed

I'm looking for the references which contains the following theorem with a proof: Theorem Let $(T_t)$ be a uniformly continuous semigroups with generator $A \in B(H)$, where $H$ is a Hilbert space. ...
1
vote
2answers
90 views

Self-adjoint Hilbert Space operators

Let $H$ be a Hilbert space and $T$ is a self adjoint continuous operator in $\mathcal B(H)$. Show that $\|T^{2^{k}}\| = \|T\|^{2^{k}}$. Does this equality hold for all operators? Now it is clear that ...
3
votes
1answer
648 views

In a Hilbert space, every bounded and closed set is weakly relatively compact.

My aim is to prove that in a Hilbert space, any sequence has a weakly convergent subsequence. To prove this, I'm trying to prove that: ...
2
votes
1answer
86 views

Superspace as the Hilbert Space for Quantum Gravity

This is a question I've asked in physics.stackexchange, but have obtained no answers: Let $\mathcal{A}$ be the Ashtekar connection. Since $^{(3)}g_{AB}=i\frac{\delta}{\delta\mathcal{A}^{AB}}$ (see R. ...
0
votes
1answer
68 views

Convergent series in Hilbert space

I am looking for a proof of the following theorem. Consider a countable orthonormal set in Hilbert space $H :\ \ u_1, u_2, ...$ $\sum_{j=1} ^{\infty} r_ju_j$ is convergent in $H \iff \sum_{j=1} ...
0
votes
1answer
30 views

Check this is a hilbert norm: $ \ell^2 $ with norm $\| \cdot \| := \| \cdot \|_{\ell^2} + \| \cdot \|_{\ell^p}$

Clearly $ p \geq 2 $ so it gains sense calculating the $\ell^p $-norm. According to my calculation this norm is equivalent to the $\ell^2 $ norm, in fact given a cauchy sequence w.r.t $\| \cdot \| $ ...
2
votes
1answer
84 views

Hilbert space, orthonormal system, compact set of vectors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...
0
votes
1answer
36 views

Hilbert space continuous linear map, one dimensional subspace

Could you help me with the following exercise? Let $H$ be a Hilbert space, $\alpha : H \rightarrow \mathbb{C}$ a linear continuous mapping, $\alpha \neq 0$. Prove that the orthogonal complement ...
1
vote
1answer
47 views

Isometry from closed operator

I have a following problem Let $H$ be a Hilbert space. We have a closed densely defined operator $A \colon D \subset H \rightarrow H$, we know that $\|Ax\| = \|x\|$ for all $x \in D$, can we extend ...
1
vote
1answer
67 views

Question about normal operators

I have a question about definitions and theorems because I am a little bit confused. By definition we say that a (possibly unbounded) operator $T$ on a Hilbert space $H$ is normal if $D(T)$ is dense ...
0
votes
1answer
152 views

absolutely convergent series in Hilbert space

Is it possible to find an infinite dimensional Hilbert space, where every convergent series is absolutely convergent? I could not find any clue to find an example of such type or to disprove that. ...
2
votes
1answer
110 views

Hilbert Space and Projections

If $M$ is a closed subspace of the Hilbert space $H$ and $x$ $\in$ $H$, prove that: $$\underset{y \in M}{\min} ||x-y|| =\max\{|\langle x,z\rangle|:z \in M^{\perp}, ||z||=1\}.$$ There isn't a ...
2
votes
1answer
47 views

Proof of an equivalence in Hilbert spaces

Let $H$ be a Hilbert space. Prove that the following are equivalent: a) the algebraic dimension of $H$ is finite; b) each closed, not empty subset $C$ has an element of minimum norm (that is the ...
1
vote
1answer
412 views

Spectrum of operator

Like my previous question, I'm considering the same space and operator: Hilbertspace adjoint But this time I am trying to determine the spectrum of $T$. I feel like I'm messing up my definitions a ...
0
votes
1answer
103 views

Proving Density of Subset of Hilbert Space

Suppose we have a subspace, $M$, of Hilbert space $H$. Prove the first statement implies the second statement: 1) If $<f,g> = 0$ for any $g\in M$, then $f=0$ in $H$. 2) $M$ is dense in $H$. I ...
1
vote
1answer
178 views

Hilbertspace adjoint

Im doing the following excercise: Ok, so let $(e_n)$ be a orthonormal basis of $l^2$, and fix arbitrary complex numbers $(\lambda_n)$ and define $T:l^2\to l^2 $ as $$T(\sum x_ne_n)=\sum ...
3
votes
1answer
362 views

Double orthogonal complement of any closed subspace is it self

Let $H$ be a pre-Hilbert space such that any closed sub space $M \subset H$ has the property $M^{\bot \bot}=M$. Prove that $H$ is a Hilbert space (ie, prove that $H$ is complete) My attempt: As ...
1
vote
1answer
106 views

$ H $ hilbert space: Hamel dimension of $ H $ = Hilbert dimension of $ H $ $ \Leftrightarrow$ dim $ H $ is finite

Clearly $\Leftarrow $ is a trivial trivial application of G-Schmidt algorithm. I've experienced some trouble in proving the other direction. I focused my self on the fact that span($ A $)=$ H $ (it ...
2
votes
1answer
307 views

Relationship between different topologies of bounded operators on a Hilbert space

I am self-studying functional analysis. Given that $B(H)$ are the bounded operators on a Hilbert space, $H$. I would like to ask how to formally prove that the weak topology is weaker than the ...
3
votes
1answer
81 views

Sesquilinear Forms: Reals

Given a real Hilbert space $\mathcal{H}$. Consider symmetric forms: $$s:\mathcal{H}\times\mathcal{H}\to\mathbb{R}:\quad s(\psi,\varphi)=s(\varphi,\psi)$$ By polarization one obtains: ...
2
votes
0answers
36 views

Is this dual-spaced norm based on $L_2$ norm

I am reading the book of Claes Johnson about Numerical Solution of Partial Differential Equations by the Finite Element Method and particularly pages 34 and 98. I wrote these notes to my craft Is ...
1
vote
1answer
110 views

Multiplication operator on Hilbert space

i looked to the question Spectrum and point spectrum of this operator. I will go further with asking. We know that $T$ is well-defined iff $(\lambda_n)\in\ell^{\infty}$. But if ...
0
votes
2answers
122 views

$\|.\|_2$ closure of a set which is dense in $L^2[0,2\pi].$

The following is an exercise of Conway's Functional analysis, chapter 1, section 5. Let $L=\{f\in C[0,2\pi]|f(0)=f(2\pi)\}$ and show that $L$ is dense in $L^2[0,2\pi]$.
9
votes
3answers
222 views

Conjecture: the function $d(x, y):=\frac{||x-y||}{\max(||x||, ||y||)}$ is a distance

I make the following conjecture: the function $$ d(x, y):=\frac{||x-y||}{\max(||x||, ||y||)} $$ is a distance on $H$, where $H$ is a normed vector space or a Hilbert space, and $x, y \in H$ (the ...
3
votes
0answers
84 views

Does a “typical” reproducing kernel on a manifold generate an infinite-dimensional RKHS?

Let $M$ be a finite dimensional manifold and $k:M\times M\rightarrow\mathbb{R}$ a smooth reproducing kernel. Is the set of all such $k$ with the property that the reproducing kernel Hilbert space ...
3
votes
1answer
96 views

Is every Hilbert space an $L^2$ space?

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you
4
votes
2answers
395 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
3
votes
2answers
129 views

Dense Graph $G(T)\subset H\times H$

The following construction appears to yield a dense Graph in $H\times H$ where $H$ is a seperable Hilbert-space. Take $\{x_n\}$ a countable dense subset of $H$. Let $\{e_n\}$ an orthonormal basis of ...
2
votes
1answer
94 views

Approximate point spectrum and left topological zero divisors

Recall that a left topological zero divisor in a Banach algebra $A$ is an element $a\in A$ such that there exists a sequence of unit vectors $(a_{n})$ in $A$ with $\lim_{n\rightarrow\infty}aa_{n}=0$. ...
0
votes
3answers
53 views

linearly independent in Hilbert Space

Please help me to solve the linearly independent of functions in Hilbert Space how i can show that the functions $\sin(t)$ and $\cos(t)$ are linearly independent in Hilbert Space (L^2[0,pi])?
1
vote
3answers
99 views

Prove a space is Hilbert [duplicate]

I got stucked in this problem and get no clue to solve this. Can any one please help me? Thanks Suppose $X$ is an inner product space. If for every bounded linear function $f$, there exists $z \in ...
1
vote
1answer
140 views

Gelfand triple for tensor product of Hilbert spaces

Is there any dense embeding $\to$ that makes $H^1_0(D) \otimes L^2(\Gamma) \to L^2(D) \otimes L^2(\Gamma) \to (H^1_0(D) \otimes L^2(\Gamma))^{*}$ a Gelfand tripe? In fact we may only answere to the ...
1
vote
1answer
74 views

Explicit operator in separable Hilbert space

This is a question about (possible unbounded) operators. We know that $\mathcal{D}(T^*)=\{0\}$ iff $\mathcal{G}(T)$ is dense in $\mathcal{H}\times\mathcal{H}$, where $\mathcal{H}$ is a separable ...
2
votes
0answers
132 views

Conditional expectation on the space of bounded linear operators

In the paper from the link http://arxiv.org/pdf/0906.0139.pdf the author uses a diagonal conditional expectation. We take a seperable Hilbert space $H$ and fix an orthonormal basis $(e_n)_{n \in ...
-1
votes
2answers
122 views

Normal compact operator commute with bounded self adjoint operator in Hilbert space.

Suppose $H$ is a Hilbert space and $A:H\rightarrow H$ is a normal compact operator such that $\ker(A)=0$. show that if $B$ is a bounded self adjoint operator that commutes with $A$ then the spaces in ...
0
votes
1answer
57 views

Different types of continuity in $\ell^2$

Consider the following functional $J$ on $\ell^2$ which for $x = \{x_n\}$ is defined by $$J(x) = \sum_{n=1}^{\infty}n^{1/n}x_{n}^{2}.$$ Is $J$ continuous? Is $J$ lower semi-continuous? Is $J$ ...
2
votes
2answers
82 views

Clarifying the definition of essential self-adjointness

If a Hilbert space operator $T$ has a unique self-adjoint extension, must the extension be the closure of $T$?
1
vote
1answer
38 views

Scalar product in L2(0,1)?

Is $s(f,g) = \int_0^1 f(x)g(1-x)dx$ a valid scalar product in $L^2(0,1)$?
0
votes
0answers
40 views

Prove that a give sequence of function is a base of $L^2([0,1])$

Consider $(\phi_k)_{k \geq0} \in \mathcal C^{\infty}([0,1])$ with $\phi_k \not\equiv 0 $ such that $$\int_0^1 \phi_k(s) ds = 0, \quad \forall k\geq 1$$ and $$\sup_{ t \in [0,1]} \left | \frac{d}{dt} ...
1
vote
1answer
224 views

Construction of Gaussian Hilbert spaces

I am reading the very first chapter of "Gaussian Hilbert Spaces" by S. Janson. Definition: A Gaussian Hilbert space is a closed subspace of $L^2(\Omega, \mathcal{F}, P)$ consisting of centered ...
4
votes
3answers
99 views

$A^2$ self-adjoint and Compact, prove $A$ has an eigenvalue

Suppose $H$ is a Hilbert space and $A \in L(H)$ is such that $A^2$ is compact and self-adjoint. Prove that $A$ has an eigenvalue. (Here $L(H)$ is the set of bounded linear operators on a Hilbert ...
1
vote
1answer
55 views

A problem about projective operater

Let $P$ and $Q$ be projective on a Hilbert space $H$. Show that $P+Q$ is projective if and only if $\mbox{ran }P \perp \mbox{ran }Q$. The sufficiency is easy. About the necessity, suppose $P+Q$ is ...
0
votes
4answers
234 views

Orthonormal basis in Hilbert space - 2 questions

I know there have been a number of questions on Hilbert spaces and orthonormal basis, but I can't find any answers to these two questions: 1) Let $H$ be a Hilbert space, and say we found a Hilbert ...
2
votes
1answer
68 views

Understanding problems of space

I've been trying to understand the concept of space for some time now, but I still can't grasp the essence of it. In high school math we've been using 2D- and 3D- Euclidean space. Now that I am ...
1
vote
1answer
33 views

The space $\mathcal{D}((0,T);V)$ and its norm/embeddings?

Let $V$ be a Hilbert space. Define $\mathcal{D}((0,T);V)$ to be the set of functions $u:(0,T) \to V$ such $u$ is compactly supported on $(0,T)$ and is a $C^\infty$ test function. What is the norm ...
2
votes
1answer
71 views

determinant identity for invertible finite rank operators

I am currently reading a paper where the following identity, valid for an invertible finite - rank operator $T \colon \mathscr{H} \to \mathscr{H}$ on a separable Hilbert space, is given: $$ \log \det ...