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

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2answers
111 views

Is $H^2\cap H_0^1$ equipped with the norm $\|f'\|_{L^2}$ complete?

Let $-\infty<a<b<+\infty$. Consider the norms $\|\cdot\|_{L^2}$, $\|\cdot\|_0$ and $\|\cdot\|_{H^1}$ defined on suitable spaces and given by ...
1
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2answers
65 views

Chart of how the mathematical spaces are related? (soft question)

When dealing with specific function spaces e.g. Sobolev, Hilbert, etc., I find it easy enough to accept the properties of that space and work with them; however, I have a hard time visualizing how ...
4
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1answer
42 views

Is this a correct argument for why this space of sequences is a Hilbert space?

I am assigned the following problem (a piece of Exercise 6.5 in Brezis's book): Let $(\lambda_n)$ be a sequence of positive numbers such that $\lim_{n\to\infty}\lambda_n=+\infty$. Let $V$ be the ...
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1answer
166 views

The Trace Class Operators Form a Banach Space

I want examining the trace class operators $L_1(H)$ of a separable Hilbert space $H$ with the norm $||A||_1=\sum\limits^{\infty}_{n=1}\lambda_n$ where $\lambda_n$ are the eigenvalues of ...
2
votes
2answers
190 views

Linear operators with no adjoint

Here is a standard theorem about bounded operators: Let $H$ be a Hilbert space. For any bounded linear operator $A:H\to H$ there is a unique bounded operator $A^*$ s.t $\langle Au,v\rangle=\langle ...
4
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3answers
147 views

Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
0
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0answers
24 views

Inner product spaces, examples for subspaces with certain properties [duplicate]

Let $H$ be a inner product space. Give examples for a subset $U\subset H$ so that (a) $\overline{U}\neq U^{\bot\bot}$ (b) $\overline{U}\oplus U^{\bot}\neq H$ I have thought about $H=C[0,1]$ with ...
5
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1answer
121 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
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1answer
136 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
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1answer
132 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 ...
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0answers
38 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. ...
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0answers
32 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. ...
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2answers
82 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 ...
2
votes
1answer
585 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
79 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
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1answer
66 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
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1answer
29 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
83 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
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1answer
34 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 ...
0
votes
1answer
44 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 ...
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1answer
65 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
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1answer
133 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
108 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
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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 ...
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1answer
361 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
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1answer
92 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 ...
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1answer
177 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 ...
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1answer
290 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 ...
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1answer
96 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
250 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 ...
2
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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 ...
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1answer
95 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
119 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
220 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
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0answers
79 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 ...
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1answer
80 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
342 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
127 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
57 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
51 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])?
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3answers
93 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
134 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 ...
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1answer
71 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
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0answers
124 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 ...
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votes
2answers
109 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
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1answer
56 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
78 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 $T$?
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1answer
34 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 $L^2(0,1)$?
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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} ...
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1answer
196 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 ...