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

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$\|.\|_2$ 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
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3answers
188 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 ...
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0answers
68 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
68 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
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2answers
103 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
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2answers
118 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
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1answer
35 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 $\lim_{n\rightarrow\infty}aa_{n}=0$. ...
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0answers
45 views

Hilbert space — show a subspace is closed

Consider a Hilbert space $\mathcal{L}^{2}=\lbrace X: X-\text{real-valued random variable}, \mathbb{E}(X^{2})<\infty \rbrace$ with the inner product $<X,Y>=\mathbb{E}(XY)$. Let ...
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3answers
47 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
51 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 ...
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1answer
103 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
57 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 ...
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0answers
76 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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2answers
73 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 ...
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1answer
52 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 $J$ ...
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2answers
64 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
27 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
36 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
55 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 ...
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3answers
75 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 ...
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1answer
46 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 ...
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4answers
78 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 ...
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1answer
62 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 ...
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1answer
28 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 ...
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1answer
55 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 ...
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0answers
10 views

Does the product of two kernels define the joint distribution?

If I have two kernels, $k(x, x')$ and $k(y, y')$ and I'd like to represent the product of the marginals, is that achieved by creating a new kernel defined by the product between the two kernels, or is ...
2
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1answer
275 views

Orthonormal Basis for Hilbert Spaces

The following is the definition of orthonormal base that I am using: The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space H, an ...
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2answers
78 views

Finding an isometry between two subspaces of a Hilbert space

So, I'm given a Hilbert space which is the direct sum $H=H_1\oplus H_2$ of two separable Hilbert spaces $H_j$. There is a closed subspace $D\subseteq H$ which satisfies that it is not a subspace of ...
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0answers
45 views

Verify that the operator $T$ defined by $T( \varphi _{k})=\frac{1}{k}\varphi _{k+1}$ is compact, but has no eigenvectors.

Let $H$ be a Hilbert space with basis $\left \{ \varphi _{k} \right \}_{k=1}^{\infty }$ .Verify that the operator $T$ defined by $$T( \varphi _{k})=\frac{1}{k}\varphi _{k+1}$$ is compact, but has no ...
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1answer
40 views

Index map defines a bijection to $\mathbb{Z}$?

In the book "Spin Geometry" by Lawson and Michelsohn, page 201, proposition 7.1(chapter III), it asserts that the mapping which assign a Fredholm operator from one Hilbert space to another its index ...
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1answer
65 views

Properties of subspaces of Normed Vector Spaces

How does it follow that a subset of a normed vector space cannot be open if it does not contain an open ball $B_{\epsilon}(0)$ where $\epsilon > 0$? I just want to confirm also that for normed ...
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1answer
34 views

Reproducing Kernel and the continuous of the evaluation functional $e_{t}$

I'm working on Reproducing Kernel Hilbert Spaces and I had a problem proving the the continuity of the evaluation functional $e_{t}$ ($e_{t}(\phi) = \phi(t)$). Theorem A Hilbert Space of complex ...
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2answers
111 views

Is fractional Sobolev space $H^s$ Hilbert?

For $s \in (0,\infty)$ a fractional number, define $H^s(\Omega) = W^{s,2}(\Omega)$ on good domain $\Omega$. Every textbook doesn't say that $H^s$ is Hilbert. Is it? I have only seen this fact when ...
2
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1answer
41 views

Riesz Representation Theorem: isomorph

Riesz' Representation Theorem states that every linear functional can be represented by a vector. This shows that the Dual can be ANTILINEARLY and norm preserving identified with the Hilbert Space ...
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0answers
42 views

How to find a hilbert basis of a given subspace considering a given inner product

Let $X$ be the space of continuous functions on $[-1;1]$ to $\mathbb{R}$ with the inner product: $$\langle f,\ g\rangle = \int_{-1}^{1} \! f(x)g(x) \, dx$$ and let $U$ be a subspace of $X$ with $U := ...
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1answer
45 views

showing that a sequence is converging.

suppose $\left \{ T_{k} \right \}$ is a collection of bounded operators on Hilbert space $H$ ,with $\left \| T_{k} \right \|\leq 1$ for all $k$ .suppose also that $$T_{k}T_{j}^{*}=T_{k}^{*}T_{j}=0 ...
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0answers
32 views

Please help understand a particular proof of basic theorem

I read the following proof that in a vector space $V$ of dimension $n$, a set of orthonormal vectors $\{\phi_1, ..., \phi_m\}$, with $m<n$, is not complete : Among the linear combinations ...
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1answer
94 views

Dense subspace of $L^{2}[0,1]$

I know that $C[0,1]$ is dense in $L^{2}[0,1]$ but is $\{f\in C^{2}[0,1]:f(0)=f(1)=0\}$ dense in $L^{2}[0,1]$?
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1answer
36 views

Are eigenspaces in a Hilbert space rays?

It may sound as a dumb question but I just want to be sure that I understand all the terminology: The eigenspaces corresponding to a (non-degenerate) eigenvalue of a operator on a Hilbert space are ...
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0answers
57 views

Gelfand triple in Tensor product spaces

Let's define $A^k={\rm span} \{\phi_1,\dots,\phi_k\}$ and $B^k={\rm span} \{\psi_1,\dots,\psi_k\}$ where $\phi \in H^1(X)$ and $\psi \in L^2(X)$ simple functions and $X$ is closed subset of ...
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2answers
48 views

Upper bound for norm of Hilbert space operator

It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $||T||=\sup_{||x||=1}|\langle Tx,x\rangle|:=M$. It seems that for any bounded operator on ...
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0answers
35 views

$\gamma-$radonifying operators.

I am reading about $\gamma$-radonifying operators, and came across 2 similar definitions. And I hoped to understand why they are equivalent. Let $H$ be a seperable real Hilbertspace, $E$ banach ...
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1answer
49 views

Closed range in Hilbert Space

If $H$ is a Hilbert Space. Let $A: H \rightarrow H$ be a one-to-one bounded operator with the additional property that $\beta||u|| \leq ||Au||$. How would you show that $R(A)$ (the range of A) is ...
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1answer
50 views

About eigenspaces

In the context of a Hilbert space $H$, when an operator $A$ is diagonalizable we usually decompose the Hilbert space into direct sum of eigenspaces $$H=\bigoplus\limits_{n=1}^\infty E_n$$ where $E_n$ ...
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33 views

Help on a Hilbert Space theory utilization.

I need some help here concerning the Hilbert Spaces theory. Below, you can see a part of Olivier Chapelle's paper: "Training a Support Vector Machine in the Primal". As you can see below, in Eq.(8) ...
2
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1answer
48 views

Norm of the dual of the Tensor product of Hilbert spaces

Let $V$ and $W$ be Hilbert spaces, we can define inner product and induced norm on Tensor product of these spaces as: Let $v_1,v_2 \in V$,and $w_1,w_2 \in W$. then $(v_1 \otimes w_1, v_2 \otimes ...
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1answer
247 views

Approximation of bounded and continuous mappings

Does anyone know if we can approximate a bounded (i.e. bounded sets in V are mapped to bounded sets in V': for every bounded $U\subseteq V$ and $x\in U$, there exists $K_U>0$ such that ...
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0answers
23 views

Proof of Rayleigh trace

I found the following statement without proof: Let us given a self-adjoint Operator $T\colon L^2 \to L^2$ which has n eigenvalues $ \lambda_n \leq \dots \leq\lambda_{n-1} < \lambda_1 =1$ counted ...
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1answer
32 views

If $f$ is identically zero then the coefficients are all zero

I am looking at the space: $$A:=\left\{f(x)=\sum_{k\in\mathbb{Z}}{a_ne^{inx}}:(a_n)_{n\in\mathbb{Z}}\in l^1(\mathbb{Z})\right\}$$ I want to say the following: if $f\equiv0$, then $a_n=0$ for all ...
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1answer
51 views

Show this subpace of a Hilbert space is dense

This is part of an exercise in Rudin's Functional Analysis, in the chapter on Unbounded Operators. Let $H$ be a Hilbert space with orthonormal basis $\{e_n\}$. Let $X$ be the set of all finite sums ...