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

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1answer
176 views

Minimum maximum identity

Let $x_0\in H$. $H$ is an Hilbert space, $M$ is a closed subspace of $H$. In my lecture notes about functional analysis, there is the following identity $$\min\{\|y-x_0\|: y\in M\}=\max\{|\langle ...
2
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2answers
239 views

What is the norm of the operator $L((x_n)) \equiv \sum_{n=1}^\infty \frac{x_n}{\sqrt{n(n+1)}}$ on $\ell_2$?

Let $(x_n) \subset \ell_2$ and let operator $L:\ell_2\to \mathbb R$ be defined by: $\displaystyle L((x_n)) := \sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n(n+1)}}$. Find the norm of L.
2
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1answer
101 views

Diagonal operators on non-separable Hilbert space

Let $H$ be a non-separable Hilbert space with an orthonormal basis $(e_\alpha)_{\alpha<\omega_1}$. To each $f=(f_\alpha)\in c_0(\omega_1)$ associate an operator on $H$ defined by $T_f ...
3
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1answer
211 views

eigenvalue question

I think this question isn't that hard, but I am a bit confused. Define the linear operator $T_k:H\mapsto H$ by \begin{align} T_ku=\sum^\infty_{n=1}\frac{1}{n^3}\langle u,e_n\rangle e_n+k\langle ...
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2answers
158 views

Orthogonal projections

Let $H$ be a Hilbert space, and $V_{1}, V_{2}$ are two finite dimensional subspaces of $H$. If $P_1,P_2$ are two orthogonal projections, $P_1:H\to V_1$ and $P_2:H\to V_2$, and $P_2\circ P_1=P_2\circ ...
0
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1answer
60 views

System in Hilbert Space

Let $\phi_1, \phi_2,\dots$ be a complete orthonormal system in a Hilbert space. Define vectors by $$\psi_n=C_n(\sum_{k=1}^n \phi_k-n\phi_{n+1})$$ $(n=1, 2, \dots)$. (i) Show that $\psi_1, \psi_2, ...
3
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0answers
53 views

Existence of an ergodic-looking limit in a Hilbert space

This is part of a problem from Reed & Simon's Functional Analysis -- I'll write the problem first. Let $V$ be a linear transform on the Hilbert space $H$, such that its powers are uniformly ...
1
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1answer
104 views

Properties on Hilbert space

$H$ is real Hilbert space. $a\colon H\times H \to \mathbb R$ is a bilinear form on $H$ with $\lvert a(x,y)\rvert \leq C\lVert x\rVert \lVert y\rVert$ and $a(x,x) \geq \alpha \lVert x\rVert^2$. I would ...
4
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1answer
462 views

Showing the basis of a Hilbert Space have the same cardinality

I am trying to show that if we have two orthnormal families, $\{a_i\}_{i\in K}$ and $\{b_j\}_{j\in S}$ and these are the basis of some Hilbert Space H then they have the same carnality. So If I ...
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1answer
70 views

Show that $C([0,1],\mathbb{R})$ with the $L_2$ inner product norm is not a Hilbert space.

I need to prove that all continuous functions on the closed set $[0,1]$ is not a Hilbert space. Given the $L_2$ norm. I guess I need to show that every Cauchy sequence in the space, does not ...
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1answer
185 views

Orthonormal Family in a Hilbert Space

If we have an orthonormal family, $\{u_n\}_{i=1}^\infty$ in a Hilbert Space $H$, I need to show that for $x\in H$ we have the following inequality: $$\left|\left\{n|\langle x, u_n \rangle > ...
2
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1answer
381 views

Compact operator on $l^2$

Let A be a bounded linear operator on $l^2$ defined by A($a_n$)=($\frac{1}{n} a_n$). Would you help me to prove that A is compact operator. I guess the answer using an approximation by a sequences of ...
1
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1answer
116 views

Compact Operator defined by inner product

Let $H$ be a Hilbert space and $y,z \in H$. Define bounded linear operator $Ax=\langle x,y\rangle z$ where $\langle,\rangle$ is inner product. Would you help me to prove that $A$ is compact operator.
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1answer
177 views

Composition of two commutative projection

Let $P$ and $Q$ be projection on a Hilbert Space. If $PQ=QP$, would you help me to prove that $% PQ$ is also a projection.
5
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2answers
307 views

Unit Ball of $\mathcal{l}_2$

Let $B(\mathcal{l}_2) :=\{x \in \mathcal{l}_2 : \|x \| \leq 1 \}$ and $S(\mathcal{l}_2) :=\{x \in \mathcal{l}_2 : \|x \| = 1 \}$ be the unit ball and the unit sphere of $\mathcal{l}_2$, respectively. ...
0
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1answer
400 views

Orthogonal Projection on Hilbert Space

Let $A$ be a non-empty subset of a Hilbert space $H$. Suppose that $T$ is a linear operator on $H$ such that $T(H) \subseteq A$ and, for every $x \in H, (x-Tx) \perp A$. Then $T$ is bounded. $A$ is ...
3
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1answer
98 views

Orthogonal matrix

Let $A$ be a real $n \times n$ matrix. I'm trying to prove that if $A$ maps orthogonal vectors into orthogonal vectors and $\lVert A \rVert = 1$, then, for every $x,y \in \mathcal{l}_2^n$, ...
0
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1answer
45 views

Closest point to the origin in $\mathcal{l}_2$

Suppose $K$ is the closed convex hull generated by the canonical basis $\{e_n\}$ in $\mathcal{l}_2$. How do you find the unique closest to $0$ in $K$? I don't have the foggiest idea about how to do ...
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1answer
100 views

Show that $U \subset V \Leftrightarrow V^\bot \subset U^\bot$ for $U,V$ subspaces in a Hilbertspace

Let $(\mathcal H, \langle\cdot,\cdot\rangle)$ be a Hilbertspace, $U,V \subset \mathcal H$ are closed subspaces. I want to show $$U \subset V \Leftrightarrow V^\bot \subset U^\bot$$ $\Rightarrow$ is ...
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1answer
87 views

Proving $L^2$ convergence (application of dominated convergence?)

For any $f\in L^2(\mathbb{R}^d)$ prove \begin{align}\left\lVert \int_{\mathbb{R}^d} e^{i |x-y|^2}f(y) dy-\int_{\mathbb{R}^d}e^{i |x-y|^2} e^{-|y|^2/a}f(y) dy \right\rVert_{L^2} \rightarrow 0\ \ \ ...
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0answers
138 views

Show that its a Generalized Eigenvalue problem

Show that the minimizer is obtained by a generalized eigenvalue problem. $$\alpha=\underset{1^TK\alpha=0; \ \alpha^TK^2\alpha=1}{\text {arg min}} \gamma ||f||_{K}^2+f^TLf$$ Details: $K$ ...
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1answer
159 views

Riesz, Hilbert and Hamel bases

I was surprised to read both at PlanetMath and in Wikipedia (apparently copied from PlanetMath) that If $H$ is a finite-dimensional [Hilbert] space, then every basis of $H$ is a Riesz basis. I ...
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1answer
119 views

Finding Riesz basis

Let H be a Hilbert space .Is there always a non orthogonal Riesz basis $D$ on it such that following holds? $$\sup_{g\in D }\sum_{g'\in D,g'\not=g}|\langle g,g'\rangle|<1/3 $$ And is there Riesz ...
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2answers
660 views

Riesz basis in Hilbert space

We know that a collection of vectors $\{x_{k}\}$ in a Hilbert space called Riesz basis if it is an image of orthonormal for H under invertible linear transformation. How to prove that there is ...
8
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1answer
705 views

Transforming a distance function to a kernel

Fix a domain $X$: Let $d : X \times X \rightarrow \mathbb{R}$ be a distance function on $X$, with the properties $d(x,y) = 0 \iff x = y$ for all $x,y$ $d(x,y) = d(y,x)$ for all $x,y$ Optionally, ...
8
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0answers
406 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
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0answers
48 views

geometric charaterization of complex interpolation spaces $(H,Y)_\theta$ where $H$ is a Hilbert space?

Let $C$ be the class of Banach spaces $X$ such that there exists $0<\theta<1$, a Hilbert space $H$ and a Banach space $Y$ such that $$ X=(H,Y)_\theta $$ (complex interpolation of Calderon). ...
2
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1answer
165 views

Dual space norms and equivalence

Suppose $S(r)$ is set parametrised by $r \in [0,T]$. Let $\phi_t^s : H^1(S(t)) \to H^1(S(s))$ is a linear homeomorphism. Suppose $\lVert \cdot \rVert_{H^1(S(t))}$ and $\lVert \phi_t^s(\cdot) ...
3
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1answer
2k views

Volterra Operator is compact but has no eigenvalue

Volterra operator is defined as operator $V:L^2[0,1]\rightarrow L^2[0,1]$ by \begin{eqnarray} (V)(f(x))=\int_0^xf(y)dy \end{eqnarray} Would you help me to prove that this operator is compact but has ...
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2answers
200 views

Hilbert dual space (inequality and reflexivity)

Let $V \subset H$ where $H$ is Hilbert space. Let $T:H^* \to V^*$ be the canonical map that restricts the domain of a functional in $H$ so that it's a functional in $V$. How do I show that $$\lVert ...
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2answers
261 views

Finding the min of an integral

So I have to find the following $$\min_{a,b,c\in\mathbb{R}}\int_{-1}^{1} |x^3-a-bx-cx^2|^2dx$$ I have a hint at a solution which says to consider $X=\{\mbox{polynomials of degree} \leq 2\}$. So then ...
0
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1answer
158 views

Diagonalizable operator is compact [duplicate]

Possible Duplicate: How to prove that an operator is compact? Let $H$ be a separable Hilbert space with bases $\left\{ e_{n}\right\} $, $% \left\{ \alpha _{n}\right\} \subset \mathbb{F} $ ...
1
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1answer
138 views

Compact operator in $L^2$

Let $\left( X,\Omega ,\mu \right) $ be a measure space and $k\in L^{2}\left( X\times X,\Omega \times \Omega ,\mu \times \mu \right) $, define \[ \left( Kf\right) \left( x\right) =\int k\left( ...
7
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4answers
719 views

An idempotent operator is compact if and only if it is of finite rank

Would you help me to solve this problem. Show that an idempotent operator on hilbert space is compact if and only if it has finite rank.
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0answers
97 views

Reproducing Kernels are Positive Definite. Does the converse hold true?

Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...
3
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2answers
1k views

Every Hilbert space has an orthonomal basis - using Zorn's Lemma

The problem is to prove that every Hilbert space has a orthonormal basis. We are given Zorn's Lemma, which is taken as an axiom of set theory: Lemma If X is a nonempty partially ordered set with the ...
2
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1answer
216 views

Notation: Representer Theorem for Reproducing kernel hilbert spaces

Am studying the basic concepts of RKHS and the representer theorem: In $f(x_i)=<f,k(x_i,\mathbb{.})>$, what does $ f$ on the r.h.s denote? What is its structure-is it a vector? I was thinking ...
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1answer
76 views

Banach space geometry without bounded operators?

I understand that $B(X)$ can be think of as the collection of symmetries of a Banach space $X$, and that they provide important information concerning the geometric structure of the space. But I am ...
4
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2answers
955 views

$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
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1answer
1k views

$\ell_p$ is Hilbert space if and only if $p=2$

Can anybody please help me to prove this.. Let p greater than or equal to 1,show that the space of all p-summable sequences is an inner product space if and only if p=2
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1answer
138 views

Show the existence and uniqueness of a closed ball containing a bounded subset of a Hilbert space

The problem: Assume $A$ is a bounded subset of a Hilbert space $H$. Let $r$ be the infimum of the radii of closed balls containing $A$, so $r = \inf \{s \geq 0 $ $\vert$ there exists $x \in H$ such ...
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2answers
1k views

Uniqueness of symmetric positive definite matrix decomposition

We know that any symmetric positive semi-definite matrix $K$ can be written as $K= AA^T$, where $A$ has real components. One way to get to $A$ is to compute eigen value decomposition of $K= P^T DP$ ...
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1answer
387 views

In a separable Hilbert space, how to show that the orthogonal projection onto a subspace of $n$ orthonormal basis elements converge?

Could anyone help me with this problem? I don't know where to start. Let $\{ e_n \}_{n=1}^\infty$ be an orthonormal basis in a separable Hilbert space $H$. Denote by $P_n$ the orthogonal ...
5
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2answers
518 views

Orthonormal basis for Sobolev Spaces

Sobolev spaces of order 2 are known to form a Hilbert space. Consider such a Sobolev space of (order 2) functions on the domain $f:\mathbb{R}\rightarrow \mathbb{R}$. What is an example for the basis ...
4
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1answer
63 views

Help with proof of closedness of a set

Let $u_n$ be a sequence in Hilbert space such that $\|u_n\|=1$ for all $n$, and $\langle u_n|u_m\rangle=0$ whenever $n\neq m$. Why is the following set closed: $\{0\}\cup \{u_n \mid n\geq 1\}$? Thanks ...
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1answer
192 views

weak convergence condition

Let $l^{2}=\left\{x=(x^{(1)},x^{(2)},...):\sum_{i=1}^{\infty }\left\vert x ^{(i)}\right\vert ^{2}<\infty \right\} $. Would you help me to prove that $({\vert|x_n |\vert})$ is bounded sequence and ...
1
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1answer
120 views

Frames for Hilbert space

A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0<A\leq B<\infty$ such that $$ A\|f\|^{2} \leq \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq ...
2
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1answer
81 views

Question about linear operators on Hilbert spaces

I have the following question: "Let $V$ be a Hilbert space and let $T$ be a linear operator on $V$. If $S$ is any linear operator on $V$ that satisfies $\langle Tv,w \rangle = \langle Sv,w \rangle$ ...
2
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0answers
105 views

Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...
0
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1answer
29 views

Checking the completeness of a given space

Let $X$ be the vector space of all real sequences with finite support (i.e., there are only finitely many non-zero elements) with the scalar product $$ \langle x,y\rangle=\sum_{i=1}^{\infty}x_iy_i $$ ...