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

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2
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1answer
215 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 ...
1
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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
910 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 $p\neq 2$. ...
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
136 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
380 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
477 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 ...
1
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1answer
186 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
118 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
80 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 $\langle Tv,w \rangle = \langle Sv,w \rangle$ ...
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0answers
101 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
28 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 $$ \langle x,y\rangle=\sum_{i=1}^{\infty}x_iy_i $$ ...
5
votes
1answer
204 views

Conditions for the sequence being weakly convergent

Let $H=\ell_2$ be the Hilbert space of the square-summable sequences where $$ \langle x,y\rangle=\sum_{i=1}^{\infty}x_iy_i, \quad \|x\|=\sqrt{\langle x,x\rangle}. $$ Let $F: H\rightarrow H$ be an ...
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2answers
72 views

How do functions form a Hilbert space

I'm trying to wrap my head around function spaces. I get that you can define the inner product as the integral the multiplication of two functions over the entire domain because it satisfies the ...
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2answers
50 views

Why is the sequence $(\langle x_n,a \rangle)$ Cauchy when $(x_n)$ is?

Let $\mathcal H$ a Hilbert space over $\mathbb R$ and $A = \{x\in \mathcal H : \langle x, a \rangle \geq 1 \}$. I'm trying to prove that $A$ is closed. Let $(x_n) \subset A$ be a Cauchy-sequence. ...
0
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1answer
104 views

About the positivity of the inner product on $L^2[0,1]$

My textbook on Hilbert space theory claims that the map $$\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}~\mathrm{d}x$$ is an inner product on $C[0,1]$. But I am not sure whether ...
1
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1answer
247 views

Unique extension to a bounded operator

Suppose $\left\{ e_{1},e_{2},\ldots\right\} $ is an orthonormal basis for a Hilbert space $\mathcal{H}$ and for each $n$ there is a vector $Ae_{n}$ in $\mathcal{H}$ such that $\sum\left\Vert ...
4
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1answer
238 views

On the regularity of the Laplace equations and tensor products and such

To start with, let me apologize for my ignorance as I know next to nothing about partial differential equations. My question is about the tensor product of Banach spaces but actually I do not ...
4
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1answer
126 views

Showing a representation is irreducible by showing that a degenerate subspace has codimension one.

Throughout $\phi$ be a continuous character from a locally compact abelian group $G$ to the circle. I'm trying to understand this implication. Basically we want to show that a certain representation ...
2
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2answers
131 views

Theorem about orthogonal system in inner product space.

It is known that "If $\{x_n\}$ is a sequence in a real Hilbert space $H$ satisfying $$ \langle x_n, x_m\rangle =0 \quad\forall n\ne m, $$ then $\displaystyle\sum_{n=1}^{\infty}x_n$ is convergent if ...
2
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1answer
133 views

Finding a linear mapping in a special Hilbert space

Let $H=\ell_2$, the real Hilbert space whose elements are the square-summable sequences of real scalars, i.e., $$ H=\left\{u=(u_1,u_2,\ldots,u_i,\ldots): ...
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1answer
214 views

Countable Hilbert Spaces

I have seen a simple proof that no banach space over $\mathbb{R}$ can be of countably infinite dimension. However since the space of all square integrable functions on the unit interval forms a ...
3
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1answer
178 views

What is meant by `element $x\in H$ of minimal norm'

I do not seek a proof of the following exercise. I just want to understand this question in order to solve it myself. Let $H$ be a Hilbert space over $\mathbb R$ and let $a, b\in H$ be such that ...
2
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0answers
156 views

Convergence of orthogonal projections

Suppose that $a_m$, $m \in \mathbb{N}$, is a sequence of bounded linear operators on a Hilbert space converging strongly to an bounded linear operator $a$. If U is a finite-dimensional subspace of H, ...
3
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6answers
190 views

Is $\operatorname{range} =\ker^\perp$ only true for projection?

Let $P$ be a linear operator on a Hilbert space $H$. If $\operatorname{range} P=(\ker P)^\perp$, is $P$ necessarily a projection, i.e., $P^2=P$?
3
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1answer
634 views

The relation between bounded invertible and surjective operators

Please, answer me that how is the set of all bounded invertible operators (for example on a Hilbert space) clopen (closed and open) in the set of all bounded surjective operators? In fact, which ...
2
votes
2answers
138 views

$f'$ is in $L^2[0,1]$

Let $f$ is absolutely continuous function on $[0,1]$, $f(0)=0$ and $f' \in L^2[0,1]$. Would you help me to prove that there is constant $c$ such that $$|f(t)| \leq c \left( \int_0^1 |f'(t)|^2 dt ...
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1answer
69 views

Convergent series with coefficient in $\ell^2$.

Let $z$ denote a complex number and $\{\alpha_n\}$ be a sequence in $\ell^2$. Would you help me to prove that series $\sum_{n=0}^{\infty} \alpha_n z^n$ has radius of convergence greater than or equal ...
5
votes
1answer
496 views

Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
6
votes
3answers
385 views

A complete orthonormal system contained in a dense sub-space.

Let H be a separable complex Hilbert space. Let A be a dense sub-space of H. Is it possible to find a complete orthonormal system for H that is contained in A?
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1answer
122 views

Contractibility of the sphere and Stiefel manifolds of a separable Hilbert space

Why are the sphere $$S=\lbrace |x|=1\rbrace$$ and the Stieffel manifolds of orthonormal $n$-frames $$V_n=\lbrace (x_1,\dots,x_n)\in S^n\mid i\neq j\Rightarrow\langle x_i|x_j\rangle=0\rbrace$$ of a ...
8
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2answers
3k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
4
votes
2answers
175 views

Orthogonality checking in Kreyszig exercise

Let $H$ be inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $\lVert \cdot\rVert$. Let $x,y \in H$. Would you help me to prove that $\langle x,y\rangle=0$ if and only if ...
0
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1answer
389 views

is a direct sum of Hilbert spaces a Hilbert space.?

I have read in proofwiki that a direct sum of Hilbert spaces is a Hilbert space. However, Wikipedia Page about direct sum says it is not necessarily true, that is, the direct sum of Hilbert spaces is ...
0
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0answers
117 views

Set of Bounded linear Operators on $l_2$ is dense on the set of bounded operators on $l_2$?

Let $l_2^{+}$ be the Hilbert space of all square summable sequences $\{x_n\}, n \in \mathbb{N}$ under some definition of inner-product $\langle,\rangle_l$. Define $B[l_2^{+}]$ as the set of all ...
1
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1answer
230 views

Proving Fréchet differentiability

Am learning about Fréchet differentials and was wondering if for a real matrix $X$ and positive semidefinite real matrices $A,B$ the function $f(X)=TrX^TAX-X^TBX$ is twice Fréchet differentiable or ...
0
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1answer
198 views

Solution for a Frobenius norm inequality

Am trying to find a real scalar $\gamma$ such that for a given pair of real rectangular matrices $X,Y$ the following holds: $\frac{||Y||_{F}^{2}}{5} \leq ||\gamma X||_{F}^{2}\leq ||Y||_{F}^{2}$ ...
0
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1answer
87 views

Scalar multiplication and Frobenius norm

Was wondering on what would be the real number (scalar) $\gamma$ that needs to be multiplied with each entry in a real rectangular matrix $X_{m\times n}$ such that the Frobenius norm of $X$ equals a ...
4
votes
1answer
83 views

Invertible operator not preserving Hilbert dimension

It is known that for a bijective linear operator $T:X\to Y$ the algebraic dimensions of the linear spaces $X$ and $Y$ coincide. I am asking for an example of an invertible (bounded) linear operator ...
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0answers
579 views

Proof of Isometry: Inner Product Preserving Map

For known points $x_i,x_j,\ldots,x_k$, in $\mathbb{R}^n$, consider a mapping $y_i,y_j,\ldots,y_k$ in $\mathbb{R}^n$ produced by minimizing the function $f(y)=\sum_{i,j} \left \langle x_i,x_j \right ...
2
votes
2answers
78 views

Fourier Coefficients in arbitrary Hilbert Spaces

Say we have an orthonormal basis $\{e_n\}$ for a infinite Hilbert Space $H$. I want to prove that any vector $x=\sum_{n=1}^\infty\langle x, e_n\rangle e_n$. I don't know where to start. Could I ...
2
votes
1answer
92 views

Can this Lemma be extended a little?

Consider this lemma (my question are below): Lemma Given three pairwise orthogonal subspaces $X$, $Y$, $Z$ of a Hilbert space $H$ that span the whole space, any vector $\nu\in H,\ ||\nu||=1$, can ...
4
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1answer
824 views

Formula for trace of compact operators on $L^2(\mathbb{R})$ given by integral kernels?

Given an appropriate function $K: \mathbb{R}^2 \to \mathbb{C}$, say continuous of compact support, we obtain a compact operator $T$ on the Hilbert space $L^2(\mathbb{R})$ by the formula $$ (T h)(t) = ...
2
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0answers
154 views

A doubt about tensor product on Hilbert Spaces

An operator is a bounded (i.e., continuous) linear transformation between Hilbert spaces. Let $\mathcal{B}[\mathcal{H}]$ be the set of all operators in the Hilbert space $\mathcal{H}$. Let ...
2
votes
1answer
248 views

orthogonal subspaces in a Hilbert space

Is it true that if $A,B$ are closed subsets of a Hilbert space $H$, such that $A\perp B$, we have $A+B+(A\cup B)^{\perp} =H$ ? What if $A,B$ are closed subspaces ?$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...
0
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1answer
72 views

Rearragement of a series in Hilbert space

Let $H$ be a Hilbert space and $\sum_k x_k$ a convergent infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation ...
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1answer
47 views

Can this type of series retain the same value?

Let $H$ be a Hilbert space and $\sum_k x_k$ a countable infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation ...
5
votes
2answers
1k views

Relationship of Fourier series and Hilbert spaces?

I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...