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

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95 views

Prove that $A\geq I$ implies that $A$ is invertible.

Here's the question: Let $A$ be a positive operator on a (possibly infinite dimensional) Hilbert space. Let $I$ denote the identity operator. Suppose that $A \geq I$, which is to say that $A - I$...
3
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1answer
160 views

Compact diagonal operator

Suppose $A : H \to H$, where $H$ is a Hilbert space, is bounded. Also, $A$ is a diagonal operator with diagonal $\{a_n\}$. Show: If $A$ is compact, then $a_n \to 0$ as $n \to \infty$. Should I prove ...
3
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2answers
66 views

Second differential of the norm in an infinite dimensional Hilbert space

Let $f: E \to \mathbb{R}$ sending $x \mapsto \|x\|$ and make some simple hypothesis $E$ is a Hilbert Space Let's say that the norm $\|\cdot\|$ is derived from a scalar product [solved] So we can ...
3
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2answers
45 views

If $H$ is a Hilbert space, does the coordinate projection $\pi :H\oplus H\rightarrow H$ take closed subspaces to closed subspaces?

Here $H\oplus H$ has the product topology, which is induced from the "$\ell^2$-norm" $\|(x,y)\|:=\sqrt{\|x\|^2+\|y\|^2}$. This is indeed a Hilbert space via the inner product $\langle (x,y), (x',y')\...
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2answers
191 views

An orthonormal subset of a Hilbert space is closed.

In Rudin Real and Complex Analysis there is an exercise (6, Ch. 4) that asks to show that a countably infinite orthonormal set $\{u_n:n\in\mathbb{N}\}$ in a Hilbert space $H$ is closed and bounded but ...
3
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1answer
124 views

Closed Subspaces of Hilbert Spaces

I read the following statements. But I do not know how to show it or any example to support it. Could anyone provide some explanation and examples, please? Thank you! The subspace $C^\infty$ ...
3
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1answer
115 views

Definition of unitary operators

Let $\phi, \psi \in \mathcal{H}$ be some element from a hilbert space $\mathcal{H}$ and $U$ a linear operator $U: \mathcal{H} \rightarrow \mathcal{H}$. Does $$ \forall \phi: \| U \phi \|^2 = \| \phi \...
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1answer
57 views

Are translates of Gaussians an overcomplete set in $L^2(\Bbb R)$?

Consider the Gaussian $\exp(-t^2/2)$. Is it the case that any function in $L^2(\Bbb R)$ can be written as a limit of a sum of scalings and translations of Gaussians? That is, for any $f\in L^2(\Bbb R)$...
3
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2answers
76 views

Find norm of operator

I have a linear functional $$A: L_2[0,2] \to \mathbb R, Ax = \int_0^2(t^2+2)x(t)dt$$ I need to find $C$, trying to measure $C$ and $||Ax||$ to find it, but how can I do it in this problem?
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1answer
132 views

Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
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2answers
179 views

A dense subset of a Hilbert space

I am curious about the following problem: Consider the Hilbert space (a weighted $L^2(\mathbb{R})$ space): $$\mathscr{H}=\bigg\{f: \mathbb{R}\to\mathbb{R}\text{ Lebesgue measurable}\,\bigg|\,\int_\...
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2answers
155 views

exercise on the closed subspaces of an Hilbert spaces

I have a question regarding exercise 3.1.13 of "Analysis Now" by Pedersen volume 118 of the Springer GTM. The exercise aim to show that any closed subspace $X$ of $L^2([0,1])\cap L^{\infty}([0,1])]$ ...
3
votes
2answers
328 views

Is it a unitary, self adjoint and normal operator?

Let $A\colon H\to H$ be a bounded linear operator on a complex Hilbert space such that $\|Ax\|=\|A^*x\|\forall x$, given that there is a nonzero $x$ for which $A^*x=(2+3i)x$. Then I need to know ...
3
votes
1answer
675 views

The unit ball in a Hilbert space

I have a request for any ideas to prove: If $H$ is a Hilbert space, then any unit vector is an extreme point of the unit ball of $H$. Every isometry is an extreme point of the unit ball of the ...
3
votes
1answer
104 views

Is $L^2(0,T;V_f) \subset L^2(0,T;V)$ closed if $V_f \subset V$?

Let $V$ be an infinite-dimensional separable Hilbert space and let $V_f$ be a subspace of $V$ that is finite dimensional. It follows that $V_f$ is closed. Is it true that $L^2(0,T;V_f)$ is closed as ...
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2answers
65 views

Question about finding minimum-Hilbert spaces

How to find $$\min_{a,b,c\in\mathbb{C}}{\int_0^{\infty}} |a+bx+cx^2+x^3|^2 e^{-x} dx = ?$$ Thanks in advance.
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2answers
326 views

What is my operator norm (cannot get good enough bounds).

Given a space of square integrable functions $x(t)$ over the interval $[0;1]$ one can introduce a norm $$\|x(t)\|= \sqrt{\int_0^1 (x(t))^2 \, dt};$$ Then what is a norm of the transformation below (...
3
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1answer
225 views

Hahn-Banach theorem (second geometric form) exercise

Let $X$ be a vector normed space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that $$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F).$$ Apply the Hahn-Banach theorem (second ...
3
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1answer
239 views

The span of the orthorgonal projections is norm dense in $B(H)$

This is a question in my functional analysis book. How to use the spectral theorem to prove that the span of the orthogonal projections is norm dense in $B(H)$?
3
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2answers
131 views

Show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ is not closed in $\ell^2$

How to show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ where $e_n=(\delta_{nk})_{k\in\mathbb N}$ is not closed in $\ell^2$?
3
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1answer
328 views

Bounded operators on separable Hilbert spaces

Let $H$ be a separable Hilbert space. Show that every bounded operator from $H$ to itself can be approximated in the strong operator topology by a sequence of finite rank operators. Im not sure what ...
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6answers
209 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., $P^2=P$?
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2answers
391 views

norm of a normal operator using projections

Let $H$ be a Hilbert space and $T$ a normal operator on $H$. In the sequel, ${\rm tr}$ denotes the trace for trace class operators. Do we have $$ \vert\vert T \vert\vert= \sup |{\rm tr} (TP)| $$ ...
3
votes
2answers
188 views

Minimization problem in Sobolev spaces

This is a homework problem and I don't know how to solve it: Consider the following minimization problem on the real-valued sobolev space $H^{1,2}(\Omega)$ with dimension $n=1$ and $\Omega=(0,1)$: $$...
3
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1answer
175 views

Convexity of a set in Hilbert space

Let $H$ be a Hilbert space and $\left\{ e_{i}\right\} _{i=1}^{\infty}$ an orthonormal system. I need to prove that the following set is a convex set: $$C=\left\{ x\in H\,:\,\sum_{n=1}^{\infty}\left(1+...
3
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4answers
131 views

reference for strongly continuous semi-groups

At the moment I am trying to understand the proof of the Fredholm property in Salamon's notes on Floer homology. There I came across the notion of an unbounded operator on a (real) Hilbert space which ...
3
votes
1answer
34 views

Do the holomorphic or meromorphic functions on a domain $D \subseteq \mathbb{C}$ form a Hilbert space $\mathcal{H}$?

In a physics paper I am reading that the meromorphic functions on $\mathbb{C}$ with $f(x) = f(\overline{x})$ form a Hilbert space. $$ \mathcal{H} = \{ f(x) : f(x) = f(\overline{x}) \}$$ Even let $f(...
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2answers
40 views

Does $A+A^\perp=\scr H$ in a Hilbert space imply $A$ is closed?

This is just the converse to the Hilbert projection theorem, which says that if $A$ is a closed subspace of a Hilbert space $\scr H$ then $A+A^\perp=\scr H$. If $A$ is a linear subspace of $\scr H$ ...
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1answer
68 views

Proving Toeplitz matrix defines bounded operator on $ l^2 $

I should first mention this: I have asked this question previously but I only got a partial answer that does not suit the actual assumptions but only the related ones, it reads as follows: Define ...
3
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2answers
72 views

Show $ \langle Tx,x \rangle \in \mathbb R$ for all $x \in H$ implies $T$ is self-adjoint

Show that a linear operator $T: H \rightarrow H$ is self adjoint if and only if $\langle Tx, x \rangle \in \mathbb R$ for all $x \in H$. You may use that the equality that for all $x,y \in H$ $4\...
3
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1answer
38 views

Identity Operator can be uniformly approximated by orthonormal basis

Let $H$ be a separable Hilbert space with orthonormal basis $e_1, e_2, ...$. I know that for any $x \in H$, we have $$\|x\|^2 = \sum\limits_n \|\langle x, e_n \rangle\|^2$$ and in fact $x = \lim\...
3
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1answer
35 views

Projection on closed subspace of $L^1$, $L^{\infty}$

For $p=1,\infty$ let $K$ be a closed subspace of $L^p(\mathbb{R},m)$. According to this question, it should be easy to find examples of $K$ and $f\in L^p(\mathbb{R},m)$ such that there exists a non-...
3
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2answers
32 views

If the scalar product are equal then the operators are equal.

I want to show the following: Let H be a $\mathbb C$ -hilbert space and $S,T\in L(X)$ If $\langle Sx,x \rangle = \langle Tx,x \rangle$ for all $x\in H$, then $S=T$ Any hints for me?
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1answer
93 views

Orthogonality of projections on a Hilbert space

Assume that $p$ and $q$ are (orthogonal) projections on Hilbert space $\mathcal{H}$. I want to prove: $pq=0$ iff $p+q\leq1$ I had the following in mind: Assume $pq=0$. Then $qp=0$, hence $p+q$ is a ...
3
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2answers
284 views

How can I show $U^{\bot \bot}\subseteq \overline{U}$?

Let $H$ be a Hilbert space and $U$ a subspace. Let $U^{\bot}$ denote its orthogonal complement. I had no trouble showing $\overline{U}\subseteq U^{\bot\bot}$. But now I'm stuck for $\supseteq$. ...
3
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1answer
163 views

Questions about the Fourier transform as a unitary transform

As far as I know, the Fourier transform is a (linear) unitary transform: $T: \textbf{L}^2(-\infty, +\infty) \rightarrow \textbf{L}^2(-\infty, +\infty)$ where the basis functions {$e^{i \omega x} | \...
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1answer
271 views

How to prove the sum of RKHS (Reproducing Kernel Hilbert Space)?

$k,k_1$ and $k_2$ are kernels on $\mathcal{X}\times\mathcal{X}$, and $k=k_1+k_2$, then we have the following properties for the RKHS (Reproducing Kernel Hilbert Space) $\mathcal{H}$, $\mathcal{H}_1$ ...
3
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3answers
85 views

On the intuition behind the projection theorem.

I have recently proved the projection theorem in a Hilbert space setting. The statements were: If $M$ is a closed subspace of a Hilbert space $H$ and $x \in H$, then: There is a unique element $\...
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3answers
192 views

Selfadjointness of the Dirac operator on the infinite-dimensional Hilbert space

I am a physicist, so my background in functional analysis is limited only to basics. However, I would like to prove that the free Dirac operator is selfadjoint (or Hermitian, or neither). The free ...
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1answer
110 views

What is the image of operator exponential?

Given a Banach space $V$ and a bounded linear operator $A:V\to V$, the operator $e^A$ is bounded and invertible. When $V$ is finite dimensional, every invertible operator is of the form $e^B$ (one can ...
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2answers
179 views

Can Hilbert spaces generalize non-Euclidean geometry by having the sum of the angles of a triangle not be equal to pi?

I am an amateur mathematician learning new things. Let A and B be vectors in a Hilbert space. The three vectors A, B and A-B form a triangle. The idea of the angle between two vectors can be ...
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1answer
141 views

No trace on $B(H)$ if $H$ is infinite dimensional

Let $H$ be an infinite dimensional Hilbert space and $B(H)$ the bounded linear operators on $H$. Then thre is no ultra weakly continous non-zero positve trace $tr:B(H)\rightarrow \mathbb{C}$. I ...
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1answer
151 views

Strengthened Cauchy-Schwarz inequality

I'm looking for some simple proof of the following consequence of the "strengthened" Cauchy-Schwarz inequality: Let $\mathcal{H}$ be a real Hilbert space such that $\mathcal{H}=\mathcal{V}\oplus\...
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1answer
90 views

Prove that $A^2$ is an Hilbert Space.

We denote by $A^2$ the space of analytic functions on $B_1=\{z=x+iy\in \mathbb{C}, x,y\in \mathbb{R}||z|<1\}$, such that $$\left(\int\int_{B_1}|f(z)|^2 dx \, dy\right)^{1/2}<+\infty$$ In $A^2$, ...
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2answers
490 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 u,A^...
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1answer
926 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: ...
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1answer
1k 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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1answer
40 views

name of matrix of inner products $\langle f_i, f_j\rangle$

Given a Hilbert space $H$ and a number of elements $\phi_i\in H$, does the matrix $M$ with $$ M_{i,j} := \langle\phi_i, \phi_j\rangle $$ have any particular name?
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3answers
90 views

What am I doing wrong? inner product

The general form of an inner product in $\mathbb{C}^n$ is $\langle x,y\rangle=y^{*}Bx$ where B is a Hermitian positive definite matrix. Then for any square matrix $A$ we have $\langle Av,w\rangle=w^{*}...
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2answers
226 views

Equivalent norms imply equivalent inner products?

Let $H$ be Hilbert and let it have two innner products $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$. If the norms $|\cdot|_1$ and $|\cdot|_2$ are equivalent, does this ever imply: there exist constants $...