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

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459 views

From weak convergence to strong convergence

Let $H$ be a real Hilbert space and $F:H\rightarrow H$ be a mapping such that $$ (A)\qquad\qquad(u_n\rightharpoonup u_*, F(u_n)\rightarrow F(u_*))\; \Longrightarrow\;(u_n\rightarrow u_*) $$ We are ...
3
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1answer
181 views

Operator norm and spectrum

Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$? ...
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1answer
82 views

How to show projection of $L^2$ function converges to that $L^2$ function

My teacher said that if $P_n f = \sum_{j=0}^n(f,w_j)w_j$, where $w_j$ is orthonormal basis of $L^2$, then $|P_n f- f|_{L^2} \to 0$ for $f \in L^2$. How do I prove this? I thought $$|P_nf - f| = ...
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611 views

Estimate on the norm of a self-adjoint operator

EDIT: thks to Martin's comment I realize the previous version was wrong. Here is the correct version of what I need to show: I am trying to show that if $A$ is a self - adjoint operator in a Hilbert ...
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2answers
317 views

Find adjoint operator of an operator T

I would like to find the adjoint operator of $$ T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds. $$ Here $H^1([0,1])$ is the Sobolev space $W^{1,2}([0,1])$. I tried to find ...
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3answers
217 views

Functional analysis-Hilbert spaces

Let $ X$ be an inner product space. Show that $ X$ is a Hilbert space if and only if for each continuous linear functional $ L$ on $ X$,there exists $ z\in X$ such that $ L(x)=\langle x,z\rangle $ . ...
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317 views

Compute spectral/projection-valued measures explicitly?

Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following: ...
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2answers
1k views

No Nonzero multiplication operator is compact [duplicate]

Let $f,g \in L^2[0,1]$, multiplication operator $M_g:L^2[0,1] \rightarrow L^2[0,1]$ is defined by $M_g(f(x))=g(x)f(x)$. Would you help me to prove that no nonzero multiplication operator on $L^2[0,1]$ ...
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234 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 ...
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2answers
101 views

Equiangular sequence in $\ell^2$

I have a countable infinite normed, equiangular sequence $a_n \in \ell^2$, i.e $\langle a_n, a_m \rangle=\theta$ for $n\not=m$ and $\langle a_m, a_m \rangle =1$ for some $\theta <1$. It's clear ...
3
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91 views

Function space in QM

I need to understand how one can think of a function as a vector (in Hilbert space, more specifically) so I can follow along QM texts. I've read this question's answers, but they were uninspiring to ...
3
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1answer
127 views

Contractive Operator and Realization Theorem

Good morning, I have searched, by using google for a time, a proof of the following theorem : Let $\pmatrix{A&B \\ C&D}\colon H \oplus K \to H\oplus K$ be a contractive operator of a Hilbert ...
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1answer
770 views

Dual of $C[0,1]$, Hilbert space and Riesz representation.

Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. I need help proving the following claim: ...
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1answer
69 views

Algebraic description of coisometries

Let $H$ be a Hilbert space. An operator $T\in\mathcal{B}(H)$ is called coisometric if it maps open unit ball of $H$ onto open unit ball of $H$. Please tell me how to prove that condition $T$ is ...
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1answer
192 views

A property of Hilbert sphere

Let $X$ be (Edit: a closed convex subset of ) the unit sphere $Y=\{x\in \ell^2: \|x\|=1\}$ in $\ell^2$ with the great circle (geodesic) metric. (Edit: Suppose the diameter of $X$ is less than ...
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1answer
37 views

The weak topology on $H$ is the weak* topology on $H^*$ pulled back via $\Phi$

I'm reading the following in Analysis Now by Pedersen: The map, $H$ a Hilbert space $$\Phi:H\to H^*: x\mapsto(\cdot\mid x)=[y\mapsto (y,x)]$$ is a conjugate linear isometry. Then define the weak ...
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75 views

Hilbert Space multiplication Operator, shift operator

I have this problem and am not sure how to even approach it.. Hilbert space $l^2(\mathbb{Z})$ with orthonormal basis$ $$(e_n)$ and Hamiltonian operator $He_n=i(e_{n+1}-e_{n-1})$ a)I need to use ...
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2answers
74 views

Is $B(H)$ sot separable

To prove that the unit ball of $B(H)$ is separable in strong operatior topology using the fact that $K(H)$ is separable and also is sot- dense in $B(H)$. I think we can conclude that $B(H)$ is also ...
3
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1answer
54 views

How to rigorously understand continuous bases?

In Quantum Mechanics it is quite common to see the idea of a continuous basis of a Hilbert space. In truth if $\mathcal{H}$ is the state space of a quantum system and if $X : U\subset \mathcal{H}\to ...
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130 views

Hierarchy of Mathematical Spaces

I really got lost among all those many different spaces in mathematics, and I got really confused what is special case of what. For example, I knew for long time vector spaces, then Hilbert spaces, ...
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46 views

Spanning set is closed.

Suppose $\{e_1,e_2,\ldots,e_n\}$ is an orthonormal set in $\mathscr{H}$ (Hilbert space) and define $$M \equiv \operatorname{span}\{e_1,e_2,\ldots,e_n\}.$$ Show that $M$ is closed. Can I show that ...
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1answer
47 views

Find an inner product that makes a given set of linearly independent vectors orthogonal

I need to find an inner product such that given a set $S$ of linearly independent vectors in a Hilbert space $H$, $S$ will be orthogonal with these product. I thought Gram -Schmidt Process would help ...
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63 views

Why is the additive category of Hilbert spaces not abelian

As an answer to this post Additive category that is not abelian it was said that the additive category of Hilbert spaces is not abelian. Why is that? Also what category of Hilbert spaces is this? ...
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91 views

Modulus: Invariant Domain

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$M:\mathcal{H}\to\mathcal{K}:\quad \|M\|=1$$ Regard dense domains: ...
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1answer
37 views

$\langle f, \phi_n \rangle = 0 \implies f = 0$ is equivalent to the definition of orthonormal basis

Is there an "easy" way to see that if $\{\phi_n\}_{n=1}^\infty$ is a set of orthonormal functions in a Hilbert space then showing $\langle f, \phi_n \rangle = 0$ for all $n$ implies $f = 0$ is ...
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2answers
43 views

Hilbert space Inequality

Let $K ⊂H$ be a nonempty closed convex set, $H$ Hilbert. Let $f ∈H$ and let $u=P_Kf$. Prove that $$||v − u||^2 ≤ ||v − f ||^2 − ||u − f ||^2, ∀v ∈ K$$ I've tried to use the parallelogram identity and ...
3
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1answer
117 views

$L_2$ norm and RHKS norm in Hilbert spaces $\mathcal{H}$

According to this paper (just right below the Theorem 3 and above the section 3) Reproducing Kernel Hilbert Space(RKHS) $\mathcal{H}$ on $\mathcal{X}$ is a Hilbert space of functions from ...
3
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1answer
153 views

The inner product of the Cartesian Product space

I want to know how can one define the inner product in the Cartesian product of spaces, i.e. let $A,B$ two hilbert spaces. Let $a_1, a_2 \in A$ and $b_1, b_2 \in B$, how can one express the inner ...
3
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141 views

Is the intersection between two $n$-spheres an $(n-1)$-sphere?

It is true that the intersection between two $n$-sphere in $\mathbb{R}^n$ is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and ...
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2answers
153 views

Book for Hilbert spaces.

Which book either on functional analysis or specifically for Hilbert spaces has the best way of explaining with most examples and to the point without much applications. I studied Limaye's book and ...
3
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42 views

Question on Completeness of Derived Inner Product Space

Let $(\mathcal{H},\langle{,}\rangle)$ be a separable, infinite-dimensional Hilbert space. Let $\mathcal{X}''$ denote the space of bounded sequences in $\mathcal{H}$. For a Banach limit $L$, define a ...
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48 views

Matrix space, with $\langle A,B\rangle=\text{tr}(AB^*)$ isn't Hilbert space, how can i find a counter example?

Generally, I'm having quite troubles thinking about counter examples. So I would love if someone could guide me into finding the example for the following question by myself (and not just giving it ...
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47 views

Finding the maximum of an integral of a function with given constraints.

This comes from Rudin's Real Analysis text. The first part of the problem asks us to compute $\displaystyle\min_{a,b,c}\int_{-1}^1|x^3-a-bx-cx^2|dx$ (which I have done). Now it asks us to find ...
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1answer
105 views

Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 ...
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2answers
233 views

Show that the trace class operators on a Hilbert space form an ideal

Let $(H, (\cdot, \cdot))$ be a separable Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $\{\phi_n\}_{n=1}^\infty$ is an orthonormal basis for $H$. Let $\mathcal{B}(H)$ ...
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2answers
134 views

Properties of a set in $\ell^2$ space

Let $\ell^2 = \{x= (x_1,x_2,x_3,\ldots): x_n\in \mathbb C\text{ and } \sum_{n=1}^\infty |x_n|^2 < \infty\}$ and $e_n \in \ell^2 $ be the sequence whose $n$-th element is 1 and all other elements ...
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1answer
85 views

Linear and monotone mapping

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and monotone, i.e., $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \geq 0$$ for all $x,y \in \mathbb{R}^n$. Say for which matrices $A ...
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66 views

Is it Hermitian or not?

I have a question, since I'm realy consfused. I am doing quantum information theorem, and there's a theorem that says, that for Hermitian $A\in L(H)$ ($H$ some finite dimensional Hilbertspace) then ...
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96 views

Determining if the span of a set is dense in L^2(0,1)

I am trying to determine whether or not the following statement is true: If $f \in L^2(0,1)$ and $\int_0^1 x^nf(x) = 0$ for all positive integers $n$. Then $f(x) = 0$ I have already verified this ...
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1answer
733 views

Functional weakly lower-semicontinuous [duplicate]

If $X$ is a topological space, then a functional $\varphi:X\to\mathbb{R}$ is lower-semicontinuous (l.s.c) if $\varphi^{-1}(a,\infty)$ is open in $X$ for any $a\in\mathbb{R}$. If $X$ is a Hilbert ...
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141 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 ...
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1answer
135 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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102 views

Properties of orthogonal projections

I have a question about orthogonal projections on a Hilbert space $H$. Can we say that the range of a projection $P\in B(H)$ is closed? Thus we have to ask if the range is a Hilbert space. Moreover: ...
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1answer
74 views

Orthonormal bases for Hilbert spaces

In Reed and Simon (Functional Analysis) Theorem II.6 states that, given an orthonormal basis $\{ x_\alpha \}_{\alpha \in A}$ (not necessarily countable)for a Hilbert space $H$, every $y \in H$ can be ...
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1answer
70 views

Hilbert space - probability measure: st. norm. variables

I am considering the following homework. Let $\Omega=\ell_2$ be the Hilbert space of square summable sequences, $\mathcal A$ the Borel $\sigma$-algebra and $\{e_n: n\mbox{ natural}\}$ the natural ...
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1answer
320 views

Equivalent norm in sobolev space H^2

I consider space $H^{2}(0,a)=\{ f\in L^{2}(0,a): f',f''\in L^{2}(0,a) \}$ I define norm $\Vert w \Vert_{H^{2}}:=b\Vert w''\Vert_{L^{2}}$, where b is positive constant. I couldn't proof that it is ...
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1answer
84 views

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
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1answer
38 views

Extension of family of operators

Let $A(z)$ where $z\in \mathbb{R}$ be a family of (bounded) operators on some Hilbert space. Assume we know these operators have a meromorphic extension to all of $\mathbb{C}$. Assume moreover that we ...
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1answer
103 views

A certain Hilbert space projection operator; verification needed

Let $V \subset H$ be separable Hilbert spaces with dense and continuous embedding. For each $n$, let $V_n$ and $H_n$ be finite-dimensional subspaces of $V$ and $H$ respectively with dimension $n$. ...
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1answer
54 views

An element of $L^2(0,T;V_n)$.

Let $V$ be Hilbert with basis $w_j.$ Let $V_n = \text{span}(w_1, ..., w_n)$. Is it true that every element $v \in L^2(0,T;V_n)$ can be written as $$v(t) = \sum_{j=1}^n a(t)w_j?$$ I think so. But my ...