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

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1answer
62 views

Please check proof of convergence in $L^2(0,T;L^2)$ of a composition (uses Nemytskii operator)

Suppose we have a continuous function $g:\mathbb{R} \to \mathbb{R}$ that satisfies $|g(x)| \leq C|x|$. Let $u_{n} \to u$ in $L^2(0,T;L^2)$. I want to show that $g(u_{n'}) \to g(u)$ in $L^2(0,T;L^2)$. ...
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1answer
101 views

Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1 $ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
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1answer
29 views

subsets of closed ball of Hilbert space [closed]

Let the sequences $(x_n)$ and $(y_n)$ be subsets of unit closed ball $\overline{B}(0,1)$ of Hilbert space H. If $\lim_{n\to \infty}\langle x_n,y_n \rangle=1$, then $$\lim_{n\to ...
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1answer
19 views

I need help showing something is a linear continuous operator.

Define $T:C([0,1])\rightarrow C([0,1])$ by $T(f)(x)=f(0)+\int_0^xtf(t)dt$ I want to show that $T$ is a continuous linear operator.Showing the linear part is easy enough, but I am not quite sure how to ...
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1answer
64 views

Concluding that a linear operator on a Hilbert space is invertible

Setting: Let $H$ be a Hilbert space with two inner products, $\langle \cdot,\cdot\rangle$ and $[\cdot, \cdot]$, and $S:H\to H$ be a bounded linear operator such that for all $x,y\in H$, we have ...
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1answer
30 views

About what happens to eigenspace under functional calculus for Unbounded Operator

Let $T$ be an unbounded self adjoint positive operator on a Hilbert Space $\mathcal{H}$. Let $x \in \mathcal{H}$ be a vector such that $Tx = x$. Is it true that $T^{\frac{1}{2}} x = x$. For what $f$ ...
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1answer
70 views

Extending a compact operator to the entire Hilbert space

In a course I'm taking we defined compact operators as a linear mapping $H\rightarrow H$, where $H$ is a Hilbert space, that maps bounded sets to relative compact ones. The lecturer mentioned that the ...
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2answers
40 views

Proof: adjoint map of projection is a projection and …

Let $V$ be a pre hilbert space and $\pi \in \mathrm{End}(V)$. Show: the adjoint map $\pi^+$ of a projection (meaning: $\pi^2 = \pi$) is a projection itself. Show then: a projection $\pi$ is ...
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1answer
46 views

Isomorphism between Hilbert spaces

I want to show that the function $$ L^2(\Omega,\mathcal{O})\longrightarrow L^2(\widetilde{\Omega},\mathcal{O}) \colon f \longmapsto f|_{\widetilde{\Omega}}$$ is a isomorphism, where ...
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1answer
46 views

Determining the exact form of a projection in a Hilbert space

Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$ where $\mathcal{L}^2[0,T]$ is the set of Lebesgue ...
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2answers
99 views

Find the minimum distance that equal maximum inner product

If $x_0 \in$ $H$ (Hilbert Space) and $M$ is a closed linear subspace of $H$, prove that $$\min \{\|x - x_0\|: x \in M\} = \max \{\langle x_0, y\rangle : y \in M^\perp, \|y\| = 1\}.$$ I suppose $P$ ...
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1answer
84 views

Prove that this space is not Banach

Let $\Omega\subset\mathbb{R}^n$ be an open, bounded set with boundary $\partial\Omega$ of class $C^1$. $$\mathcal{A}:=\{u\in C^2(\bar\Omega):u=0\text{ on }\partial\Omega \}$$ endowed with the scalar ...
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1answer
60 views

Normal Compact Operator: not diagonalizable!

To proposition 5.17 in Weidmann's 'Lineare Operatoren in Hilberträumen' (german version) it is noted that the expansion of compact operators that are normal rather than self adjoint doesn't apply in ...
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1answer
43 views

Verification of conclusions regarding duality maps

I have two conclusions drawn from two results. I want to know how valid these two conclusions are. Firstly Consider the duality mapping(set-valued) $J:X \rightrightarrows X^{*}$ defined: $J(u) := \{ ...
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2answers
128 views

Symmetric Operator vs. Real Spectrum

For symmetric operators one has a characterization: $$A\text{ symmetric}:\quad A=A^*\iff\sigma(A)\subseteq\mathbb{R}$$ (I want to investigate to what extend symmetry is a necessary assumption.) ...
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1answer
47 views

When $M$ is closed $M^\perp$ is one-dimensional vector space

If $M=\{ x: Lx=0\}$, where $L$ is continuous linear functional on $H$ (Hilbert Space). Prove that $M^\perp$ is vector space of one-dimensional unless $M= H$. I know $M$ is closed so that $M^\perp$ is ...
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1answer
92 views

Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
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1answer
49 views

Maximal set at Hilbert space

I need to prove the following: If B orthonormal set in a Hilbert space X, then B is maximal if and only if $ B^{\perp} =\{0\} $ I tried the following : $ B^{\perp} =\{0\} $ => B maximal $ ...
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1answer
39 views

$T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$

The question goes as follows: $T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$. Given is the data: $X$ is a Hilbert space with an orthonormal ...
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73 views

A question on functional analysis

Let $H_i$, where $i = 1,2$ be Hilbert spaces and $T_i : H_i \rightarrow H_i$ be closed operators, such that $T_i$ have positive spectrum. Let $\phi : H_1 \rightarrow H_2$ is an isometric isomorphism ...
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1answer
39 views

$Te_n$ converging to zero

I have the following question in my functional analysis book I dont understand: $X$ is an infinite dimensional Hilbert space with an orthonormal basis $(e_n)$. Show that if $T \in K(X)$, then $Te_n ...
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36 views

Spectral theory - how to prove this lemma?

in Anver Friedman, Foundations of Modern Analysis I found a lemma (6.7.3): If A is a self-adjoint operator and $\{E_\lambda\}$ is a spectral family such that $A=\int_m^{M+\varepsilon} \lambda ...
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4answers
397 views

Square root of a Hermitian operator exists

There are a lot of questions here about square root operators, but none of them addresses the basic question of existence, and I didn't find a very beefy section in Wikipedia talking about this, so ...
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1answer
42 views

Proving $||f+g||\cdot||f-g|| \le ||f||^2+||g||^2$ in a Hilbert Space

Let $f$ and $g$ be vectors in a Hilbert space $H$. Show that $$||f+g||\cdot||f-g|| \le ||f||^2+||g||^2$$ My question is, do i have to rewrite $||f+g||$ as $\sqrt{\langle f+g,f+g\rangle}$ and same ...
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1answer
155 views

What is the dual space in the strong operator topology?

Let $X$ be a Banach space, the strong operator topology on the space of bounded linear operators $\mathcal{B}(X)$ is defined by the family of continuous semi-norms $A\to\|Ax\|$, $x\in X$. What is the ...
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2answers
161 views

Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...
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1answer
26 views

$\displaystyle ||T||= \sup_{||f||=1} |\langle Tf,f \rangle|$ for complex hilbert spaces

Let $H$ be a hilbert space. Let $T:H\to H$ be a linear bounded operator prove that $\displaystyle ||T||= \sup_{||f||=1} |\langle Tf,f \rangle|$ Obviously $\sup_{||f||=1} |\langle Tf,f| \le ||T||$ ...
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0answers
49 views

Let $T$ be a bounded operator such that $<Tf,f>=0$ then $T=0?$

Let $H$ be a hilbert space. Let $T:H\to H$ be a linear bounded operator such that $<Tf,f>=0$ for all $f\in H$. It is necesarily true that $Tf=0 ?$ When I mean Hilbert space over a field ...
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1answer
45 views

Inner product on Hilbert Spaces

It's an open question. How could you define an inner product for a product of noncontable Hilbert spaces?
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112 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
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1answer
81 views

Creation and Annihilation Operators: Norm Estimate

Given the Fock space: $$\mathcal{F}(\mathcal{h}):=\bigoplus_0^\infty\mathcal{h}^{n}\text{ with } \mathcal{h}^{n}:=\bigotimes_1^n \mathcal{h},\mathcal{h}^0:=\mathbb{C}$$ Define the creation and ...
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1answer
65 views

Positive Operator: Norm Estimate

In class we encountered the statement: $$H\geq C1\quad(C>0)\implies\|\mathrm{e}^{-\beta H}\|<1\quad(\beta>0)$$ How does one prove this? Moreover, what about the weakened version: $$H\geq ...
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3answers
207 views

Is this space a Hilbert Space?

I have a space of continuously differentiable functions on [a, b] with the dot product defined in this way: $ x \cdot y = \int_a^b \! [x(t)y(t) + x'(t)y'(t)] \, \mathrm{d}t. $ Is this space a Hilbert ...
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1answer
64 views

Positive operators in Hilbert spaces

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property: $$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever ...
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1answer
42 views

Completeness: $\mathcal{l}^2(S)$

Surely, for countable index sets this is just the diagonal trick: $\#S<\infty$ However for arbitrary index sets how do I prove that the limit will actually have only countable non vanishing ...
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1answer
33 views

Proof that solution of $\lambda$-affine, linear ODE is entire in $\lambda$

Suppose $F(\lambda)~(\lambda\in\mathbb{C})$ is a linear ordinary differential operator (with, say, domain $D$ dense in some Hilbert space), and is also affine-linear in $\lambda$. Is there a proof ...
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56 views

Formula for trace of particular operators

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$. View the Fourier transform as a unitary operator $\mathcal{F} \in B(\mathcal{H})$. For each function $f \in C_0(\mathbb{R})$, let $T(f) \in ...
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1answer
47 views

Nonseperable Hilbert space: Explicit ONB?

Every Hilbert space admits an ONB by axiom of choice. For separable Hilbert spaces this can in fact be constructed by Gram-Schmidt. For nonseparable Hilbert spaces there can be no general construction ...
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1answer
81 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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1answer
41 views

help,example about disjoint operators

$T\colon L^2[0,1]→L^2[0,1]$ is given by $$ Tx(t)=∫_0^1 tx(s)\,ds $$ How can we find adjoint operator of $T$ in this space? $\langle Tx,y\rangle= \langle x,T^*y\rangle$ should be okay.But what ...
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0answers
30 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
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0answers
38 views

The relationship between CPTP maps and quadratic forms

Let $H$ be a finite-dimensional Hilbert space (so there is a canonical isomorphism $H\cong H^*$). For a Hilbert space $H$ define $B(H)$ to be the space of linear operators on $H$; we have $B(H)\cong ...
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1answer
22 views

Normal bounded operator

Let $T$ be a bounded normal operator on a Hilbert space. Now I have to show that $T$ is self-adjoint if and only if $\sigma(T) \subset \mathbb{R}$. I already know that for an Abelian unital ...
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1answer
89 views

Dual of $div$ on spaces where the tangential value is fixed

Say $\Omega$ is a domain in $\mathbb R^3$ with a smooth boundary $\Gamma$. Consider the spaces $$ H_{n,0}=\{v\in H^1(\Omega):n\cdot v \bigr |_{\Gamma} = 0\} $$ and $$ H_{t,0}=\{v\in ...
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1answer
83 views

Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
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1answer
89 views

Is the image of a closed subspace under a bounded linear operator closed?

This seems obvious, but I can't get the proof straight, and I made up the statement myself, so I'm not sure if it's true in the stated generality. Given a bounded linear operator $T$ in Hilbert space, ...
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53 views

Hilbert space without the projection theorem

One succinct statement of the projection theorem in Hilbert space is $A+A^\bot=\scr H$, where $A\in\scr C$, the set of closed subspaces of $\scr H$. (We will also denote the set of all subspaces by ...
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0answers
56 views

Prove that if $X$ is a Hilbert space, then $B(X)$ is not a Hilbert space

I`m having a homework question that goes like this: X is a Hilbert space, a complete inner product space, show that B(X) is not a Hilbert space. I`m quite stuck and I would love to understand this ...
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1answer
89 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
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1answer
74 views

I dont understand this notation

I`m having a homework question that goes like this: $X$ is a Hilbert space, a complete inner product space, show that $B(X)$ is not a Hilbert space. My only question for now is what does $B(X)$ ...