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

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1answer
25 views

Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
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23 views

Spectral Measures: Completeness

Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. A spectral measure can be completed $\overline{E}$. ...
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1answer
38 views

How to show that the operator $T(\{x_n\})=\{n x_n\}$ has closed graph?

Consider the subspace $$D=\left\{x\in \ell^2 \ \big|\ \sum_{n\in\mathbb N} n^2 |x_n|^2<\infty\right\}$$ of $\ell^2$, and let $T:D\to\ell^2$ be defined by $T(\{x_n\})=\{n x_n\}$. I need ...
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1answer
38 views

Fréchet derivatives of $\sum_{n=1}^\infty x_n^2/n^3 -\sum_{n=1}^\infty x_n^4$

I read that the second order Fréchet derivative $F''(0)$ of linear functional $F:\ell_2\to\ell_2$, where $\ell_2$ is the separable real Hilbert space, defined by ...
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1answer
20 views

Show that this operator is linear

Let $\Bbb H$ is a Hilbetr space and $T:\Bbb H\to\Bbb H$ be a operator such that $$<x,Ty>=<Tx,y>$$ $\forall x,y\in\Bbb H.$ I want to show that $T$ is linear and bounded. If I can show that ...
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45 views

Derivative of norm in Hilbert space

I read (p. 485 here) that the Fréchet derivative of norm (non-linear) functional $p:H\to\mathbb{R}$, $x\mapsto\|x\|$ is $\frac{x}{\|x\|}$ for all $x\ne 0$, which I think to be intended as the linear ...
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24 views

Sufficient conditions for weak continuity

Are there any "easily verifiable" sufficient conditions for weak (equivalently, weak*) continuity of (not necessailry linear) maps on the unit ball of $\ell^2$, mapping into $\ell^2$? Apologies for ...
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1answer
17 views

Subspace of a Hilbert space with a distinct inner product

I don't really know where to begin with the following question: Let $ (H_0, \langle \cdot \rangle_0)$ be a closed subspace of $ (H, \langle \cdot \rangle )$ such that norms induced by $ \langle \cdot ...
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1answer
22 views

Proving an orthogonal projection of the Hilbert adjoint is just the adjoint

I'm facing the following problem: let $ H_0 \subset H $ be a $ T$-invariant closed subspace of Hilbert space $ H $ (i.e. $ T(H_0) \subset H_0 $) and $ P$ - an orthogonal projection of $ H $ onto $ ...
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1answer
76 views

Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
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1answer
21 views

If $E$, $\overline{E}$ are orthogonal projections such that $\mathrm{range}(\overline{E})=\overline{\mathrm{range}(E)}$, then is $E\ge\overline{E}$?

I feel like this should be true. Let $\mathrm{range}(E)=A$ and $u$ be an arbitrary vector in a Hilbert space $H$, it is sufficient to show $\langle (E-\overline{E})u,u\rangle=0$. By Cauchy-Schwartz: ...
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1answer
36 views

Finite rank approximation of bounded operators on Hilbert space

Let H be a (finite dimensional) Hilbert space. The approximation property states that every bounded operator from H to itself can be approximated by a sequence of finite rank operators. My question ...
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1answer
17 views

About the von Neumann decomposition

The von Neumann theorem states that for any symmetric operator $f$, the domain $D_{f^\dagger}$ of its adjoint $f^\dagger$ is the direct sum of the three subspaces $D_{\bar{f}}$, $\aleph_z$, and ...
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1answer
42 views

CAR- & Weyl-Algebra: Uniqueness?

Given a Hilbert space: $\mathcal{h}$ Consider representations of the CAR-algebra: $\mathcal{A}_\text{CAR}^{(\prime)}(\mathcal{h})$ In Bratelli & Robinson it is stated the uniqueness: ...
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14 views

continuos spectrum of $R+L$, where $R$ and $L$ are the right and left shift of sequences in $l_2$

consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I'm trying ...
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29 views

Any example of non-closed operator?

I cannot think of one. By the way, is there any good exercise book on functional analysis or hilbert space?
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1answer
44 views

Unit sphere weakly dense in unit ball

I'm studying for an exam and came across a problem: I want to prove that the unit sphere in a Hilbert space $\mathcal{H}$ is weakly dense in the unit ball. I already had to prove that the unit ball ...
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1answer
91 views

Spectral Measures: Lebesgue

Preface This thread deals with dominated convergence for functional calculus: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ Framework Given a Borel space $\Omega$ ...
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3answers
36 views

Why cannot a densely defined operator be extended to an everywhere defined operator?

I am a physicist learning functional analysis because of its fundamental role in quantum mechanics. There are so many bizarre facts. One is, there are densely defined operators which seem cannot be ...
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1answer
44 views

Are all Banach spaces also Hilbert spaces?

We have the well-known "polarization identity" $$(x,y)=\frac{1}{4}\left(\|x+y\|^2-\|x-y\|^2+i\|x+iy\|^2-i\|x-iy\|^2\right)\tag{1}$$ that works in any Hilbert space. Hence, is every Banach space also a ...
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1answer
18 views

Why only densely defined operators can have an adjoint operator?

Why is it impossible or making no sense to define an adjoint operator for a non-densely defined operator?
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3answers
155 views

An example of non-closed subspace of a Hilbert space?

I am reading a book on Hilbert space. It seems that the author assumes that a linear subspace of a Hilbert space can be non-closed. I cannot think of an example. I am still used to the ...
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33 views

Properties of Hilbert Spaces- Contrasting Two Different Topological Spaces

Let H be the space of real sequences x = $(x_1 , x_2, ... )$ with $\sum(x_n^2)$ finite. (This is $l_2$ in fact.) I wish to show the following: The topology on H is ...
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1answer
27 views

Measurability of inner integral $x \mapsto \int f(x,y)\, d\mu(y)$

Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing ...
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1answer
26 views

Orthonormal Hamel Basis is equivalent to finite dimension

Consider a Hilbert space which is infinite dimensional. If it is separable, it is well known that an orthonormal basis will be countable, while a hamel basis will be uncountable (since it is a ...
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19 views

Let $H$ be a Hilbert space, $A$ is unitary and $S=\{Ax:x\in H\}$. Does $S^{\perp}=\operatorname{Null}(A)$?

Let $H$ be a Hilbert space, and $S=\{Ax:x\in H\}$. Does $S^{\perp}=\operatorname{Null}(A)$? What I have is if $x\in S^{\perp}$ then $x\perp A(A^*A^*Ax)$ then $(x,A^*Ax)=(Ax,Ax)=0$, so $x$ is in ...
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0answers
16 views

On existence of an element whose is orthogonal with given $n$ elements of a Hibert space of infinite dimension

Let $H$ be a Hilbert space of infinite dimension with the scalar product $\left\langle {.,.} \right\rangle $. Given $u_1,...,u_n\in H$. Is there a $u\in H$, $u\ne 0$ for which $\left\langle {u,{u_j}} ...
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1answer
25 views

0 limit point of spectrum of completely continuous operator $H\to H$

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 475 here) that 0 is an accumulation point for the spectrum of a completely continuous operator $A:H\to H$ where $A$ ...
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1answer
79 views

Reiterate Volterra integral operator is a contraction

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 472 here) the statement that Volterra operator $A:L_2[a,b]\to L_2[a,b]$ defined ...
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1answer
27 views

Calculation of operator norm

$H$ is a Hilbert space, $T: H \to H$ linear bounded operator, $||T||$ is the norm of $T$ given by $$||T||=\sup\{||T(x)||;||x||\le 1 \}. $$ Is it true that $$||T||=\sup\{|\langle ...
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1answer
31 views

$\ker (I-A)=\{0\}\Rightarrow\text{im }(I-A)=H$ for $A:H\to H$ compact

Let $T$ be the operator defined by $T:=I-A$ where $I:H\to H$ is the identity and $A:H\to H$ is a compact operator defined on Hilbert space $H$. In such a case, if we defined the chain of ...
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0answers
33 views

the spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?
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1answer
38 views

Direct sum of kernel and image of the adjoint operator

Let $H$ be a separable Hilbert space and $T:=I-A$, where $A:H\to H$ is a compact operator. If $T^\ast$ is the adjoint operator of $T$ it can be proved that $\ker T$ and $\text{im } T^\ast:=T^\ast (H)$ ...
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1answer
22 views

Degenerate Hilbert-Schmidt operators

Let us define a Hilbert Schmidt operator $A:L_2[a,b]\to L_2[a,b]$ by $$A\varphi:=\int_{[a,b]} K(s,t)\varphi(t)d\mu_t$$where $\mu_t$ is the linear Lebesgue measure. A degenerate case is represented by ...
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3answers
75 views

Need countereample : If a sequence $(a_n) \in l^2 $ , then the sequence $(1/a_n) \notin l^2 $

I want to know the counterexample for the following statement : Given a sequence $(a_n)$ such that $a_i\ne 0 $ for any $i$ : If the sequence $(a_n) \in l^2 $ , then the sequence $(1/a_n) \notin l^2$. ...
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0answers
19 views

Limits of trajectory of gradient flow in Hilbert space

I have been studying about gradient flow in Hilbert space of a Morse function $f$. Specifically, let $X$ be a Hilbert space and $f : X\to \mathbb R$ be $C^3$ function. The gradient flow here is ...
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1answer
24 views

CAR-Algebra: Nontriviality?

Given a Hilbert space $\mathcal{h}$. Consider the abstract CAR-algebra $a:\mathcal{h}\to\mathcal{A}_\text{CAR}$. Then their actually isometries: $$a:=a(f):\quad ...
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1answer
27 views

How to show that a vector space is closed?

I am trying to complete a proof which requires me to prove that a subspace $H$ of $L^2{(\Omega,\mathcal{F}},\mathbb{P})$ is closed vector space in $L^2{(\Omega,\mathcal{F}},\mathbb{P})$ What do I ...
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2answers
91 views

Dense subspaces, closed subspaces and unbounded operators in Hilbert spaces

Let $\mathcal{H}$ be a Hilbert space, and let $N\subseteq\mathcal{H}$. I found two interesting statements (without proof): if a closed subspace $N$ is such that $N^{\perp}=\{0\}$ (which is ...
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1answer
33 views

$H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$

When $\Omega$ is a bounded open set of $\mathbb{R}^N$ with the help of Poincare inequality, we know that $H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$ is a Hilbert space. ...
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1answer
11 views

If $H$ is a Hilbert space and $T$ an isometric operator, then $\overline{R(T-I)}=H \implies N(T-I)=\{0\}$?

Let $H$ be a Hilbert space. Let $T$ be a linear operator and $R(T)$, $D(T)$, $N(T)$ the range, domain and kernel of $T$, respectively. If $\|Tx\|=\|x\|$ for all $x \in D(T)$, then $T$ is called an ...
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20 views

Subdifferential of a continuous function is non-empty

Prove that if $X$ $-$ normed vector space, $x_0 \in \text{int }A$ and convex function $f$ is continuous in $x_0$ then $\partial f(x_0) \neq \emptyset$. $\partial f(x_0) = \{x^*\in X^*:\forall x \in ...
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2answers
143 views

Hilbert Spaces: Tensor Product

Attention The question has been modified! (Previous answers were perfectly correct then.) (But they're not up-to-date anymore.) Reference Build-up on: Vector Spaces: Tensor Product Problem ...
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2answers
27 views

Which Sobolev-Space to use to formulate weak biharmonic equation, $H^2_0$ or $H_0^1\cap H^2$?

For the weak formulation of the biharmonic equation on a smooth domain $\Omega$ $$ \Delta^2u=0\;\text{in}\;\Omega\\ u=0, \nabla u\cdot \nu=0\; \text{on}\; \partial\Omega $$ why does one take ...
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27 views

Characterization of Hilbert spaces among Banach spaces

Let $H$ be a complex Hilbert space. I know that $H\simeq \overline{H^*}$ by the Riesz representation theorem, where $\overline{X}$ means the complex conjugate space of $X$. I want to prove the ...
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53 views

Dichotomy for global existence or blow up for solutions of evolution problems.

Consider the problem (Nonlinear Schrödinger equation) \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u\mp u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ ...
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0answers
23 views

Compact operator space Hilbert

Let $H_1$ and $ H_2$ Hilbert space and $T:H_1\rightarrow{H_2}$ a compact operator. Shows $N(T)^\perp \subseteq H_1$ is a subspace separable of $H_1$. indeed as $N(T)^\perp$ is a closed subspace of ...
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1answer
44 views

double Orthogonal complement is equal to topological closure

So I'm in an advanced Linear Algebra class and we just moved into Hilbert spaces and such, and I'm struggling with this question. Let $A$ be a nonempty subset of a Hilbert space $H$. Denote by ...
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1answer
28 views

Does Convergence of Maps Evaluated at Points Imply Convergence in Operator Norm?

Suppose that I have $T,T_n \in B_H$, for some Hilbert space $H$. Is the following implication true? $$ \|(T-T_n)x\| \rightarrow 0 \ \forall x\in H \ \Rightarrow \ \|T-T_n\| \rightarrow 0, \ \text{ie} ...
4
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1answer
79 views

Uncountable series without axiom of choice

Consider a sequence of positive real numbers $(\alpha_i)_{i\in I}$ for some (suppose maybe wellordered for now) set $I$. Using axiom of choice, it is easy to see that $\sum_i \alpha_i$ is infinite if ...