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

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Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
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16 views

What can one assume about $T^*$ when showing that $T$ is normal?

Consider a continuous and linear operator $T$ such that $$ T : l^2 \to l^2 $$ where $(a_n) \mapsto (\alpha_n a_n)$ Moreover $(\alpha_n)$ is a sequence of complex numbers that converges to zero. Now, ...
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27 views

Isolated Eigenvalue

What does it mean that an eigenvalue is "isolated"? My intuitive understanding says it is when one can find an open ball around it such that there is no other eigenvalue in that open ball. However, I ...
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17 views

Norm of operator matrix

I'm having trouble with the following: suppose H is a Hilbert space and $f_{i, j}, g_{i, j} : H \rightarrow H$, $1 \leq i, j \leq n$ are bounded operators. Then we have operators $(f_{i, j}) , (g_{i, ...
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32 views

Hilbert space L2 - inner product

I have a problem with one exercise. I have to prove that $L^2$ space is Hilbertian. So I think that the best way is to check out inner product by definition of norm, so: \begin{equation*} ...
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18 views

an objective question from functional analysis [on hold]

Let $A$ and $B$ be bounded operators on a Hilbert space $H$ such that $AB=BA$. Let $\lambda$ be an eigenvalue for $A$. Then it must be that a)$B$ has no eigenvalue b)$B$ has at least one ...
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1answer
28 views

Showing that an operator generates a contraction semigroup

Let $A$ be the infinitesimal generator of a contraction semigroup $(T(t))_{t\ge 0}$ on the Hilbert space $X$, and $D\in\mathcal{L}(X)$. I want to show that the operator $A+D-2\|D\|I$ with domain ...
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28 views

Prove Operator is a Projector

Let $\mathscr{H}$ be a complex Hilbert space. A projector is a linear map $P:\mathscr{H}\to\mathscr{H}$ such that $P\circ P = P$. I'm trying to prove the following claim, from the information given ...
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1answer
23 views

Is this a metric on $\mathbf P\mathcal H$?

Let $\mathcal H$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. On the projective space $\mathbf P\mathcal H = \left(\mathcal H\setminus\{0\}\right)\big/{\sim}$ ...
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18 views

Hamiltonian: Scattering Spaces

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a family of projections: $$1(r)^2=1(r)=1(r)^*\quad(r\geq0)$$ Denote for shorthand: ...
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41 views

Powers of compact operators

Consider a Hilbert space $H$ and a compact self-adjoint operator $T : H \to H$. I want to prove that all positive powers (especially fractional powers) of $T$ are compact. From the spectral theorem, I ...
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Lemma 3.3-7 and Theorem 3.6-2 in Kreyszig's “Introductory Functional Analysis With Applications”: What if completeness is lost? [duplicate]

Let $X$ be an inner product space, and let $M$ be a non-empty subset of $X$. Then we have the following: (a) If the space of $M$ is dense in $X$, then $M^\perp = \{0 \}$, that is, $x \in X$, $x ...
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1answer
27 views

On the subset of a closed vector subspace

Theorem: Let $H$ be a Hilbert space, and let $U$ and $V$ be closed subspaces of $H$ such that $U\subset V$. Then there exists a nonzero vector $v\in V\backslash U$ such that $v\bot U$. The fact that ...
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1answer
31 views

Need help understanding compact embedding of hilbert spaces

I am trying to understand the following statement, and I would like some clarification Consider a Hilbert space $H$ which is compactly embedded in a Hilbert space $L$, with $H^*$ being the dual ...
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18 views

Norm of a linear continuous form

Let $E=\{f\colon[0,2]\to\mathbb{R} \mid f \text{ continuous} \}$ be a prehilbert space equipped with inner product: $$\langle f,g\rangle=\int_0^2 f(t)g(t)\, dt$$ And let : $$U\colon E ...
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1answer
23 views

Proving an integral identity

I'm dealing with the Hermitian operator, and I've been asked to prove that all $f(x) = x^n e^{\alpha x}$ belong to $L^2(-\infty,\infty;e^{-x^2/2})$ by showing that: $$\int_{-\infty}^{\infty}x^m ...
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44 views

Intersection of Hilbert spaces

Consider two Hilbert spaces $H_1$ and $H_2$ with inner products $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$ generating norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ ...
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19 views

Theorem 3.3-1, Lemma 3.3-2, and Theorem 3.3-4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to write these as one?

I'm trying to prepare some ancilliary material on the following three results in sec. 3.3 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig: (First, I'm giving ...
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1answer
35 views

Prob. 2, Sec. 3.3 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: How to minimise the norm?

Let $z$ be a given complex number. Let $M \subset \mathbb{C}^n$ be given by $$M \colon= \left\{ (\xi_1, \ldots, \xi_n ) \in \mathbb{C}^n \mid \sum_{i=1}^n \xi_i = z \right\}.$$ Then $M$ is convex ...
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29 views

Hamiltonian: Invariant Core

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote its evolution by: ...
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18 views

Reducing Subspaces: Hamiltonian

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a projection: $$P\in\mathcal{B}(\mathcal{H}):\quad P^2=P=P^*$$ Then one has: ...
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113 views

If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n)(b_n)$ converge?

Could someone help me to solve this or at least give me a hint?, I have tried using Cauchy's criterion, the Dirichlet test for convergence, etc, but I can´t prove it.Honestly I don´t know where to ...
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1answer
19 views

Selfadjoint Operators: Weak Convergence

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Regard a sequence: ...
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39 views

The property of closed subspace

We know that a set is closed if and only if every convergent sequence with elements in the set has a limit point in the set. I am reading a paper, and the paper claims that the following is due to S ...
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14 views

Operator norm of symmetric Matrix in Hilbert Space with Hermitian Inner Product

Assume we have a postive definite real matrix $P$ and we define an inner product on a finite dimensional hilbert space $\langle x, y \rangle = x^\top P y$ and clearly the induced norm is $|| x || = ...
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35 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 $x,y \in H$ ...
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22 views

Is there a pseudocontractive mapping that is not strictly pseudocontractive?

Given a Hilbert space $H$, a mapping $T:H\rightarrow H$ is said to be pseudocontractive if $$\|Tx-Ty\|^2\leq \|x-y\|^2+\|(x-Tx)-(y-Ty)\|^2\,\,\, \forall x,y\in H,$$ and it is strictly ...
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12 views

A rescaled inner product inequality

I was wondering if the following inequality is true: Let $\xi_1,...,\xi_n$ be vectors in a Hilbert space $H$ and let $x_{i,j}$ be complex numbers such that $\prod x_{i,j}$ is real and $$\prod ...
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1answer
23 views

A question on Isometry between the orthogonal subspaces of Hilbert spaces

I was reviewing my class-notes on Functional analysis and the professor had mentioned that given a closed proper subspace $U$ of an hilbert space $\mathcal{H}$, $\exists $ a closed subspace ...
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80 views

$\sin$ and $\cos$ are the basis of what space?

When learning Fourier expansions, we learn that $\{\sin(mx), \cos(mx)\}_{m \in \Bbb N}$ is an orthonormal basis for our space and thus we can expand functions in it. My question is what space is this ...
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23 views

Cauchy sequence in reproducing kernel Hilbert space

Consider a positive definite kernel $K:\mathbb N\times \mathbb N\rightarrow \mathbb R$. Denote the unique RKHS associated with $K$ by $\mathcal H_K$. The RKHS $\mathcal H_K$ consists of \begin{align} ...
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$P$ and $Q$ are unitarily equivalent iff dimensions of ranges and kernels are the same

Two projections $P,Q$ are unitarily equivalent if and only if $$dim(randP)=dim(ranQ)$$ $$dim(kerP)=dim(kerQ)$$ How can we show this? One directionn seems easy: If $P$ and $Q$ are unitarily eqv, ...
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63 views

How do I prove a differential operator has no purely imaginary eigenvalues?

Anyone who has taken a course in linear algebra knows how to prove the eigenvalues of a self-adjoint operator are real or the eigenvalues of a skew-self-adjoint operator are purely imaginary. This is ...
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57 views

Prob. 8, Sec. 3.5 in Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications

Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications Prob. 8, Sec. 3.5 $\DeclareMathOperator{\span}{span}$Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$, and let $M = ...
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44 views

A question about Fourier coefficients.

Is it true that the sequences $ (A_{n})_{n \in \Bbb{N}} = (0)_{n \in \Bbb{N}} $ and $ (B_{n})_{n \in \Bbb{N}} = \left( \dfrac{1}{\sqrt{n}} \right)_{n \in \Bbb{N}} $ are the Fourier coefficients of ...
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9 views

Existence of Schauder base for given operator

Suppose $A: l_2 \rightarrow l_2$ is a finite-rank linear bounded operator of dimension $k$. Is it true that there exists a Schauder orthonormal base for which only first $k$ columns will be nonzero ...
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24 views

is $\langle\lim_{n\to \infty}u_n,g\rangle = \lim_{n\to\infty} \langle u,g\rangle $ valid for bounded linear operators?

Suppose M is any linear manifold in H. H is a hilbert space. Define the orthogonal complement of M to be $$M' =\{f \in H | \langle f,g\rangle= 0 ,\forall g\in M\}.$$ To see that M' is a closed ...
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18 views

Singular Spectrum: Techniques?

Given a Hilbert space $\mathcal{H}$. Let the Lebesgue measure be $\lambda$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its spectral measure by: ...
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14 views

Are all linear basis functions a reproducing kernel hilbert space?

Do any linear basis function like for instance linear b-splines form a reproducing kernel hilbert space? is it sufficient for the kernel to be semi-positive definite and have a positive Fourier ...
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Prove that $\bigcap_n K_n \neq ∅$.

Let $H$ be a Hilbert space. Discuss the validity of the following statement: If ${K_n}$ is a decreasing sequence of nonempty, bounded, closed convex sets in $H$, then $\bigcap_n K_n \neq ∅$. ...
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If I want to prove that $M^{\perp}$is a closed

If I want to prove that $M^{\perp}$is a closed Can I say because it is the inverse image of $0$ by continuos function ( projection operator )
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59 views

How can I prove the following theorem with explanation? please

How can I prove the following theorem with explanation. please For any nonempty subset $M$ of a Hilbert space $H$, the span of $M$ is dense in $H$ if and only if $M^{\perp}=\{0\}$ I read the prove ...
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19 views

ONB of Hilbert dual $H'$

Let $H$ an arbitrary Hilbert space, $\{ e_i \}_{i \in I}$ ONB of $H$. Is there an ONB $\{ e^j \}_{j \in I}$ of the Hilbert dual $H'$, s.t. $e^j(e_i)=\delta_{ij}$? If so, is $\{e_i \otimes e^j\}_{i,j ...
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1answer
30 views

Proving that this space is not Hilbert.

Consider $E$ the space of all the functions defined on $\Bbb R$ which admit a representation of the form $x(t) = \sum_{r \in \Bbb R}^* c_r e^{irt}$, where $\sum^*$ indicates that only a finite number ...
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1answer
26 views

check if a linear operator is bounded

show that $Tf = f(0)$ is not a bounded linear functional on the space of continuous functions measured with the L2 norm, but it is a bounded linear functional if measured using the uniform norm. ...
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15 views

Diagonal non-compact operator

Suppose we have an operator $I:l_2 \rightarrow l_2$ which is diagonal but not compact. Does that follow: there exists a constant $C$ such that infinite number of diagonal terms $>C$?
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Dual of Riesz basis with opt. stab. const. $\lambda_\min$, $\lambda_\max$ has opt. stab. const. $\frac1{\lambda_\max}$ and $\frac1{\lambda_\min}$.

Consider a Hilbert space. Consider a Riesz basis $\phi_k$, $k \in \mathcal{K}$ of this space, where $\mathcal{K}$ is an appropriate set of indices. By definition, the Riesz basis fulfils the ...
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84 views

Theoretical Basis for Eigenvalue transformation on Bessel's Equation

The method I've been taught for finding all of the eigenvalue solutions to Bessel's operator $$b(f)\equiv f''(x)+\frac{1}{x}f'(x)$$ goes as follows. Let $g(a)=f(\sqrt{\lambda}x)$. Then $$b(g)=\lambda ...
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1answer
27 views

Is it true that $M$ is complete?

If $H$ is a Hilbert space and $M$ is a nonempty,closed, bounded and convex subset(not necessarily a subspace)of $H$, then is it true that $M$ is complete? If it is, then can we use it without proof? I ...
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resource on integral operators

Can you please suggest for me a good resource on integral operators.These are the specific topics that I am looking for: Bounded linear operators in hilbert space. Compact operators Spectral theory ...