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

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642 views

Sum of Closed Operators Closable?

Let $A$ and $B$ be closed operators on a (separable complex) Hilbert space with dense domains $D(A)$ and $D(B)$ respecitvely. Then, we may define the operator $A+B$ on $D(A)\cap D(B)$. In general, ...
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1answer
521 views

Reproducing kernel Hilbert spaces and the isomorphism theorem

A reproducing kernel Hilbert space is a Hilbert space in which the evaluation functional $L_x : f \rightarrow f(x)$ is continuous. By continuity, the Riesz representation theorem says that this ...
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2answers
76 views

$P+Q-PQ$ is a projection if and only if $PQ=QP$.

Let $\mathcal H$ is a Hilbert space and $P,Q:\mathcal H \to \mathcal H$ are projections. I want to show that $P+Q-PQ$ is a projection if and only if $PQ=QP$. If $PQ=QP$ clearly $P+Q-PQ$ is a ...
6
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1answer
211 views

The strong operator limit of a sequence of unitary operators

If $\mathcal H$ is a Hilbert space and $U_n \in B(\mathcal H)$ is a strong-operator convergent sequence of unitary operators, say $U_n\rightarrow U$, is it true that $U$ is unitary? More explicitly, ...
6
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1answer
2k views

What is the difference between an isometric operator and a unitary operator on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is the identity operator) What is the difference between ...
6
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1answer
628 views

Continuity of scalar product

In a Hilbert space $H$ with inner product and associated norm, why would if $\|x-x_n\| \longrightarrow 0$ and $\|y-y_n\| \longrightarrow 0$ also $\langle x_n,y_n\rangle \longrightarrow\langle ...
6
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2answers
1k views

Parallelogram law valid in banach spaces?

It is known that the parallelogram law $\|x-y\|^2+\|x+y\|^2 = 2(\|x\|^2 + \|y\|^2)$ holds in any space with an inner product (the norm being induced by this inner product). Is this formula valid in ...
6
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1answer
194 views

Convex subset of Hilbert space as intersection of closed balls

How does one prove that any closed, convex, and bounded subset of a Hilbert space is the intersection of the closed balls that contain it?
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1answer
147 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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1answer
126 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 ...
6
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1answer
243 views

Matrix Representation of Trace Class Operators

Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e: $A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
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3answers
501 views

Why isn't it a Hilbert space

Let $X$ be the vector space of all the continuous complex-valued functions on $[0,1]$. Then $X$ has an inner product $$(f,g) = \int_0^1 f(t)\overline{g(t)} dt$$ to make it an inner product space. But ...
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2answers
860 views

Recognition of an orthonormal complete set in an Hilbert Space

I recently came across this exercise: Let $u_n:[0,1]\to \mathbb R$ be the sequence of functions defined by: $$(u_n):=\text{sign }(\sin(2^n\pi x)),\qquad n=0,1,2,\dots$$ a)Prove that this set of ...
6
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1answer
73 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm ...
6
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1answer
106 views

Why are inner product spaces only defined on $\Bbb R$ or $\Bbb C$?

A vector space $V$ makes sense over any field $F$, or even a division ring. So why does adding an inner product suddenly not make sense without taking the $F=\Bbb R$ or $\Bbb C$? What are the primary ...
6
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1answer
409 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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2answers
439 views

Example of a non-algebraic $\ell^2$-function in two variables

Let's call an $\ell^2$-function $\mathbb{N} \times \mathbb{N} \to \mathbb{C}$ algebraic if it is in the image of the natural algebra homomorphism $\ell^2(\mathbb{N}) \otimes \ell^2(\mathbb{N}) \to ...
6
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1answer
280 views

Definition of resolvent set

I'm having trouble understanding some subtlety of definition of resolvent set for given bounded operator A everywhere defined on some Hilbert space. Book I use (and many other sources) give the ...
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2answers
86 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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2answers
128 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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1answer
52 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
6
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1answer
317 views

Approximation of bounded and continuous mappings

Does anyone know if we can approximate a bounded (i.e. bounded sets in V are mapped to bounded sets in V': for every bounded $U\subseteq V$ and $x\in U$, there exists $K_U>0$ such that ...
6
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1answer
245 views

A baby version of the Stein-Cotlar almost-orthogonality lemma

The following is an exercise from Stein and Shakarchi's Real Analysis. Suppose $\{T_k\}$ is a collection of bounded operators on a Hilbert space $H$, each with norm at most $1$. Suppose also that ...
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2answers
769 views

Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces

Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e: $$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in ...
6
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1answer
717 views

Every Hilbert-Schmidt is an integral operator?

Let $(X,\mu)$ be a $\sigma$-finite measure space. If $K\in\mathcal{L}^2(X\times X,\mu\times\mu)$ then the map $A_K:\mathcal{L}^2(X,\mu)\to\mathcal{L}^2(X,\mu)$ defined by\begin{equation} ...
6
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2answers
711 views

Orthonormal basis for Sobolev Spaces

Sobolev spaces of order 2 are known to form a Hilbert space. Consider such a Sobolev space of (order 2) functions on the domain $f:\mathbb{R}\rightarrow \mathbb{R}$. What is an example for the basis ...
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3answers
5k views

How to prove that square-summable sequences form a Hilbert space?

Let $l^2$ be the set of sequences $x = (x_n)_{n\in\mathbb{N}}$ ($x_n \in \mathbb{C}$) such that $\sum_{k\in\mathbb{N}} \left|x_k\right|^2 < \infty$, how can I prove that $l^2$ is a Hilbert space ...
6
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1answer
45 views

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$, looking at $C_{(2)}[-1,1]$, with $L_2$ norm. I tried to look at a general polynomial $\sum_{i=0}^{98} ...
6
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1answer
203 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
6
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1answer
373 views

Measure on a separable Hilbert space

Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have $$ ...
6
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1answer
155 views

A complete eigenvector basis for the restricted operator

Let $X$ be a (not necessarily bounded) selfadjoint linear operator on a Hilbert space $H$ and let $M$ be a closed subspace such that $X(M) \subset M$. Suppose that $X$ admits an orthonormal basis ...
6
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1answer
384 views

Two different definitions of ellipticity

This is a question originating in another mathematics forum, matematicamente.it (in Italian). In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
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1answer
41 views

Completeness of derivatives of Hilbert basis with respect to a parameter

Let us take a Hilbert basis $\left|x_\lambda\right >$ in a Hilbert space $\mathcal{H}$, i.e. the $\left|x_\lambda\right >$ are a complete, orthonormal set of vectors. The subscript indicates ...
6
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1answer
111 views

Show that an unbouned normal operator is closed

A linear operator $A$ is called nomal if $\mathcal{D}(A)=\mathcal{D}(A^{*})$ and $\lVert A\phi\rVert =\lVert A^{*}\phi\rVert$ for every $\phi\in \mathcal{D}(A)$. Show that normal operators are closed. ...
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114 views

If the Fourier transform of a measure is zero then the measure is zero

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ be such that $$\hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle} \Bbb d \mu _{(y)} = 0, \ \forall ...
6
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2answers
99 views

Properties of matrices $M=UDU^*$ with $UU^*=Id$

I recently came across some matrices of the form $M=UDU^*$ (the superscript $*$ denotes the conjugate transpose), where $U \in \mathbb{C}^{r\times n}$ with $r<n$, $D \in \mathbb{C}^{n \times n}$ a ...
6
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0answers
56 views

Two Hilbert spaces $V \subset H$, a basis for both spaces?

Let $V \subset H$ be a pair of Hilbert spaces (with different inner products). The embedding is continuous and dense, and both spaces are separable. Is it always the case that one can I find a ...
6
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1answer
319 views

the basis for the Sobolev space $H^1_0([0,1],\mathbb{R})$

According to the Sturm-Liouville theorem, for any continuous function $p\in\mathcal{C}^0([0,1],\mathbb{R})$, there is a Hilbert basis (normlised) $(\psi_n)_{n\geq1}$ of $L^2([0,1],\mathbb{R})$ such ...
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321 views

Hilbert spaces - equivalent norm

Let $H$ be a Hilbert space with a norm $\| \cdot \|_1$. Let $\| \cdot \|_2$ be another norm on $H$ which is equivalent with $\| \cdot \|_1$. It is easy to see that $(H, \| \cdot \|_2)$ is a Banach ...
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517 views

Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
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2answers
2k views

Elegant proof that $L^2([a,b])$ is separable

Is anybody aware of, or can provide at least an outline, of a proof that the Hilbert space of Lebesgue functions square-integrable on the closed real interval [a,b], equipped with the $L^2$ norm, is ...
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3answers
2k views

Convergence in weak topology implies convergence in norm topology

In Hilbert space why does convergence in weak topology $x_n$ to $x$ imply that $x_n$ converges to $x$ in norm? Thank you very much for your answers. What if I put a condition on weak convergence ...
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1answer
233 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
4k views

Volterra Operator is compact but has no eigenvalue

Volterra operator is defined as operator $V:L^2[0,1]\rightarrow L^2[0,1]$ by \begin{eqnarray} (V)(f(x))=\int_0^xf(y)dy \end{eqnarray} Would you help me to prove that this operator is compact but has ...
5
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2answers
270 views

Hilbert space with all subspaces closed

Does there exist an infinite-dimensional Hilbert space with all subspaces closed?
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1answer
1k views

Proof Complex positive definite => self-adjoint

I am looking for a proof of the theorem that says: A is a complex positive definite endomorphism and therefore is A self-adjoint. Does anybody of you know how to do this?
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1answer
2k views

$\ell_p$ is Hilbert space if and only if $p=2$

Can anybody please help me to prove this.. Let $p$ greater than or equal to $1$, show that the space of all $p$-summable sequences is an inner product space if and only if $p=2$.
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3answers
543 views

Soft Question Hilbert Space Geometry

Just a quick question about the geometry of Hilbert spaces from an intuitive standpoint. Maybe just assuming we're working with $L^2$ would simplify the situation. Basically, in something like ...
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1answer
809 views

The Hahn-Banach theorem for Hilbert spaces follows from Riesz's theorem

How does the Hahn-Banach theorem for Hilbert spaces follow from Riesz's representation theorem?
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854 views

$f$ an isometry from a hilbert space $H$ to itself such that $f(0)=0$ then $f$ linear.

This question was on an exam and I am not sure how to answer it. I mostly tried writing zero in different ways and tried lots of algebra to get something out. I also tried to use the fact that $H$ is ...