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

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2
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2answers
238 views

Stone Weierstrass Overkill in the Measurable Setting?

If $\mu$ is Lebesgue measure on the Borel sigma algebra $\mathcal{B}$ of $[0,1]$. Establishing that the linear span in $L^{2}([0,1]\times[0,1],\mathcal{B}\otimes\mathcal{B},d(\mu \times \mu))$ of the ...
1
vote
1answer
72 views

Functional Analysis Question

I am trying to prove a statement in Folland real analysis textbook but I am having troubles with it. Any help is greatly appreciated. It says when X is a Hilbert space, weak convergence and weak * ...
1
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0answers
103 views

Nice tutorial for Reproducible kernel hilbert space

I am looking for some nice tutorials related to Kernel Hilbert space. I went through lot of them but I couldn't figure out even why it is called reproducible. Any suggestions guys?
0
votes
4answers
704 views

Hilbert Spaces as Euclidean Space generalization

A very (hopefully) simple question: So basically Hilbert Spaces are just Euclidean N-dimensional spaces with complex numbers instead of real numbers. Is that it, a fancy name just for this? (And of ...
2
votes
1answer
216 views

Projection of vector on subspaces in a Hilbert space

This may be a vague title and I think that this question must have an easy answer. Let $\mathcal{H}$ be a weighted $\ell^2$ space of complex sequence $\{x(n)\}_{n \geqslant 1}$ such that $$\|x\|^2 = ...
1
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1answer
78 views

Algebraic and topological complements in a Hilbert space

Every closed subspace of a Hilbert space has a topological complement, namely its orthogonal complement. I'm wondering if every algebraic complement of a closed subspace of a Hilbert space is ...
3
votes
1answer
150 views

How is the inner product in $H^{-1/2}$ defined?

Since $H^{1/2}$ is a Hilbert space, $H^{-1/2}$ must also be a Hilbert space by the isomorphism of Riesz representation theorem. How is the inner product defined there? We know there is a nice ...
12
votes
4answers
414 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
0
votes
1answer
62 views

Hilbert valued $L^p$ functions

Let $\{e_i\}_{i\in N}$ be an orthonormal basis in $L^2(R^m)$. Take an arbitrary $\varphi_{1}\in {\rm L}^{s}(R^{d};{\rm L}^2(R^m))$, $s>1$. Does it hold \begin{equation} \lim\limits_{M\to ...
3
votes
2answers
538 views

Uncountable basis and separability

We know that a Hilbert space is separable if and only if it has a countable orthonormal basis. What I want to ask is If a Hilbert space has an uncountable orthonormal basis, does it mean that it is ...
2
votes
0answers
52 views

Yosida approximation, solution is $C^\infty$? (Explanation of passage needed)

I'm reading Brezis' book Functional Analysis, Sobolev Spaces and PDEs, Lemma 7.1 page 186. Let $w \in C^1([0,\infty);H)$ satisfy $$w' + A_{\lambda}w = 0$$ where $A_\lambda$ is the Yosida ...
3
votes
1answer
128 views

Is my proof that a function is measurable correct?

Let $V$ be separable and Hilbert. Let $\mathcal V = L^2(0,T;V)$. Assume for each $t \in [0,T]$, $$a(t;\cdot,\cdot):V \times V \to \mathbb{R}$$ is continuous and bilinear. Or equivalently, we have ...
2
votes
1answer
92 views

Calculating the norm of an infinite vector

I'm reading "Introduction to Hilbert Spaces" by N. Young. Right in the first chapter, after introducing inner products and norms in general linear spaces, it asks to show that the norm of the vector: ...
1
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1answer
78 views

Characterization of Hilbert spaces [duplicate]

Let $X$ be a Banach space for which there exists a constant $\beta<\infty$ such that for every finite-dimensional subspace $B$ of $X$ , $d(B,\ell_2^n)\le\beta$ (where $\dim B=n$). Then $X$ is ...
0
votes
0answers
42 views

Equation holding on dense subset and passing to limit (Hilbert space basis)

We have that $w_j$ is a (Schauder) basis for the separable space $V \subset H$. We have $$\frac{d}{dt}(z(t),w_j) = (f(t),w_j)$$ Because $j$ here is arbitrary and since finite linear ...
5
votes
0answers
169 views

Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...
2
votes
1answer
152 views

Norm of Hilbert's operator $H f(x)= \int_0^{\infty} \frac{f(y)}{x+y}\, dy$ [duplicate]

Hilbert's operator $$H f(x)= \int_0^{\infty} \frac{f(y)}{x+y}\, dy \quad\text{ for all } f \in {L}^2(0,+\infty) \text{ and } x \in(0,+\infty),$$ is regular integral operator on $L^2(0,+\infty)$ ...
3
votes
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 ...
2
votes
1answer
483 views

Invertibility of a linear operator on a Hilbert space.

Let $H$ be an infinite dimensional Hilbert space over $\mathbb C$, $T$ be a continuous linear operator of $H$, $r(T)=\sup_{||x||=1}|(Tx|x)|$ be the numerical radius of $T$, and $z\in \mathbb C$, such ...
3
votes
1answer
99 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 $n$. ...
2
votes
1answer
131 views

Spectrum of linear operators

I can't solve the following: i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$. ii) Let $S : l^2 ...
2
votes
1answer
124 views

Want to show an operator is compact

With $V=L^2(0,T;H^1(\Omega))$, let $A:V \to V^*$ with $$\langle Au,v \rangle = \int_0^T \int_{\Omega} \nabla u(t) \cdot \nabla v(t).$$ I want to show that $A$ is a compact operator. So, one way to ...
2
votes
1answer
126 views

If $V \subset H$ compact, is $L^2(0,T;V) \subset L^2(0,T;H)$ compact too?

As the question states, if we have the compact embedding of Hilbert spaces $V \subset H$, is $L^2(0,T;V) \subset L^2(0,T;H)$ compact too? If not true in general, is it true for $V=H^1(\Omega)$ and ...
2
votes
1answer
99 views

Question about Hilbert Schmidt theory

Let $V \subset H \subset V^*$ be a Gelfand triple with all spaces being Hilbert and separable. Suppose $A:V \to V^*$ is such that $$\langle Au,u \rangle_{V^*,V} \geq C\lVert u \rVert^2_{V}$$ and $A$ ...
1
vote
1answer
146 views

Is B(H) a Hilbert space?

If H is a Hilbert space, Is B(H) under the operator norm a Hilbert space? If not, is there exists any norm on B(H) that makes it a Hilbert space?
3
votes
2answers
381 views

Eigenfunctions of Laplacian and orthonormal basis (with different inner products)

Suppose I have $L^2(\Omega)$ which has two inner products that are both norm-equivalent. The eigenfunctions of the Laplacian $\Delta$ we know forms an orthonormal basis of $L^2(\Omega)$ -- with ...
0
votes
1answer
34 views

Confusion about domains and range (operator in $L^2$)

Suppose we have bounded linear maps $F:L^2(A) \to L^2(B)$ and $G:L^2(A) \to L^2(A)$. Let $f \in L^2(B)$ and $u \in L^2(A)$. In fact suppose $f$ is smooth. Is $fF(G(u)) = F(G(fu))$? I want to say ...
0
votes
0answers
147 views

Forming the tensor product of a `real' vector space with a 'complex' vector space.

I have a question that I am hoping someone could clarify for me. Context: Consider the algebra $A = (B,\circ)$, given by: \begin{align} B = \{ \begin{pmatrix} a & f\\ \overline{f} & ...
2
votes
1answer
121 views

Is the limit of compact operators again compact?

Let $(T_n)_{n \in \mathbb{N}} \subset \mathcal{L}(\mathcal{X}, \mathcal{Y})$ where $T_n$, $n \in \mathbb{N}$, is compact. Now, assuming that $(T_n)_{n \in \mathbb{N}}$ has a limit $T \in ...
2
votes
1answer
69 views

Why is closeness of an ideal useful?

In the GNS-construction for an $C^*$-algebra $\mathcal A$ (see this script on page 30) one starts with a state $\phi:\mathcal A\rightarrow \mathbb C$ (positive linear functional with $\|\phi\|=1$). ...
3
votes
1answer
410 views

Compact operator between Hilbert spaces: range and orthogonal complement of the kernel are separable

Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be Hilbert spaces and $T: \mathcal{H}_1 \rightarrow \mathcal{H}_2$ a compact operator. I want to show that $(\ker T)^\perp$ and $\text{ran}\ T$ are separable. ...
1
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2answers
216 views

Hilbert space with two inner products; separability and orthonormal basis

Let $H$ be a separable Hilbert space with inner product $(\cdot,\cdot)_H$. So it has an orthonormal basis $h_j$. (You can consider $H=L^2(\Omega)$). Suppose I know that $(\cdot,\cdot)_G$ is an inner ...
1
vote
1answer
67 views

Finite dimensional subspace of Hilbert space and basis

Let $H$ be infinite-dimensional Hilbert space with basis functions $b_i$. Let $B_n = \text{span}\{b_1, ...,b_n\}$. So $\text{dim}(N) = n$. Let $c_i$ be another basis for $H$. Is it true that ...
1
vote
2answers
83 views

Linear algebra in Hilbert space

Let $M,N$ be closed subspaces of a separable Hilbert space. How to prove rigorously the following: $\operatorname{dim} M >\operatorname{dim} N => \exists u\neq0 \in M, u\in N^\perp$ ...
2
votes
0answers
127 views

Function of a completely continuous operator

I would be most thankful if you could help me with this question. If $A$ is a completely continuous Hermitian operator on a Hilbert space $H$, for what class of functions $f$ can one define a function ...
-1
votes
1answer
41 views

Range of $Df(a)$ contained in the subspace $\{f(a)\}^{\perp}$ with $f$ differentiable

Let $A$ a open set in a Hilbert Space $H$, suppose that $f:A\to F$ is differentiable at $a\in A$ and that $||f(x)||=c$ forall $x\in A$. Show that range of $Df(a)$ is contained in the subspace ...
1
vote
2answers
200 views

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just ...
5
votes
1answer
214 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 ...
3
votes
1answer
571 views

Direct sum of orthogonal subspaces

I'm working on the following problem set. Let $\mathcal{H}$ be a Hilbert space and $A$ and $B$ orthogonal subspaces of $\mathcal{H}$. Prove or disprove: 1) $A \oplus B$ is closed, then $A$ and $B$ ...
3
votes
1answer
102 views

Questions about $B(H)$ and $B(H)/K(H)$ as Banach space

I am trying to investigate the relation between Uniformly Convexity and existence of Schauder Basis for a Banach space. I read in a Handbook article that $B(H)$ (the algebra of all bounded operators ...
2
votes
1answer
77 views

Hilbert space on line bundle

Suppose that $L$ is a complex line bundle on a manifold $M$ with measure $\mu$, How can we prove, $L^2(M,L,\mu)$ is Hilbert space?
2
votes
1answer
80 views

Sufficient condition for self-adjoint subset of bounded linear operators on a Hilbert space being irreducible

Let $H$ be a Hilbert space and denote as $B(H)$ the bounded linear operators on $H$. Let $M$ be a subset of $B(H)$, s.t. for $A \in M$, also $A^* \in M$. How can one show that if the commutant has ...
4
votes
2answers
98 views

(From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$

Lang $SL_2$ p. 13 :Let $\{\phi_i\}$, $\{\psi_i\}$ be orthonormal bases for $L^2(X)$ and $L^2(Y)$ respectively. Let $$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$ Then $\{\theta_{ij}\}$ is an ...
0
votes
0answers
123 views

Operator identity involving square root of an operator

I would be most thankful if you could help me prove the following identity. Let $A$ and $B$ be two completely continuous Hermitian operators on a Hilbert space $H$, such that $A$ and $B$ do not ...
7
votes
2answers
277 views

Counterexample for the stability of orthogonal projections

Let $V$ be a seperable Banach space, which is dense and continuously embedded in a Hilbert Space $H$. Let $(V_m)$ be a Galerkin scheme (See definition below) for $V$. Using the embedding we can ...
5
votes
1answer
623 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?
5
votes
1answer
204 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 ...
1
vote
0answers
71 views

Are the special functions independent?

maybe the bessel functions are some complicated function of the exponential function, logarithm function... or maybe there's a relation between two or more transcendental functions. Is there a way to ...
2
votes
1answer
64 views

Is $L^2(0,T;H_n)$ compactly embedded in $L^2(0,T;H)$?

Let $H$ be a separable Hilbert space with basis $h_i.$ Let $$H_n := \text{span}\{h_1,...,h_n\}.$$ Questions: 1) Is $L^2(0,T;H_n)$ compactly embedded in $L^2(0,T;H)$? 2) Is $L^2(0,T;H_n^*)$ ...
3
votes
2answers
143 views

Equivalent norms imply equivalent inner products?

Let $H$ be Hilbert and let it have two innner products $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$. If the norms $|\cdot|_1$ and $|\cdot|_2$ are equivalent, does this ever imply: there exist constants ...