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

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1answer
148 views

Mean ergodic theorem

Let $U$ be a unitary operator and $H$ a Hilbert space. $I := \{ v \in H |  Uv = v\} $ and $A := \{ Uw - w | w \in H\}$. I would like to show that $A$ is dense in the ...
1
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1answer
112 views

Riesz sequences in Hilbert spaces

Is it true that if $\{x_{n}\}_{n=1}^{\infty}$ is a finite union of Riesz sequences in a Hilbert space H, then $\{x_{n}\}$ itself will be a Riesz sequence? What about Frames and Bessel seuences, do we ...
3
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4answers
104 views

reference for strongly continuous semi-groups

At the moment I am trying to understand the proof of the Fredholm property in Salamon's notes on Floer homology. There I came across the notion of an unbounded operator on a (real) Hilbert space which ...
1
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1answer
166 views

Show $\{u_n\}$ orthonormal, A compact implies $\|Au_n\| \to 0$

I'm having a bit trouble with this homework exercise. Let $\mathcal{H}$ be a Hilbert space and $\{u_n\}_{n=1}^\infty$ an orthonormal sequence in $\mathcal{H}$. Let $A$ be a compact operator on ...
2
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1answer
113 views

Balls in the space of bounded operators on a Hilbert space

Suppose $\mathsf{H}$ is an infinite-dimensional (non-separable preferably) Hilbert space. Consider the space $L(\mathsf{H})$ of all bounded operators on it. Is there $0\neq W\in L(\mathsf{H})$ such ...
-1
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1answer
164 views

Non-orthogonal Direct Sum?

I'm reading through a text that has been making references to "(not necessarily orthogonal) direct sums" of Hilbert spaces. What would a non-orthogonal direct sum be? Is that something like a direct ...
6
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2answers
312 views

Question about positive operators on a Hilbert space

I have the following problem. Let $\Omega \subset R^n$ have finite measure, let $H = L^2(\Omega)$ and let $S: H \to H$ be a bounded linear operator. Then it is well known that $P = SS^*$ is a positive ...
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0answers
130 views

Countable product of Hilbert spaces

Let $H_1, H_2, \ldots $ be a countable set of Hilbert spaces. Let $H\subset \prod_k H_k$ be the set where $$\|x\|^2 = \sum_k \|x_k\|^2_{H_k} < \infty.$$ Show that $H$ is a Hilbert space. It ...
5
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1answer
680 views

Hilbert Schmidt integral operator

Hilbert-Schmidt Integral operators are usually defined from $H=L_2[a,b]$ into $H=L_2[a,b]$ as $$(Tf)(x) = \int_a^b K(x,y)f(y) dy,$$ provided that $K(x,y)$ is a Hilbert Schmidt kernel, namely ...
2
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1answer
102 views

Hilbert space on a finite set

If X is a finite set, what does the Hilbert space $L^2(X)$ means? - saw this notion on The Princeton Companion to Mathematics.
3
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1answer
130 views

Is a projection operator diagonal-decreasing on a positive operator?

I just faced a obvious-looking inequality, but I didn't manage to prove it. Let $H$ be a finite-dimensional Hilbert space, $M, \rho$ positive operators on $H$, $P$ an orthogonal projector on $H$. Is ...
2
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3answers
1k views

Compact operator

If $H$ and $K$ are Hilbert spaces,show that if $T:H\longrightarrow K$ is a compact operator and $\{e_{n}\}$ is any orthonormal sequence in $H$ then $\|Te_{n}\|\to0$.Is the converse true? thanks.
5
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2answers
537 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 ...
7
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1answer
482 views

Isometric to Dual implies Hilbertable?

Let $X$ be a Banach space and suppose that $X$ is isometric to its continuous dual space $X^*$. Must $X$ be hilbertable in the sense that there exists an inner product which induces the norm on $X$? ...
7
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3answers
2k views

A linear operator on a finite dimensional Hilbert space is continuous

How do I show that a linear function from a Hilbert space $H$ to itself is continuous if $H$ is finite dimensional? Also, what would be an example of a linear function from a Hilbert space to itself ...
10
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1answer
426 views

orthonormal system in a Hilbert space

Let $\{e_n\}$ be an orthonormal basis for a Hilbert space $H$. Let $\{f_n\}$ be an orthonormal set in $H$ such that $\sum_{n=1}^{\infty}{\|f_n-e_n\|}<1$. How do I show that $\{f_n\}$ is also an ...
8
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2answers
488 views

Proving an inequality with Cauchy-Schwarz

In the "User's guide to viscosity solutions" by Crandall, Ishii and Lions (link), they make the following claim (inequality (A.4) p. 58) : Given $x$, $\xi$ $\in \mathbb{R}^n$, $A \in \cal{S}(n)$ ...
2
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1answer
114 views

Non-linear functional on $L^2$

Let $a,b,g,h$ be real numbers. How to prove that the functional $F\colon L^2 [a,b]\to \mathbb{R}$, given by $F(u)=\int_a^b (u^2(x)-gu(x)-h)\,dx$ is continuous? Thank you
2
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1answer
209 views

The commutator subgroup of the group of bounded invertible linear operators

I am curious to know what the commutator subgroup of the group of (bounded) invertible linear operators on a complex Hilbert space is? Note that by "commutator subgroup" I mean the subgroup ...
14
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1answer
674 views

Is there a constructive proof of this characterization of $\ell^2$?

I would like to revisit this question, which can be equivalently stated as: Proposition. Let $(a_n)$ be a sequence of real (or complex) numbers such that $\sum a_n b_n$ converges for every $(b_n) ...
9
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1answer
336 views

How to prove Halmos’s Inequality

How to prove Halmos’s Inequality? If $A$ and $B$ are bounded linear operators on a Hilbert space such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$ I found it from ...
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2answers
474 views

Does the vector space spanned by a set of orthogonal basis contains the basis vectors themselves always?

I used to think that in any Vector space the space spanned by a set of orthogonal basis vectors contains the basis vectors themselves. But when I consider the vector space $\mathcal{L}^2(\mathbb{R})$ ...
2
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2answers
264 views

A non complete orthogonal system whose perpendicular space is trivial

Let $Y$ be an inner-product space, and let $A$ be an orthonormal system. We're trying to find a case to demonstrate the fact that even if for any given $x$ in $Y$ there's some $u$ in $A$ such that ...
5
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1answer
352 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, ...
4
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1answer
306 views

Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces? For the sum we have the notion of a direct integral, here.
8
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2answers
464 views

Haar's base for $L^2[0,1]$

$\newcommand{\span}{\operatorname{span}}$ Define $e_{0,0}\equiv 1$, and for all $n\in \mathbb{N}$ $$e_{n,k}=\begin{cases} 2^{n/2} &\text{if } \frac{k-1}{2^n}\leq x\lt \frac{k-\frac{1}{2}}{2^n}\\ ...
5
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1answer
146 views

Addition of Unbounded Operators

Let $H$ be a (separable complex) Hilbert space and let $A$ and $B$ be two densely-defined, maximally-defined linear operators on $H$ with domains $D(A)$ and $D(B)$ respectively. (By maximall-defined, ...
2
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1answer
608 views

Riesz Lemma to the Riesz Representation Theorem

Let $H$ be a Hilbert Space and let $H^*$ be the dual space of $H$. The Riesz Lemma states that for each $T\in H^*$, there is a unique $y_T\in H$ such that $T(x)=(y_T,x)$ $\forall x\in H$. Also, ...
7
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1answer
1k views

Equivalent inner products on a Hilbert space

Take a Hilbert space $(\mathcal H,(\cdot,\cdot)_{\mathcal H})$ and two equivalent inner products $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$ on $\mathcal H$, i.e. such that there are $a,b \in \mathbb R$ ...
6
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1answer
981 views

Compactness of Multiplication Operator on $L^2$

Suppose we have an bounded linear operator A that operates from $L^2([a,b]) \mapsto L^2([a,b])$. Now suppose that $A(f)(t) = tf(t)$. Is A compact? Edit: I know $A = A^*$ but I'm not really sure ...
4
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1answer
254 views

How to characterize self-adjoint operators in terms of orthogonal diagonalizability

Have a look at the following excerpt of Tosio Kato (taken from Zeidler Applied functional analysis vol. I): The fundamental quality required of operators representing physical quantities in ...
3
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2answers
277 views

Is a complex space more “advanced” than a “generic” real space?

For instance, does taking the square root of a complex number and its complex conjugate create a metric that "automatically" makes it an inner product space? Is a complex space more complete than a ...
7
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2answers
750 views

Contexts For Bessel's Inequality?

Bessel's inequality appears to be about orthonormal sequences. But (in the context of inner product spaces), I've thought of this inequality as being a demonstration that the hypotenuse of triangles ...
4
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1answer
207 views

Set of all compact operators $K(H)$ is the unique ideal in $B(H)$?

I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
4
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0answers
143 views

When functions, analytically continued, carry over certain properties

Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
3
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0answers
263 views

How does the parallelogram law imply the existence of an inner product for a given norm? [duplicate]

Possible Duplicate: Norms Induced by Inner Products I am trying to prove to that if a norm of a vector space satisfies the parallelogram law ($\| \vec x + \vec y \|^2 + \| \vec x - \vec ...
1
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2answers
2k views

A proof of the Riesz representation theorem

I'm having trouble filling the steps in this guided proof of Riesz's representation theorem. (I already have a proof I can understand, but I'd like to understand this one too.) Let $H$ be a Hilbert ...
6
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1answer
244 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 ...
5
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1answer
280 views

Is this functional weakly continuous?

Take a $C^1$ function $G \colon \mathbb{R}\to \mathbb{R}$ and define a functional $$\mathcal{G}(u)=\int_0^1G(u(t))\, dt, \quad u \in H^1(0, 1).$$ We then have $\mathcal{G}\in C^1\big(H^1(0, 1)\to ...
7
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1answer
245 views

Existence of the Pettis integral

This is related to a question of MO: A question on the integral of Hilbert valued functions. I'm sure it's easy, but I cannot think right now, so I thought I'd ask. Let $f:[0,1]\rightarrow H$ be a ...
8
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2answers
468 views

Is compactness a stronger form of continuity?

Let $H$ be a Hilbert space. We say that a linear operator $T \colon H \to H$ is compact if it maps bounded sets to precompact ones, that is, if for every bounded sequence $(a_n)$ in $H$, $(Ta_n)$ has ...
4
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1answer
396 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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3answers
1k views

Canonical examples of inner product spaces that are not Hilbert spaces?

That is, what are some good examples of vector spaces which are inner product spaces but in which not every Cauchy sequence converges?
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1answer
379 views

Complete orthonormal sequence, Hilbert Space, Kronecker Delta

Let $H$ be a Hilbert space and $(e_n)_{n=1,2,\ldots}$ be a complete orthonormal sequence in $H$. We want to show that if $a_{np}=(e_n,f_p)$ then $\sum_{p=1}^{\infty}a_{np} ...
22
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2answers
3k views

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric ...
4
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2answers
389 views

Paradox or Error: On the inclusion of dense subspaces into Hilbert spaces

the following observations are very simple, but I suppose they contain an error, which I haven't been able to find it so far. Maybe somebody can help how to fix it: Let $H$ be a Hilbert space, $U$ be ...
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2answers
527 views

How do the solutions to the wave and heat equations converge in general?

I would like to check my understanding with someone if possible. When we cover the heat and wave equations, for instance, in "methods" courses at university, they normally restrict the initial ...
21
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3answers
744 views

If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$

I'm trying to prove the following: If $(a_n)$ is a sequence of positive numbers such that $\sum_{n=1}^\infty a_n b_n<\infty$ for all sequences of positive numbers $(b_n)$ such that ...
7
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1answer
933 views

Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
12
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1answer
643 views

Different versions of Riesz Theorems

In Wikipedia, there are three versions of Riesz theorems: 1 The Hilbert space representation theorem for the (continuous) dual space of a Hilbert space; 2 The representation theorem for ...