Tagged Questions

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

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3
votes
2answers
188 views

Distance of functions defined on a Hilbert Space

In our Topology class, we touched on Hilbert spaces for a couple of weeks. I've been studying various problems around the topics we covered, and I came across this one on a list of supplemental ...
2
votes
1answer
410 views

Relation between two orthogonal projections in a Hilbert space

Let $\mathcal{H}$ be a Hilbert space and let $P$ and $Q$ be two orthogonal projections to closed subspaces $M$ and $N$ respectively. Prove that: If $PQ$ is an orthogonal projection then it's range ...
1
vote
0answers
132 views

Addition of subspaces of a Hilbert space

Let $ \mathcal{H} $ be a Hilbert space and $ M $ be a closed subspace, prove that $ M + x_0\mathbb{R} $ is a closed subspace. It was easy enough to prove that $ M + x_0\mathbb{R} $ is a vector ...
2
votes
0answers
55 views

Intersection of a descending sequence of closed convex bounded subsets in Hilbert space [duplicate]

Possible Duplicate: Nested sequence of sets in Hilbert space $\{A_n\}$ is a descending sequence of closed convex bounded subsets in Hilbert space. Why can't the intersection be empty? I'm ...
0
votes
1answer
231 views

Nested sequence of sets in Hilbert space [duplicate]

How can I prove that nested sequence of non-empty bounded closed convex sets in Hilbert space have nonempty intersection? I just don't know where to start. Thanks
4
votes
1answer
295 views

Bounded operator from a Hilbert space to $\ell^1$ is compact

Let $H$ be any Hilbert space. How can we prove that any bounded linear operator $T\colon H \to \ell^1$ is compact? If we use the fact that the space $\ell^1$ has Schur property (norm and weak ...
3
votes
1answer
126 views

A property of Hilbert sphere

Let $X$ be (Edit: a closed convex subset of ) the unit sphere $Y=\{x\in \ell^2: \|x\|=1\}$ in $\ell^2$ with the great circle (geodesic) metric. (Edit: Suppose the diameter of $X$ is less than ...
7
votes
3answers
319 views

For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...
9
votes
4answers
852 views

Measure on Hilbert Space

On $\mathbb{R}^n$, we of course have the usual Lebesgue meausre. In many ways, separable, infinite-dimesional Hilbert space is the most natural generalization of $\mathbb{R}^n$ to ...
4
votes
2answers
513 views

Orthonormal basis in Hilbert space

I an reading a book about functional analysis and there is one thing i really don't understand. Let $\mathcal{H}$ be a Hilbert space. And $U \subset \mathcal{H}$ a closed subspace. Is it possible to ...
0
votes
1answer
152 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
vote
1answer
118 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
votes
4answers
108 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
vote
1answer
178 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 ...
3
votes
1answer
118 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
votes
1answer
168 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
votes
2answers
320 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 ...
1
vote
0answers
135 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
votes
1answer
704 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
votes
1answer
105 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
votes
1answer
136 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
votes
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
votes
2answers
580 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
votes
1answer
512 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
votes
4answers
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
votes
1answer
451 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
votes
2answers
538 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
votes
1answer
115 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
votes
1answer
213 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 ...
15
votes
1answer
697 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
votes
1answer
342 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 ...
1
vote
2answers
497 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
votes
2answers
268 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
votes
1answer
380 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
votes
1answer
322 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
votes
2answers
479 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
votes
1answer
147 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
votes
1answer
661 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, ...
8
votes
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
votes
1answer
1k 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
votes
1answer
257 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
votes
2answers
285 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
votes
2answers
821 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
votes
1answer
225 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
votes
0answers
145 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
votes
0answers
265 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 ...
2
votes
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
votes
1answer
261 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
votes
1answer
285 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
votes
1answer
254 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 ...