Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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252 views

Openness of linear mapping

Let $X$ be a topological vector space over the field $K$, where $K=\mathbb R$ or $K= \mathbb C$, and let $f:X\rightarrow K^n$ ($n \in \mathbb N$) be a linear and surjective functional. How to prove ...
6
votes
1answer
619 views

Reference request: Fourier and Fourier-Stieltjes algebras

I would like to learn the basic theory of Fourier algebras and Fourier-Stieltjes algebras. In particular, I want to know how these two objects are defined in the case of not necessarily abelian ...
5
votes
1answer
242 views

Showing that $l^p(\mathbb{N})^* \cong l^q(\mathbb{N})$

I'm reading functional analysis in the summer, and have come to this exercise, asking to show that the two spaces $l^p(\mathbb{N})^*,l^q(\mathbb{N})$ are isomorphic, that is, by showing that every $l ...
2
votes
1answer
110 views

On the sets of injective/surjective linear mappings between Euclidean spaces

Denote by $\mathcal L'(\mathrm R^n,\mathrm R^m)$ and $\mathcal L_\prime (\mathrm R^n,\mathrm R^m)$ the subsets formed by the surjective and the injective mappings, respectively, of the normed ...
10
votes
1answer
2k views

Weak-to-weak continuous operator which is not norm-continuous

Can one give a "relatively easy" example of a linear mapping $T\colon X\to X$ ($X$ a Banach space) which is a) weak-to-weak continuous b) weak*-to-weak* continuous ($X=Y^*$) but not norm-to-norm ...
10
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1answer
869 views

Nonnegative linear functionals over $l^\infty$

My purpose is a clarification of the role of the axiom of choice in constructing limits for bounded sequences. Namely, we want a linear functional of norm 1 defined on the space of all bounded complex ...
37
votes
1answer
3k views

Was Grothendieck familiar with Stone's work on Boolean algebras?

In short, my question is: Was Grothendieck familiar with Stone's work on Boolean algebras? Background: In an answer to Pierre-Yves Gaillard's question Did Zariski really define the Zariski ...
4
votes
1answer
109 views

subset relations among Sobolev spaces and their duals

This may be a rather dense question, but I would nevertheless be grateful for some guidance. The question has to do with the Sobolev spaces $H^m (\Omega)$ on an open bounded domain $\Omega$ of ...
54
votes
4answers
3k views

Is it possible for a function to be in $L^p$ for only one $p$?

I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that ...
2
votes
1answer
98 views

Approximating by smooth functions on $W^{1,p}_\mathrm{loc}$?

I'm a little bit confused about different ways of approximating by smooth functions, in particular, quasiconformal mappings. So if a map $\phi : R\to R'$ is $K$-quasiconformal map on a relatively ...
6
votes
1answer
189 views

Projecting onto the diagonal given Banach spaces with unconditional bases

Let E be a Banach space with a 1-unconditional basis $(e_n)$ (for example, $\ell^p$). Then an operator T on E can be thought of as an infinite matrix, in the obvious way. Clearly each scalar on the ...
2
votes
1answer
105 views

Limit of polygonal arcs

Given a sequence of polygonal arcs $(f_n)_{n\in\mathbb{N}}$ that has limit $f$, is $f$ continuous? I believe it should be true as I cannot think of a counter example but not sure how to prove it. ...
16
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1answer
747 views

Medial Limit of Mokobodzki (case of Banach Limit)

A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...
32
votes
2answers
1k views

The identity cannot be a commutator in a Banach algebra?

The Wikipedia article on Banach algebras claims, without a proof or reference, that there does not exist a (unital) Banach algebra $B$ and elements $x, y \in B$ such that $xy - yx = 1$. This is ...
3
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1answer
233 views

Decomposing a Bounded Linear Functional on Lp as a difference of Positive Bounded Linear functionals

I am learning Measure theory via self study of Bartle "The elements of Integration and Lebesgue Measure". I was stumped by the reasoning in one of the decomposition proofs. The point is to show that a ...
5
votes
1answer
406 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, ...
8
votes
2answers
483 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
154 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, ...
7
votes
0answers
340 views

Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
2
votes
1answer
704 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, ...
2
votes
2answers
296 views

Karhunen-Loève / Mercer's theorem. What am I missing?

I'm looking at the eigenfunction expansion of Brownian motion on the interval [0,1]: $$W_t = \sqrt{2} \sum_{k=1}^\infty Z_k \frac{\sin((k - \frac{1}{2}) \pi t)}{(k - \frac{1}{2}) \pi}.$$ One deduces ...
6
votes
1answer
302 views

Hahn-Banach to extend to the Lebesgue Measure

I remember reading an example in a textbook that went something like this: if we take a function $\ell(f) = \int_{0}^{1}f(t)\, dt$, (with this being the Riemann integral) defined only on the set of ...
10
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1answer
430 views

Isomorphisms of Fréchet Spaces

What is the proper notion of an isomorphism between Fréchet spaces? Obviously it should be a linear map. I'm just worried about the analytic structure. Should one be able to order the seminorms on ...
8
votes
1answer
196 views

Abelian sub-C*-algebras

Given a non-abelian C*-algebra $A$. I am wondering what are the possible abelian sub-C*-algebras of $A$. Let $K$ be the spectrum of $A$. Does $A$ contain an isomorphic copy (as a Banach space) of the ...
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
0answers
202 views

Two “different” adjoints of exterior derivative on manifolds with boundary in the $L^2$-setting

The follow problem appears in the setting of $L^2$-differential forms on manifolds with boundary. An abstracted operator theoretic problem is given below. Suppose $M$ is a smooth Riemannian manifold ...
4
votes
1answer
199 views

How to make sense of this integral?

Let $L$ be the Ornstein-Uhlenbeck operator on $L^2(\gamma)$ where $\gamma$ is the Gaussian measure on $\mathbf R^d$. Hille-Yosida or Lumer-Phillips can be used to prove that $L$ generates a strongly ...
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 ...
6
votes
1answer
320 views

Uniqueness of the derivative in locally convex topological vector space

I need a hint of proof of uniqueness of the derivative in locally convex topological vector space (it's asserted in Lang's "Introduction to differentiable manifolds"). Define derivative of a function ...
3
votes
1answer
64 views

Is this class of functions called a seperating set?

Suppose I have a class of functions $\mathcal{F}$ with the property that $\int f(x) g(x) = \int f(x) h(x)$ for all $f \in \mathcal{F}$ implies $g = h$. What's the correct name for this property? If ...
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vote
0answers
99 views

Does a projection valued measure (PVM) induce a PVM on a generic subspace of the Hilbert space?

Let $E:{\cal B}(X) \to Pr({\cal H})$ be a projection valued measure (PVM), where ${\cal B}(X)$ is the Borel $\sigma$-algebra of a suitable topological space $X$ and $Pr({\cal H})$ is the set of ...
2
votes
0answers
102 views

What is the connection between Hilbert modules and tangent bundles in this paper?

A paper by Cipriani and Sauvageot, available at http://dx.doi.org/10.1016/S0022-1236(03)00085-5 shows that for many Dirichlet forms on $C^*$-algebras there is a derivation $\delta$ from the domain ...
6
votes
1answer
226 views

Difficulties in solving a PDE problem

This is an exercise in "Variation et optimisation des formes", chapter 3, Ex. 3.8. The preliminaries are: $$D=(0,1)^2,\ f \in L^2(D),\ x_{ij}=(i/n, j/n),\ 0<i,j<n,$$ $$\Omega_n = D\setminus ...
4
votes
1answer
541 views

What is a Form Domain of an Operator?

I tried to look this up on Wikipedia, but I couldn't find anything. I am reading Barry Simon's book "Schrödinger Operators", where he brings up the concept of a form domain $Q(A)$ of a ...
4
votes
1answer
400 views

Finding eigenfunctions and eigenvalues

Let $K$ be the integral operator defined by $$ (Kf)(x)=\int_0^1 u(x)v(y)f(y) dy $$ for some continuous functions $u,v$ in the Hilbert space with inner product $\langle f,g \rangle = \int_0^1 f(x)^* ...
2
votes
2answers
670 views

Clarification on Continuous Matrices and reference request

I have not been able to obtain a clear definition of what is currently called a 'continuous matrix' nor 'continuous matrix operators.' It is unclear if the definition would involve a matrix with ...
5
votes
0answers
340 views

Reference for the range of possible values in Hahn-Banach Theorem

This is the usual formulation of Hahn-Banach theorem (some books use sublinear function instead, but it probably does not make much difference): Let $X$ be a vector space and let $p:X\to{\mathbb R}$ ...
3
votes
1answer
298 views

Proof that $\Delta$ generates analytic semigroup

First off, I apologize for asking a question which I'm sure has been studied to death, but I can't seem to find an answer with google. I want to see a proof that the Laplace operator $\Delta$ with ...
3
votes
1answer
281 views

How to minimize this function difference

Sorry about this somewhat lengthy introduction to my question. I thought it might be useful to know what I'm trying to do. I decided that I would like to have sequence of polynomials in $\mathbb{P}_n ...
2
votes
2answers
283 views

A Min-Max-Theorem for self-adjoint operators

I got a small question concerning the proof of a min-max theorem for selfadjoint operators that I'm currently trying to understand. The article I'm refering to is ...
7
votes
1answer
508 views

How to find an integral kernel for poisson's equation in the upper half plane

In our lecture we have shown that $\forall f \in L^2(\mathbb{R}^n_+) $ there is a unique $ u $ in the Sobolev space $ H^2(\mathbb{R}^n_+) $ satisfying $ -\Delta u = f. $ Now in our exercise sheet we ...
6
votes
1answer
249 views

Capacity theory beginner resources

I'm currently studying a book on shape optimization: Variation et optimisation de formes: Une analyse géométrique By Antoine Henrot, Michel Pierre. The book introduces at some point capacity, and uses ...
3
votes
2answers
287 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
860 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 ...
7
votes
1answer
209 views

Fourier transform of function in $L^{4/3}$

Suppose $f \in L^{4/3}(\mathbb{R}^2)$ and denote its Fourier transform by $\mathscr{F}(f)$. Is it true that the function $g:\mathbb{R}^2 \rightarrow \mathbb{C}$ defined by ...
4
votes
1answer
421 views

Double dual of the space $C[0,1]$

The second dual or double dual of the space of all continuous functions on $[0,1]$, $C[0,1]$ is von Neumann algebra. Can anyone help me identifying this space?
4
votes
1answer
252 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?
2
votes
0answers
242 views

How to prove Campanato space is a Banach space

Let $p\ge1$, $\mu\ge0$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$. Campanato space embraces all $u$'s which ...
9
votes
1answer
356 views

Origin of the name 'test functions'

This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
5
votes
1answer
361 views

Nested sets convergence

Define $\xi\in C^1([-1,1]\times[-1,1])$ such that $$ \int\limits_{-1}^1 \xi(x,y)\,dy = 1 $$ for all $x\in[-1,1]$ and $\xi\geq 0$. Put $A_0 = [0,1]$ and $$A_{n+1} = \left\{x\in ...