Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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7 votes
2 answers
2k views

Arbitrary union of meagre open sets

Let $X$ be a topological space. A subset $M$ of $X$ is called meagre (or of first category) if it is covered by the union of a countable family of closed subsets of $X$ with empty interior. Can you ...
6 votes
0 answers
321 views

Ricci-flat metrics on Cotangent bundles in adapted complex structure

greetings, Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...
8 votes
1 answer
640 views

Hahn-Banach theorem with real extended valued function

Hello to everyone, My problem is the following: I have this version of the Hahn-Banach theorem: Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any convex function. Let $W$ be a vector ...
4 votes
1 answer
414 views

Smooth functions tangent to the leaves of a foliation

Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space $$T_f C^\infty(M,N) = \...
2 votes
1 answer
235 views

BM-distances between $B(\ell_p^n)$ and $\ell^{n^2}_p$

Let $\ell_p^n$ be the $n$-dimensional real or complex $\ell_p$-space and let $\mathscr{B}(\ell_p^n)$ be the space of matrices on $\ell_p^n$ endowed with the operator norm. I am looking for any ...
3 votes
2 answers
1k views

Infinite dimensional manifold

In Hamiltonian mechanics, one essentially work with $\mathbb{R}^{2n}$. However, this is only a local description of our configuration manifold $M$. More precisely, the mechanical system is regarded as ...
5 votes
1 answer
755 views

Schwartz Kernel theorem for tempred functions

Let $T(R)$ denote the space of tempered functions on the line, i.e. the smooth functions that give Schwartz function after a multiplication by any Schwartz function, equipped with the natural nuclear ...
2 votes
1 answer
248 views

A question on Schwartz distributions

I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak* topology. Is is a Barreled space, say, a space whose convex, balanced, ...
7 votes
0 answers
284 views

Generalized Skorokhod spaces

Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
2 votes
0 answers
81 views

Description of the norm of certain interpolation space

Dear all, I suspect that there should be some detailed description of the norm (or of the unit ball) of the following complex interpolation space (for any $0< \theta < 1$): $$\Big(B(\ell_1^n, \...
1 vote
1 answer
355 views

semi group of contractions

Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative, and let $B$ is a monotone linear operator such that $D(A)\subset D(B)$. ...
11 votes
1 answer
608 views

Inequivalent complete norms and the axiom of choice

Hi, I've been wondering about the following : Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space? All the examples of inequivalent complete norms ...
2 votes
1 answer
394 views

Existence of a measure under certain condition

Hi everyone, my problem seems quite simple: I have a set $\Gamma$ along with a nice $\sigma$-algebra $\mathscr{B}$. Then I have a vector space of bounded measurable functions $A \subset \mathscr{B}_{\...
2 votes
1 answer
217 views

Subalgebras of $B(E)$

Let me fix an infinite-dimensional (complex) Banach space $E$. There is a cute result of Bracic and Kuzma which says that every maximal abelian subalgebra of $\mathscr{B}(E)$, the algebra of bounded ...
6 votes
2 answers
589 views

$\beta\mathbb{N}$ vs $\beta\mathbb{Z}$

Just started learning the Stone-Cech compactification of discrete groups this week. My motivation comes from a question on $\beta\mathbb{Z}$. Surprisingly, I realized there are muchhhh more literature ...
10 votes
0 answers
505 views

Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
11 votes
0 answers
743 views

A basic question on Stone-Cech compactification of $\mathbb{Z}$

Can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? ...
-2 votes
1 answer
1k views

holomorphic extension of a function [closed]

hi, I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{...
0 votes
1 answer
1k views

A continuous linear functional on $L^\infty(R)$ that vanishes on $C(R)$.

Intuitively, I want to construct the functional F in this way: $$F(f)=\lim_{x\rightarrow 0+}f(x)-\lim_{x\rightarrow 0-}f(x)$$ for $f\in L^\infty$. I know this is not well defined so I'd like to find ...
0 votes
1 answer
467 views

Discrete Sobolev space of $R^n$ valued maps

Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say $f:\Omega \...
14 votes
2 answers
2k views

What do we actually know about logarithmic energy ?

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
0 votes
1 answer
791 views

Can we construct a Hilbert space where the operator following differencial operator is symmetric?

I'd like to know if one can define a pertinent Hilbert space where the operator $$A_p v := -\frac{1}{2} v" + (vF + v\int_\mathbb{R} Sp + p\int_\mathbb{R} Sv )'$$ is symmetric. Here, $p$ satisfies ...
1 vote
2 answers
435 views

Analytical predicate for integers over complex numbers

A complex number $z$ is an integer if and only if $\sin(\pi z)=0$. It follows that a complex number $z$ is an integer if and only $\sin^2(\pi z) = 0$. So for a real analytic function $f$ and any real ...
4 votes
2 answers
467 views

Galois cover via C star algebras

Hello to all, here's my question, I hope it's not too trivial. I haven't found reference for it so far. We know that abelian C star algebras are the same as locally compact spaces. Now what is the ...
1 vote
1 answer
105 views

Log-nonexpansive functions: terminology and references

During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important. (Defn.). Let $h: (0,\infty) \to (0,\infty)$ be ...
1 vote
1 answer
332 views

Reflexive Besov spaces

I don't know whether the Besov space $B^1_{1,1}$ on a one dimensional torus is reflexive or not? Can someone help me please?
10 votes
1 answer
904 views

Non-probabilistic proof of the Johnson–Lindenstrauss lemma

The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...
16 votes
3 answers
760 views

Random products of projections: bounds on convergence rate?

The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
3 votes
1 answer
189 views

Sufficient (and concrete) condition for a function to satisfy some measure theoretic property

I'm interested in the following property, for a positive and locally bounded function $\omega:\mathbb{R}\to\mathbb{R}^d$, $d\ge 1$: there exists a countable sequence of open and pairwise disjoint sets ...
0 votes
1 answer
379 views

Double duals characteristic [closed]

Recall that (for $1\le p<\infty$), $\ell^p = \{\{a_n\}_{n=1}^\infty:\sum\limits_{i=1}^\infty|a_i|\lt\infty\}$, with norm $||\{a_n\}||=(\sum\limits_{i=1}^\infty|a_i|^p)^{\frac{1}{p}}$. It is well ...
1 vote
1 answer
198 views

tensor product with projective topology

There are two Banach spaces $X,Y$. These spaces have unconditional Schauder bases $\{e_i\}$ and $\{f_i\}$ respectively. Is this right that $e_i\otimes f_j$ is the unconditional Schauder basis in $X\...
6 votes
1 answer
712 views

left- and right- Folner sets

Given an amenable group, it is a standard trick to turn a left-invariant mean ( i.e. a continuous positive normalised linear functional $m:\ell_\infty(G) \to \mathbb{R}$ such that $\forall g \in G, m \...
1 vote
1 answer
1k views

Almost analytic continuation

Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
0 votes
1 answer
142 views

A special Integral Kernel

Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ?? $f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$ $K:L^2(\...
0 votes
0 answers
186 views

Properties of Eigenfunctions of a Kernel

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. I've and Kernel function $K(x,y)$ $f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$...
0 votes
0 answers
254 views

Convergence of a function in a metric space to its metric.

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric: If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...
2 votes
1 answer
257 views

Is C^{k+1}(X) compactly contained in C^{k}(X) for a closed manifold X?

Hi all, I apologize if this question is too low level for mathoverflow. I'm happy to move it to math.stackexchange if so. Let $X$ be a closed manifold, let $k$ be a nonnegative integer and let $C^k(...
2 votes
1 answer
391 views

The set of Upper semi-continuous functions as a ring.

I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set {$[a , b): a,b \in \mathbb{R} $} as it's base. If $X$ is a topological space, an upper ...
4 votes
1 answer
241 views

When is Prim(A) of an infinite discrete group hausdorff ?

Does anyone know, if the following result has been proved ? Let G be an infinite discrete group. A = L1(G) it's algebra and Prim(A) the set of prime ideals with spectral topology. The result is : ...
6 votes
1 answer
1k views

Helmholtz-Decomposition on compact Riemannian manifolds

For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that $L^...
0 votes
0 answers
162 views

Can we separate Toeplitz matrices for negative and positive eigenvalues?

Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues ...
1 vote
2 answers
629 views

A function which belongs on a concrete Besov Space

Please, anyone of you know a simple example of a function which belongs to the Besov Space with $p=q=\infty$ and $s=0$ (over $\mathbb{R}$ or $I\subset\mathbb{R}$ where $I$ is a closed interval). I ...
9 votes
0 answers
393 views

Does the algebra of bounded variation functions have a "noncommutative geometric" meaning and generalization?

According to Gelfand-Naimark theory, $C^*$-algebras of continuous functions $\mathcal{C}^0(X,\mathbb{C})$ on a compact Hausdorff topological space completely capture its topology. Furthermore, every ...
3 votes
1 answer
669 views

Is this kernel space of finite dimension ?

Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ...
2 votes
0 answers
258 views

Common eigenvector

I have little experience with functional analysis beyond an undergraduate basic course, and I'm dealing with the following problem: let $V$ be an infinite-dimensional locally convex (but not normed!) ...
26 votes
3 answers
7k views

Dual of bounded uniformly continuous functions

Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$? I ...
2 votes
0 answers
109 views

non-closed weak graph limit of symmetric operators

Hi Everyone, I was recently reading Reed & Simon's functional analysis textbook (the first volume), and it mentions casually on page 294 that weak graph limits of a sequence of symmetric ...
3 votes
2 answers
390 views

Convergence rate of an iterative process

I have the following iterative process $$a_n=a_{n-1}(1-\phi(a_{n-1})),\quad 0< a_0<1,$$ where $\phi(x)$ is a continuous increasing function, $\phi(0)=0$, and if $x\in(0,1)$ then $0< \phi(x)&...
2 votes
0 answers
175 views

Banach Algebras and the peripheral spectrum

Here is a little theorem that I'm trying to prove. I haven't seen it in literature before, but I think the applications will be quite useful, particularly in the context of Banach algebras. Denote ...
10 votes
1 answer
460 views

Which W*-algebras are the duals of C*-coalgebras?

A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...

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