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Conditional probabilities in Banach spaces

This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?. Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space ...
Tom LaGatta's user avatar
  • 8,512
4 votes
3 answers
2k views

Algebraic Dual / Continuous Dual

Let $E$ be an infinite dimensional Banach space, let $E^{\ast}$ denote its continuous (i.e., Banach space) dual, and let $E'$ be its algebraic dual. Clearly, $E^{\ast}$ is a proper vector subspace of $...
Ady's user avatar
  • 4,060
11 votes
3 answers
1k views

Continuous automorphism groups of normed vector spaces?

Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
Jason Reed's user avatar
32 votes
11 answers
23k views

A book for problems in Functional Analysis

I want to know if there's any book that categorizes problems by subjects of Functional Analysis. I'm studying Functional Analysis now a days and I really need to solve some problems in order to ...
6 votes
0 answers
639 views

Hilbert subspaces of indefinite inner product spaces

Let $E$ be a real linear space, endowed with a non-degenerate symmetric bilinear form $(.,.)$. Suppose that the [indefinite] inner product space $(E,(.,.))$ satisfies the following [sequential] ...
Ady's user avatar
  • 4,060
16 votes
0 answers
1k views

Finite Rank Commutators

My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...
Bill Johnson's user avatar
  • 31.5k
17 votes
5 answers
3k views

Conditional probabilities are measurable functions - when are they continuous?

Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
Tom LaGatta's user avatar
  • 8,512
8 votes
2 answers
1k views

Example for an integral, rectifiable varifold with unbounded first variation

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation. Recapitulation for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable ...
Elgrimm's user avatar
  • 143
10 votes
0 answers
609 views

Asymptotic non-distortion of the separable Hilbert space

By the work of E. Odell and Th. Schlumprecht, we know that the separable Hilbert space $\ell_2$ is arbitrarily distortable. But I don't know if an "asymptotic" version of their result is true. To ...
Pandelis Dodos's user avatar
2 votes
3 answers
1k views

Baire category theorem

Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = \cup_{n=1}^\infty A_n$. Let $\bar{A}$ denote the closure of $A$. Let's call the following statement (2): ...
has2's user avatar
  • 498
5 votes
3 answers
1k views

Functional calculus for direct integrals

Suppose I have a direct integral of Hilbert spaces $H = \int^\oplus H_x dx $, and suppose I have an operator $T: H \to H$ which is decomposable, and so it can be written as $T = \int^\oplus T_x$ for ...
Łukasz Grabowski's user avatar
5 votes
0 answers
417 views

Direct integrals and fields of operators

Suppose we have a measure space $(X,\mu)$ and a measurable field of Hilbert spaces $H_x$ on it. We can form the direct integral ${\cal{H}} = \int H_x \ d \mu$, which is a Hilbert space. Suppose now ...
Łukasz Grabowski's user avatar
9 votes
2 answers
1k views

Borsuk pairs of Banach spaces

Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$ is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$ $X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$ $\in$ $X$ ;...
Ady's user avatar
  • 4,060
10 votes
1 answer
776 views

Saito-Wright definition of Rickart C*-algebras

A C*-algebra is Rickart if for each $x\in A$ there is a projection $p\in A$ so that $R(x)=pA$. Here the right-annihilator $R(S)$ of $S\subset A$ is defined as $$R(S)=\{a\in A\mid xa=0\, \forall x\...
Bas Spitters's user avatar
34 votes
8 answers
9k views

When is a Banach space a Hilbert space?

Let $\mathcal{X}$ be a real or complex Banach space. It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds. ...
Teiko Heinosaari's user avatar
6 votes
3 answers
1k views

How can I embed an N-points metric space to a hypercube with low distortion?

I have a N-point metric space defined by the pairwise distance matrix. I want to encode these N points with binary strings, i.e. each point will be mapped to a vertex in a hypercube. The lengths of ...
pacificmoth's user avatar
2 votes
1 answer
168 views

Local supporting points of Lipschitz functions

Let X be a separable reflexive Banach space and f:X\to\mathbb{R} be a Lipschitz function. Say that a point x in X is a local supporting point of f if there exist x^* in X^* and an open neighborhood U ...
Anonymous's user avatar
3 votes
2 answers
416 views

Which Banach spaces have categorical duals?

I was looking carefully at all the definitions, trying to understand exactly what was going on in this question on categorical duals in Banach spaces. It seems that in the category of Banach spaces ...
Andrew Stacey's user avatar
4 votes
2 answers
4k views

Compact Convex sets and Extreme Points

There are examples that show the set of extreme points of a compact convex subset of a locally convex topological vector space need not be closed when the real dimension of the space is at least 3. ...
Mike Hartglass's user avatar
7 votes
1 answer
570 views

Categorical duals in Banach spaces

Near the bottom of the nlab page for Banach space I see "To be described: duals (p+q=pq)". Are $(\mathbb{R}^n)_p$ and $(\mathbb{R}^n)_q$ dual objects in the closed symmetric monoidal category of ...
Reid Barton's user avatar
  • 25.2k
26 votes
3 answers
2k views

Universality of zeta- and L-functions

Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
M.G.'s user avatar
  • 7,127
9 votes
1 answer
996 views

Topological "Interpolation" ?

Let E be a normed space, and let $T$:E * $\rightarrow$ E * be a nonlinear operator. Suppose that : 1) $T$ is continuous from (E *, ||.||) to itself (i.e., it is norm-continuous). and 2) $T$ is ...
Ady's user avatar
  • 4,060
29 votes
15 answers
6k views

Important results that use infinite-dimensional manifolds?

Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn up? Important ...
7 votes
3 answers
2k views

What are some interesting sequences of functions for thinking about types of convergence?

I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
6 votes
1 answer
989 views

What is the "continuity" in "absolute continuity", in general?

The wikipedia article on absolute continuity gives a delta-epsilon definition for a measure $\mu$ defined on the Borel $\sigma$-algebra on the real line, with respect to the Lebesgue measure $\lambda$:...
kweinert's user avatar
  • 208
4 votes
3 answers
6k views

Advantages of a back-propagation neural network over other function approximation methods

Hello. Let's say I have a set of input vectors $I = \{\mathbf{x_1}, \dots, \mathbf{x_k}\} \subset \mathcal{R}^m$ and a set of output vectors $O = \{\mathbf{y_1}, \dots, \mathbf{y_k}\} \subset \...
Bruno Reis's user avatar
9 votes
4 answers
1k views

Boundedness of nonlinear continuous functionals

Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$. Is it true that there exists an infinite dimensional reflexive subspace $E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ? ...
Ady's user avatar
  • 4,060
6 votes
1 answer
427 views

Subspaces of $L^{2}$

[In what follows $0^{0}$= 1 by convention.] Is there some closed infinite dimensional linear subspace $F$ of $L^{2}(0,1)$ such that $\left\lvert f\right\rvert^{\left\lvert f\right\rvert}$ belongs to $...
Ady's user avatar
  • 4,060
19 votes
7 answers
2k views

Generalizations of "standard" calculus

We have the usual analogy between infinitesimal calculus (integrals and derivatives) and finite calculus (sums and forward differences), and also the generalization of infinitesimal calculus to ...
Zev Chonoles's user avatar
  • 6,792
2 votes
2 answers
317 views

Bibliography for topologies defined by a family of seminorms

Hello I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource. Thank you very much.
Learner's user avatar
  • 143
6 votes
2 answers
1k views

Definable collections of non measurable sets of reals

Is there a definable (in Zermelo Fraenkel set theory with choice) collection of non measurable sets of reals of size continuum? More verbosely: Is there a class A = {x: \phi(x)} such that ZFC proves "...
Ashutosh's user avatar
  • 9,631
5 votes
1 answer
514 views

Request for reference: Banach-type spaces as algebraic theories.

Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight ...
Andrew Stacey's user avatar
10 votes
1 answer
635 views

What's the nearest algebraic theory to inner product spaces?

Following the references to the accepted answer to Is the category of Banach spaces with contractions an algebraic theory? one discovers that there is an algebraic theory (infinitary) which is closely ...
Andrew Stacey's user avatar
20 votes
3 answers
4k views

What is the origin of the term "spectrum" in mathematics?

The use of the term "spectrum" to denote the prime ideals of a ring originates from the case that the ring is, say, $\mathbb{C}[T]$ where $T$ is a linear operator on a finite-dimensional vector space; ...
Qiaochu Yuan's user avatar
9 votes
3 answers
763 views

Approximating with translated Gaussians and low-frequency trig functions

Defining the translated Gaussians by $f_t(x)=\exp(-(x-t)^2)$ for $t,x\in\Bbb{R}$, we showed that the linear span of $\{f_t \mid 0 \le t < \epsilon\}$ is dense in $L^2(\Bbb{R})$, for any $\epsilon&...
Axel Boldt's user avatar
6 votes
1 answer
726 views

The "ultimate" indefinite inner product space

This can be considered as a relative of Splitting a space into positive and negative parts. Is there a real (non-trivial) vector space $V$, endowed with a nondegenerate symmetric bilinear pairing $\...
Ady's user avatar
  • 4,060
4 votes
1 answer
321 views

What functorial topologies are there on the space of linear maps between LCTVS?

Setup: we consider the category of locally convex topological vector spaces with morphisms as continuous linear maps. This time, I'm explicitly allowing the axiom of choice (or at least the Hahn-...
Andrew Stacey's user avatar
13 votes
4 answers
2k views

Is the category of Banach spaces with contractions an algebraic theory?

Consider the category of Banach spaces with contractions as morphisms (weak, so $\|T\| \le 1$). Is this an algebraic theory? I suspect that this is true. The "operations" will be weighted sums, ...
Andrew Stacey's user avatar
71 votes
2 answers
6k views

Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used?

I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, ...
Andrew Stacey's user avatar
5 votes
2 answers
765 views

Can we distinguish the algebraic and continuous duals of a Banach space without choice (or HBT)?

The algebraic dual of a normed vector space is the space of all linear functionals to the ground field (either $\mathbb{R}$ or $\mathbb{C}$ for this question). The continuous dual is the subspace of ...
Andrew Stacey's user avatar
3 votes
3 answers
2k views

Conditional expectation of convolution product equals..

Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra. Consider the conditional expectation ...
student1729's user avatar
9 votes
1 answer
708 views

Hilbert spaces are induced by a bilinear form. How about n-linear forms?

A Hilbert space is a complete vector space equipped with scalar product, i.e. a symmetric positive definite bilinear form. What if we replace 'bilinear' by 'n-linear'? One might wonder, whether the $...
shuhalo's user avatar
  • 5,327
2 votes
3 answers
946 views

How can I measure the Morse index in infinite dimensions?

Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
Theo Johnson-Freyd's user avatar
2 votes
4 answers
3k views

Splitting a space into positive and negative parts

Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
Theo Johnson-Freyd's user avatar
4 votes
3 answers
609 views

When is $A : C(X) \to C(Y)$ a composition operator?

A composition operator $C\_T : C(X) \to C(Y)$ with $T \in C(Y, X)$ is defined by $C\_T f := f \circ T, f \in C(X)$. I read in the book about Composition Operators by Singh and others that a ...
santker heboln's user avatar
15 votes
2 answers
2k views

What is a projective space?

Is there a "recognition principle" for projective spaces? What categories are there with projective spaces for objects? Background: Although the title is a nod to What is a metric space?, ...
Andrew Stacey's user avatar
21 votes
1 answer
1k views

Is Dependent Choice all we really need?

http://en.wikipedia.org/wiki/Axiom_of_dependent_choice Is DC sufficient for the understanding of objects that are countable in some suitable sense? For example, is DC sufficient for the full ...
Andre's user avatar
  • 1,199
11 votes
2 answers
862 views

Monotone Lipschitz embedding ?

In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0. Thus, for some constant L, there is a map K: X --> c_0 that satisfies the ...
Ady's user avatar
  • 4,060
4 votes
5 answers
3k views

Generalize Fourier transform to other basis than trigonometric function

The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions. In how far can this ...
shuhalo's user avatar
  • 5,327
45 votes
7 answers
16k views

What is an intuitive view of adjoints? (version 2: functional analysis)

After realising that I don't have an intuitive understanding of adjoint functors, I then realised that I don't have an intuitive understanding of adjoint linear transformations! Again, I can use 'em, ...
Andrew Stacey's user avatar