Download LINEABILITY WITHIN PROBABILITY THEORY SETTINGS 1

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Vector space wikipedia, lookup

Bra–ket notation wikipedia, lookup

Dual space wikipedia, lookup

System of linear equations wikipedia, lookup

Linear algebra wikipedia, lookup

Basis (linear algebra) wikipedia, lookup

Homological algebra wikipedia, lookup

Category theory wikipedia, lookup

Representation theory wikipedia, lookup

Transcript
Manuscript
Click here to download Manuscript ultima.tex
Click here to view linked References
1
2
3
4
5
6
7
8
9
10
11
LINEABILITY WITHIN PROBABILITY THEORY
12
SETTINGS
13
14
J. ALBERTO CONEJERO, MAR FENOY, MARINA MURILLO-ARCILA,
15
AND JUAN B. SEOANE-SEPÚLVEDA
16
17
18
Abstract. The search of lineability consists on finding large vector
19
spaces of mathematical objects with special properties. Such examples
20
have arisen in the last years in a wide range of settings such as in real
21
and complex analysis, sequence spaces, linear dynamics, norm-attaining
22
functionals, zeros of polynomials in Banach spaces, Dirichlet series, and
23
non-convergent Fourier series, among others.
24
In this paper we present the novelty of linking this notion of lineabil25
ity to the area of Probability Theory by providing positive (and neg26
ative) results within the framework of martingales, random variables,
27
and certain stochastic processes.
28
29
30
31
1. Introduction
32
33
Since the beginning of the 21st century many authors have become inter34
ested in the study of linearity within non linear settings or, in other words,
35
the search for linear structures of mathematical objects enjoying certain spe36
cial or unexpected properties. Vector spaces and linear algebras are elegant
37
mathematical structures which, at first glance, seem to be “forbidden” to
38
families of “strange” objects. In other words, take a function with some
39
special or (as sometimes it is called) “pathological” property (for exam40
41
ple, the classical nowhere differentiable function, also known as Weierstrass’
42
monster). Coming up with a concrete example of such a function might
43
be difficult. In fact, it may seem so difficult that if you succeed, you think
44
that there cannot be too many functions of that kind. Probably one can45
not find infinite dimensional vector spaces or infinitely generated algebras
46
of such functions. This is, however, exactly what has been happening in the
47
last years in many fields of mathematics, from Linear Chaos to Real and
48
49
Complex Analysis [3, 5, 12], passing through Set Theory [13] and Linear and
50
Multilinear Algebra, or even Operator Theory [8, 10], Topology, Measure
51
Theory [5], and Abstract Algebra.
52
Recall that, as it nowadays is common terminology, a subset M of a topo53
logical vector space X is called lineable (respectively, spaceable) in X if there
54
exists an infinite dimensional linear space (respectively, infinite dimensional
55
56
2010 Mathematics Subject Classification. 46E10, 46E99, 60B11.
57
Key
words and phrases. lineability; spaceability; probability theory; random variable;
58
stochastic
process; martingale.
59
1
60
61
62
63
64
65