Download Common Function spaces

INFORMATION ON MASTER’THESIS
1. Full name:
Vũ Thị Tuyển
2. Sex: Female
3. Date of birth: 08- 02-1982
4. Place of birth: Ha Tay
5. Admission decision number:
Dated
6. Changes in academic process:
7. Official thesis title:
Common Function spaces
8. Major:
Probability and statistics
9. Code: 604615
10. Supervisors: PGS. TS. Phan Viết Thư
11. Summary of the finding of the thesis:
In mathematics, the Lp spaces are function spaces defined using a natural
generalization of the p-norm for finite-dimensional vector spaces. They are
sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford &
Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki
1987)
they
were
first
introduced
by Frigyes
Riesz (Riesz
1910). Lp spaces form an important class of Banach spaces in functional
analysis, and of topological vector spaces. Lebesgue spaces have applications
in physics, statistics, finance, engineering, and other disciplines.
The essay is divided into three chapters:
- Chapter I: Basic concepts of functional analysis and operater.
- Chapter II: Common function spaces.
- Chapter III: Important convergence and
Chapter I present basic concepts of function: 𝜇 − measure, beyond measure
associated with 𝜇, complete measure, 𝜎 − limited measure.
Following, it is concept of measure space: separable measure space, dense
space,
Complete space, toward norm, localizable space. Tre definitions of integrable,
uniform integrability, compacted and weak compacted…
Chapter II presents function spaces. First is concept of ℒ 0 space. It is time
to give a name to a set of functions which has already been more than once,
for the space of real valued functions f defined on conegligible subsets of X
which are virtualy measurable. Write 𝐿0 for the set of equivalence classes in
ℒ 0 under “=𝑎.𝑒 ”. Then we show 𝐿0 has linear structure, order structure,
multiplicative structure and it is Archimedian spaces Dedekind 𝜎 −
complete. The ideas of this section are often applied to spaces based on
complex- valued functions instead of real – valued functions. Next
0
L
C
will be
space of equivalence. It is easy to describle addition and scalar multiplication
rendering
0
L
C
a linear space over ℂ.
While the space 𝐿0 treated in the previous section is of very gread intrinsic
interest, its chief use in the elementary theory is as a space in which some of
the most importand spaces of functional analysis are embedded. In the next
few sections I introduce these one at a time.
The first is the space 𝐿1 of equivalence classes of intergrable functions. The
importance of this space is not only that is offers a language in which to
express those many theorems about integrable functions which do not depend
on the differences between two functions which are equal almost everywhere.
It can also appear as the natural space in which to seek solutions to awide
variety of integral equations, and as the completion of a space of continuous
functions.
The second of the classical Banach spaces of measure theory which I treat is
the space 𝐿∞ . As will appear below, 𝐿∞ is the polar companion of 𝐿1 , the
linked opposite for ordinary measure spaces it is actually the dual of 𝐿1 .
Continuing with our tour of the classical Banach spaces, we come to the 𝐿𝑝
space for 1< p < ∞. 𝐿𝑝 is the order structured spaces, Riesz space, separable
spaces and Dedekind complete. In 𝐿𝑝 , 𝐿2 has the special property of being an
inner product spaces. 𝐿2 is completed, it is a real Hilbert space. The fact that it
may be identified with its own dual can off course be deduced from this.
Next, we describe the most useful relationships between this topology anf the
norm topology of the 𝐿𝑝 spaces. For 𝜎- finite spaces, it is metrizable anf
sequential convergence of sequences of functions.
The next topic is a fairly specialized one, but it is of gread importance, for
diffirent reasons, in both probability theory an functional analysis, and it
therefor seems worth while giving a proper treatment straight away. It is
uniformly integrable. I now come to the most striking feature of uniform
integrability: it provides a description of the relatively weakly compact
subsets of 𝐿1 . I have put this into a separate section. I will try to give an
account in terms which are acssessible to novices in the theory of normed
spaces because the result is essentially measure theotetic, as well as being of
vital importance to applications in in probability theory.
In probability, Convergence in measure is Convergence in probability , Weak
convergence is convergence in distribution. Convergence in probability
implies convergence in distribution. In the opposite direction, convergence in
distribution implies convergence in probability when the limiting random
variable X is a constant. Convergence in probability does not imply almost
sure convergence.
Date:15/12/2014
Signature:
Full name:Vũ Thị Tuyển.