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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
11. Functional Analysis.
A. Weak convergence in Banach space.
B. Representation theorems in Hilbert space.
C. Fredholm alternative.
D. Spectrum of compact and symmetric operator.
E. Fixed point theorem.
F. Interpolation of Hilbert spaces.
G. Tensor product of Hilbert spaces.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Weak convergence in Banach space.


efinition

(Norm) Let $X$ be a real linear space. The norm MATH is the mapping MATH with following properties:

1. MATH .

2. MATH ., $\forall x,y\in X$ .

3. MATH , MATH .

The norm introduces the "convergence" concept: for a sequence MATH , $x_{k}\in X$ we state MATH .

Definition

(Complete normed space) The linear MATH -normed space is "complete" if every MATH -Cauchy sequence has a limit in $X$ .

Definition

(Banach space) Banach space is a linear space equipped with a norm and complete with respect to the convergence concept introduced by the norm.

Definition

(Separable and dense sets) Let $A$ is a set in a Banach space $X$ . A subset $B\subset A$ is called "dense" in $A$ if MATH

The set $A$ is "separable" if it includes a countable dense subset.

Definition

(Dual space) Let $X$ be a Banach space. The "dual space" $X^{\ast}$ is the collection MATH equipped with the norm MATH where the MATH denotes the result of MATH acting on $x\in X$ .

Proposition

$X^{\ast}$ is a Banach space.

Definition

The Banach space $X$ is called "reflexive" if $X^{\ast\ast}=X$ .

The sequence MATH "converges weakly in $X$ " to $x_{0}\in X$ if MATH

Proposition

Let $X$ be a Banach space. Any weakly in $X$ convergent sequence is bounded with respect to MATH .

Proposition

(Weak compactness of bounded set). Let $X$ be a reflexive Banach space. Let MATH be a sequence such that MATH for some constant $C$ . Then there exists a subsequence MATH and an element $x_{0}\in X$ such that MATH





Notation. Index. Contents.


















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