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 be a real linear space. The norm is the mapping with following properties:

1. .

2. ., .

3. , .

The norm introduces the "convergence" concept: for a sequence , we state .

Definition

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

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 is a set in a Banach space . A subset is called "dense" in if

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

Definition

(Dual space) Let be a Banach space. The "dual space" is the collection equipped with the norm where the denotes the result of acting on .

Proposition

is a Banach space.

Definition

The Banach space is called "reflexive" if .

The sequence "converges weakly in " to if

Proposition

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

Proposition

(Weak compactness of bounded set). Let be a reflexive Banach space. Let be a sequence such that for some constant . Then there exists a subsequence and an element such that

 Notation. Index. Contents.