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.

## Representation theorems in Hilbert space.

efinition

Let be a real linear space. The mapping is called a "scalar product" if it has the following properties:

1. ,

2. The mapping is linear for each ,

3.

4. .

Definition

The real linear space equipped with a scalar product is called "Hilbert space" if it is a Banach space with respect to the norm .

Proposition

(Riesz representation theorem). For each there exists a such that

Proposition

(Lax-Milgram theorem). Let be a Hilbert space and be a bilinear mapping: for some constants . Fix an . There exists a unique such that

 Notation. Index. Contents.