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.

## Spectrum of compact and symmetric operator.

roposition

(Spectrum of compact operator) Let be a Hilbert space, and is a linear compact operator. Then

1. .

2.

3. Either is finite or it is a sequence converging to 0.

Proposition

(Eigenvalues of compact symmetric operator) Let be a separable Hilbert space and is a linear compact operator. Then there exists a countable orthonormal basis of consisting of eigenvectors of .

 Notation. Index. Contents.