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.

## Fredholm alternative.

et and be Banach spaces and let be a Hilbert space.

Definition

The mapping is a "linear operator" if

The "range of " is the set The "null space of " is the set

A linear operator is "bounded" if

A linear operator is "adjoint" to if

An operator is "compact" if for any bounded sequence the sequence has a convergent subsequence.

For , the "resolvent set" of is the set The "spectrum" of is the complement of . The "point spectrum" is The is called "eigenvalue" and consists of "eigenvectors".

Proposition

If is compact then is compact.

Proposition

(Fredholm alternative). Let be a compact linear operator in Hilbert space . Then

1. is finite dimensional.

2. is closed.

3. .

4. .

5. .

 Notation. Index. Contents.