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.

## Tensor product of Hilbert spaces. iven two Hilbert spaces one can form pairs , , and finite linear combinations of such pairs for any choice of , , , .

Definition

(Algebraic tensor product of Hilbert spaces) Let then the quotient space is called "(algebraic) tersor product" of spaces and . The equivalence class around is denoted Proposition

(Tensor product properties 1) Let .

1. have linear operations: 2.  , , s.t. .

3. The form defined by has properties of scalar product.

Definition

(Complete tensor product of Hilbert spaces) Completion of with respect to is called "(complete) tensor product" of and and denoted .

Proposition

(Tensor product properties 2)

1. is a Hilbert space.

2. If is an (orthonormal) basis in and is an (orthonormal) basis in then is an (orthonormal) basis in .

Definition

(Tensor product of bounded operators) Given two operators and we define the tensor product according to the formula (notation of the proposition ( Tensor product properties 1 )-2): Proposition

(Tensor product of function spaces) If and are Hilbert spaces of functions on a domain  then then there is a homeomorphism from to a linear space of functions on via the correspondence Notation. Index. Contents.