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.

## Interpolation of Hilbert spaces.

et and are separable Hilbert spaces, , is dense in and the natural injection operator , is continuous ( topology is stronger: -convergence implies -convergence). We denote such relationship Consider the form Let be the set of such that the form is continuous in topology. is dense in with respect to topology. By denseness of in the form extends to . Therefore, by the proposition ( Riesz representation theorem ) for any there is an element such that The is a linear operator unbounded in topology. Furthermore, By , continuous functions of are well defined. In particular, the operator is well defined. We introduce

Proposition

(Interpolation inequality) Given we have

Proposition

(Interpolated continuity) Suppose we have the pairs and . For a linear operator we have

 Notation. Index. Contents.