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.
 12 Fourier analysis.
 13 Sobolev spaces.
 A. Convolution and smoothing.
 B. Approximation by smooth functions.
 C. Extensions of Sobolev spaces.
 D. Traces of Sobolev spaces.
 E. Sobolev inequalities.
 F. Compact embedding of Sobolev spaces.
 G. Dual Sobolev spaces.
 H. Sobolev spaces involving time.
 I. Poincare inequality and Friedrich lemma.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Sobolev spaces.

ome of the notation was introduced in the section ( Function spaces section ). The references are [Evans] and [Brenner] . A fundamentally different tool of Sobolev space analysis may be found in the section ( Averaged Taylor polynomial ).

Definition

(Sobolev space) For a subset , we define where the is the multi-index defined in the section ( Function spaces section ) and the derivative is understood in the weak sense, see the definition ( Weak derivative ).

The norm for is defined by

The space is the closure of with respect to .

We introduce the notation

Proposition

The space equipped with the operation is a Banach space.

If then is a Hilbert space.

 A. Convolution and smoothing.
 B. Approximation by smooth functions.
 C. Extensions of Sobolev spaces.
 D. Traces of Sobolev spaces.
 E. Sobolev inequalities.
 F. Compact embedding of Sobolev spaces.
 G. Dual Sobolev spaces.
 H. Sobolev spaces involving time.
 I. Poincare inequality and Friedrich lemma.
 Notation. Index. Contents.