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.

## Dual Sobolev spaces.

et be a bounded subset of .

Definition

We denote the dual space of (with respect to the topology, see the definition ( Dual space )). where represents the action of on .

Remark

It will be clear from the proposition ( Embedding of dual Sobolev space ) of this section that the operation is the extension of the scalar product from to .

Proposition

(Representation of dual Sobolev space) For an there exist functions such that

Proof

According to the proposition ( Riesz representation theorem ), for there exists a such that We set We calculate

Proposition

(Embedding of dual Sobolev space) For a bounded domain we have where the last embedding is understood as

Proof

The first embedding is trivial. The equality follows from the proposition ( Riesz representation theorem ) and being a Hilbert space. For the last embedding observe that thus

 Notation. Index. Contents.