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 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 A. Operations on sets and logical statements.
 B. Fundamental inequalities.
 a. Cauchy and Young inequalities.
 b. Cauchy inequality for scalar product.
 c. Holder inequality.
 d. Lp interpolation.
 e. Chebyshev inequality.
 f. Lyapunov inequality.
 g. Jensen inequality.
 h. Estimate of mean by probability series.
 i. Gronwall inequality.
 C. Function spaces.
 D. Measure theory.
 E. Various types of convergence.
 F. Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.
 G. Lebesgue differentiation theorem.
 H. Fubini theorem.
 I. Arzela-Ascoli compactness theorem.
 J. Partial ordering and maximal principle.
 K. Taylor decomposition.
 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.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Cauchy and Young inequalities.

roposition

Let are any real numbers and are real numbers connected by the relationship . Then

 (Cauchy inequality)
 (Cauchy inequality with epsilon)
 (Young inequality)

Proof

We use the last result:

 Notation. Index. Contents.