Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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.

Representation theorems in Hilbert space.


efinition

Let $H$ be a real linear space. The mapping MATH is called a "scalar product" if it has the following properties:

1. MATH ,

2. The mapping MATH is linear for each $v\in H$ ,

3. MATH

4. MATH .

Definition

The real linear space $H$ equipped with a scalar product MATH is called "Hilbert space" if it is a Banach space with respect to the norm MATH .

Proposition

(Riesz representation theorem). For each MATH there exists a $v\in H$ such that MATH

Proposition

(Lax-Milgram theorem). Let $H$ be a Hilbert space and $B$ be a bilinear mapping: MATH for some constants $\alpha,\beta>0$ . Fix an $f\in H$ . There exists a unique $u\in H$ such that MATH





Notation. Index. Contents.


















Copyright 2007