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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.
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.

Fredholm alternative.


et $X$ and $Y$ be Banach spaces and let $H$ be a Hilbert space.

Definition

The mapping $A:X\rightarrow Y$ is a "linear operator" if MATH

The "range of $A$ " is the set MATH The "null space of $A$ " is the set MATH

A linear operator is "bounded" if MATH

A linear operator $A^{\ast}$ is "adjoint" to $A$ if MATH

An operator $A$ is "compact" if for any bounded sequence MATH the sequence MATH has a convergent subsequence.

For $A:X\rightarrow X$ , the "resolvent set" of $A$ is the set MATH The "spectrum" MATH of $A$ is the complement of MATH . The "point spectrum" MATH is MATH The MATH is called "eigenvalue" and MATH consists of "eigenvectors".

Proposition

If $A:H\rightarrow H$ is compact then $A^{\ast}$ is compact.

Proposition

(Fredholm alternative). Let $A$ be a compact linear operator in Hilbert space $H$ . Then

1. MATH is finite dimensional.

2. MATH is closed.

3. MATH .

4. MATH .

5. MATH .





Notation. Index. Contents.


















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