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Numerical Analysis
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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.

Tensor product of Hilbert spaces.


iven two Hilbert spaces $H_{1},H_{2}$ one can form pairs $\left( f,g\right) $ , $f\in H_{1}$ , $g\in H_{2}$ and finite linear combinations of such pairs MATH for any choice of MATH , MATH , MATH , MATH .

Definition

(Algebraic tensor product of Hilbert spaces) Let MATH then the quotient space MATH is called "(algebraic) tersor product" of spaces $H_{1}$ and $H_{2}$ . The equivalence class around $\left( f,g\right) $ is denoted MATH

Proposition

(Tensor product properties 1) Let MATH .

1. $H,\otimes$ have linear operations: MATH

2. $\forall f\in H$ MATH , MATH , MATH s.t. MATH .

3. The form MATH defined by MATH has properties of scalar product.

Definition

(Complete tensor product of Hilbert spaces) Completion of MATH with respect to MATH is called "(complete) tensor product" of $H_{1}$ and $H_{2}$ and denoted $H_{1}\otimes H_{2}$ .

Proposition

(Tensor product properties 2)

1. $H_{1}\otimes H_{2}$ is a Hilbert space.

2. If MATH is an (orthonormal) basis in $H_{1}$ and MATH is an (orthonormal) basis in $H_{2}$ then MATH is an (orthonormal) basis in $H_{1}\otimes H_{2}$ .

Definition

(Tensor product of bounded operators) Given two operators MATH and MATH we define the tensor product $A_{1}\otimes A_{2}$ according to the formula (notation of the proposition ( Tensor product properties 1 )-2): MATH

Proposition

(Tensor product of function spaces) If $H_{1}$ and $H_{2}$ are Hilbert spaces of functions on a domain $\Omega\subseteq$ $\QTR{cal}{R}$ then then there is a homeomorphism from $H_{1}\otimes H_{2}$ to a linear space of functions on $\Omega\times\Omega$ via the correspondence MATH





Notation. Index. Contents.


















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