I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 1 Calculational Linear Algebra.
 2 Wavelet Analysis.
 3 Finite element method.
 4 Construction of approximation spaces.
 A. Finite element.
 B. Averaged Taylor polynomial.
 C. Stable space splittings.
 D. Frames.
 E. Tensor product splitting.
 F. Sparse tensor product. Cure for curse of dimensionality.
 a. Definition of sparse tensor product.
 b. Wavelet estimates in Sobolev spaces.
 c. Stability of wavelet splitting.
 d. Stable splitting for tensor product of Sobolev spaces.
 e. Approximation by sparse tensor product.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Definition of sparse tensor product.

efinition

(Sparse tensor product) In context of the condition ( Sparse tensor product setup ), the sparse tensor product space is the space See the section ( Tensor product of Hilbert spaces ) for the notation .

In the span we take all possible integer sets such that and and then for each we take all possible integer sets such that . For each pair of combinations and we form a tensor product and take the linear span of such tensor products.

We proceed to calculate dimensionality of . In the following means . The constant might change its meaning in process of calculation. For we have We assume that for we have and show that in dimensions we have We calculate Note that is the size of the mesh , see the section ( Wavelet analysis ). Thus, .

Proposition

(Dimension of tensor product) Let be the size of the one-dimensional mesh (see the context ( Sparse tensor product setup )) then as .

 Notation. Index. Contents.