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.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Stable space splittings.

ondition

(Space splitting setup) Let be a separable Hilbert space. Let be a symmetric linear operator with the properties for some constants and any . We use the notation and treat the form as the principal norm on used for definition of closeness, duality and adjointness.

Let be an at most countable collection of Hilbert spaces. Let be a "Hilbert sum" of the collection :

Let be a collection of bounded linear operators . We define the operator :

We assume that is surjective:

Definition

(Stable splitting) In context of the condition ( Space splitting setup ) we say that the collection is a "stable splitting" of if there exist two constants such that where The quantities are called "lower and upper stability constants" and is "condition number" of the splitting.

Proposition

(Boundedness of surjection) In context of the condition ( Space splitting setup ) and for a stable splitting, the operator is bounded and

Proof

For any we estimate

Proposition

(Adjoint of surjection) In context of the condition ( Space splitting setup ), the adjoint operator acts where the the adjoint operator of defined by

Proof

By definition of adjointness thus , , , and, by definitions of and ,

Proposition

(Boundedness of adjoint of surjection) In context of the condition ( Space splitting setup ) and for a stable splitting, the operator is bounded and

Proof

The statement is a consequence of boundedness of , see the proposition ( Boundedness of surjection ).

Definition

(Schwarz operator) For a stable splitting we define the bounded operators The operator is called "Schwarz operator" and the operator is called "extended Schwarz operator".

Proposition

(Form of Schwarz operator) In context of the condition ( Space splitting setup ),

Proof

According to the definition of within the condition ( Space splitting setup ) and according to the proposition ( Adjoint of surjection ), We put these together:

Proposition

(Properties of Schwarz operator) If is a stable splitting (see the definition ( Stable splitting )) then the associated Schwarz operator is a symmetric positive definite operator: Thus, there exists and

Proof

The symmetry follows from the proposition ( Form of Schwarz operator ) and generic properties of taking adjoint:

For any we introduce a class . The class is never empty by surjectivity of , see the condition ( Space splitting setup ). Then, for any and we calculate where the equality takes place for and . We obtained or with equality reached at . Thus Next, we establish the upper bound for . We calculate, using the stability of splitting and , thus or

Next, we establish the lower bound for . We use again the surjectivity of , see the condition ( Space splitting setup ), to introduce a non empty class . Then, for any and any At this point we take infimum with respect to and use the definition of , We use stability of splitting, Thus or Therefore, is invertible. The formula now follows from .

 Notation. Index. Contents.