Elliptic Variational Inequalities - Computational Methods for Quantitative Finance - page 267

Appendix A
Elliptic Variational Inequalities

This appendix gives more information on elliptic variational inequalities. Some def-
initions and results from functional analysis are summarized in Sects. A.1 and A.2 .
Existence and uniqueness results are discussed in Sect. A.3 .

A.1 Hilbert Spaces

Let H be a vector space over R .An inner product (u, v) is a bilinear form from

H × H → R which is

symmetric: ∀ u, v ∈ H (u, v) = (v, u),

positive definite: ∀ u ∈ H (u, u) ≥ 0 , (u, u) = 0 ⇐⇒ u = 0 .

Each inner product ( · , · ) on H induces a norm on H via

(v, v) 2

v := ∀ v ∈ H ,

and satisfies the Cauchy-Schwarz inequality :

∀ u, v ∈ H :| (u, v) |≤ u v . (A.1)

Moreover, there holds the parallelogram law

2

2

u + v u − v 1

2 2

2

+ = u + v ∀ u, v ∈ H . (A.2)

2 2

Definition A.1.1 A vector space is a Hilbert space if it is endowed with an inner

product ( · , · ) and if it is complete with respect to the norm

H

u = (u, u) 2 .

A subset K ⊂ H is convex, if

∀ u, v ∈ K :{ λu + ( 1 − λ) v : 0 <λ< 1 }⊂ K . (A.3)