Introduction to Computational Linear Algebra

Sparse Systems to Solve Poisson Differential Equations 217

Appendix: Variational Formulations in Sobolev Spaces

For more details the reader is referred to [13], Chapter 5, p. 131.

Hilbert spaces. Orthonormal bases. Parseval’s equality

Let H be a vector space, on which one defines a scalar product (u, v), i.e., a

bilinear form from H × H → R, that is symmetric and positive definite, i.e.,

(i) (u, v) = (v, u) and (ii) (u, u) ≥ 0, with (u, u) > 0 iff u 6= 0.

The scalar product induces a norm on H:

||u|| = ||u||

H

= (u, u)

1/2

,

that verifies:

|(u, v)| ≤ ||u||.||v|| (Schwarz inequality) (7.62)

||u + v|| ≤ ||u|| + ||v|| (7.63)

It also verifies the “parallelogram” identit ...