Orthonormal Vectors

Overview

An orthogonal system is a set of vectors {% \phi_i %} in a Hilbert space such that {% \langle \phi_i, \phi_j \rangle = 0 for \; i \neq j %} An orthonormal system is an orthogonal system where {% || \phi_i || = 1 %} for any i.

Definitions

• Orthogonal Complement - Given a subset {% X %} of a Hilbert Space {% H %}, the orthogonal complement of {% X %} in {% H %} is the set
  {% {u \in H : \langle u, x \rangle =0 \; for \; x \in X} %}
• Complete - an orthonormal system {% \Theta %} is complete if {% \langle x, \phi \rangle = 0 %} for all {% \phi \in \Theta %} implies that {% x=0 %}.
  • Theorem: For any orthonormal system {% \Theta %} within a Hilber space, there exists a complete orthonormal system which contains {% \Theta %}