Hilbert Space Orthonormal Bases
1. Definition of ONB
Definition
Given a Hilbert space \(H\), a set \(\mathcal B\) of vectors in \(H\) is called an orthonormal basis when \(||e_i|| = 1\) for all \(e_i \in {\mathcal B}\), \(\left<e_i, e_j\right> = 0\) when \(e_i \neq e_j\) both belong to \(\mathcal B\), and \(H\) is the closure of all finite linear combinations of elements of \(\mathcal B\).
We will focus specifically on the situation when \(\mathcal B\) is finite or countably infinite.
2. Orthonormal Sets and Best Approximations
Proposition
Given any mutually orthogonal vectors \(v_1,\ldots,v_n\),
\[ ||v_1 + \cdots + v_n||^2 = ||v_1||^2 + \cdots ||v_n||^2. \]
Corollary
Given any vector \(x \in H\) and any orthonormal \(e_1,\ldots,e_n\),
\[{}\left| \left| x - \sum_{i=1}^n c_i e_i \right| \right|^2{}\]
\[{}= \sum_{i=1}^n | c_i - \left<x,e_i\right>|^2{}\]
\[{}+ \left| \left| x - \sum_{i=1}^n \left<x,e_i\right> e_i \right|\right|^2.{}\]
In particular, \(\sum_{i=1}^n \left<x,e_i\right> e_i\) is the unique linear combination of \(e_i\)'s which is closest in norm to \(x\).
Proof
3. Characterization and Existence of Bases
Theorem
Suppose \(\mathcal B = \{e_n\}_n\) is a countable orthonormal set. Then the following are equivalent:
- \(\mathcal B\) is a basis of \(H\).
- For each \(x \in H\),\[ x = \sum_{n} \left<x,e_n\right>e_n \]
- For each \(x \in H\),\[ ||x||^2 = \sum_{n} |\left<x,e_n\right>|^2. \]
(In each case, note that the sum is taken over \(\{1,\ldots,\# {\mathcal B}\}\) when \({\mathcal B}\) is finite and over the natural numbers when \(\mathcal B\) is infinite.)
Proof
Theorem
A Hilbert space \(H\) has a countable (finite or countably infinite) basis if and only if it has a countable dense subset (such a Hilbert space is called separable.)
Proof