Orthonormal sets

Definition 1: an orthogonal set \(M\) in an inner product space \(X\) is a subset \(M \subset X\) whose elements are pairwise orthogonal.

Pairwise orthogonality implies that \(x, y \in M: x \neq y \implies \langle x, y \rangle = 0\).

Definition 2: an orthonormal set \(M\) in an inner product space \(X\) is an orthogonal set in \(X\) whose elements have norm 1.

That is for all \(x, y \in M\):

\[ \langle x, y \rangle = \begin{cases}0 &\text{if } x \neq y, \\ 1 &\text{if } x = y.\end{cases} \]

Lemma 1: an orthonormal set is linearly independent.

Proof:

Will be added later.

In the case that an orthogonal or orthonormal set is countable it can be arranged in a sequence and call it can be called an orthogonal or orthonormal sequence.

Theorem 1: let \((e_n)_{n \in \mathbb{N}}\) be an orthonormal sequence in an inner product space \((X, \langle \cdot, \cdot \rangle)\), then

\[ \sum_{n=1}^\infty |\langle x, e_n \rangle|^2 \leq \|x\|^2, \]

for all \(x \in X\).

Proof:

Will be added later.

Theorem 1 is known as the Bessel inequality, and we have that \(|\langle x, e_n \rangle|\) are called the Fourier coefficients of \(x\) with respect to the orthonormal sequence \((e_n)_{n \in \mathbb{N}}\).

Orthonormalisation process

Let \((x_n)_{n \in \mathbb{N}}\) be a linearly independent sequence in an inner product space \((X, \langle \cdot, \cdot \rangle)\), then we can use the Gram-Schmidt process to determine the corresponding orthonormal sequence \((e_n)_{n \in \mathbb{N}}\).

Let \(e_1 = \frac{1}{\|x_1\|} x_1\) be the first step and let \(e_n = \frac{1}{\|v_n\|} v_n\) be the \(n\)th step with

\[ v_n = x_n - \sum_{k=1}^{n-1} \langle x_n, e_k \rangle e_k. \]

Properties

Proposition 1: let \((e_n)_{n \in \mathbb{N}}\) be an orthonormal sequence in a Hilbert space \((X, \langle \cdot, \cdot \rangle)\) and let \((\alpha_n)_{n \in \mathbb{N}}\) be a sequence in the field of \(X\), then

the series \(\sum_{n=1}^\infty \alpha_n e_n\) is convergent in \(X\) \(\iff\) \(\sum_{n=1}^\infty | \alpha_n|^2\) is convergent in \(X\).

if the series \(\sum_{n=1}^\infty \alpha_n e_n\) is convergent in \(X\) and \(s = \sum_{n=1}^\infty \alpha_n e_n\) then \(a_n = \langle s, e_n \rangle\).

the series \(\sum_{n=1}^\infty \alpha_n e_n = \sum_{n=1}^\infty \langle s, e_n \rangle e_n\) is convergent in \(X\) for all \(x \in X\).

Proof:

Will be added later.

Furthermore, we also have that.

Proposition 2: let \(M\) be an orthonormal set in an inner product space \((X, \langle \cdot, \cdot \rangle)\), then any \(x \in X\) can have at most countably many nonzero Fourier coefficients \(\langle x, e_k \rangle\) for \(e_k \in M\) over the uncountable index set \(k \in I\) of \(M\).

Proof:

Will be added later.