notes/docs/mathematics/functional-analysis/inner-product-spaces/orthonormal-sets.md

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.

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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.

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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

1. 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.
2. 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.
3. 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.

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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.

??? note "Proof:"

Will be added later.