Complete Orthonormal Basis and Complete Orthonormal Set
Theorem: Equivalence Conditions of an Orthonormal Set
Let $\left\{ \phi_{n} \right\}_{1}^\infty$ be an orthonormal set of $L^2(a,b)$ and denote $f \in L^2(a,b)$. Then, the following conditions are equivalent.
$(a)$ For all $n$, if $\left\langle f, \phi_{n} \right\rangle=0$ then $f=0$.
$(b)$ For all $f\in L^2(a,b)$, the series $\sum_{1}^\infty \left\langle f,\phi_{n}\right\rangle\phi_{n}$ converges to $f$ in the norm sense. That is, the following equation holds:
$$ f=\sum_{1}^\infty \left\langle f,\phi_{n}\right\rangle\phi_{n} $$
$(c)$ For all $f \in L^2(a,b)$, it satisfies the following equation known as Parseval’s equation:
$$ \| f \|^2 = \sum \limits_{n=1}^{\infty} \left| \left\langle f,\phi_{n} \right\rangle \right|^{2} $$
Explanation
The orthonormal set satisfying $(a) - (c)$ is called an orthonormal basis or a complete orthonormal set.
Observing these three conditions reveals that the orthonormal basis serves a role equivalent to a basis in finite-dimensional vector spaces.
When $\left\{ \phi_{n} \right\}$ is an orthonormal basis, the constants $\left\langle f, \phi_{n}\right\rangle$ are called (generalized) Fourier coefficients.
The series $\sum \left\langle f, \phi_{n}\right\rangle\phi_{n}$ is referred to as a (generalized) Fourier series.
Lemma
Assume $f \in L^2(a,b)$ and that $\left\{ \phi_{n} \right\}$ is an orthonormal set in $L^2(a,b)$. Then the series $\sum \left\langle f,\phi_{n} \right\rangle\phi_{n}$ converges in the norm sense. And it satisfies the following inequality:
$$ \left\| \sum \left\langle f,\phi_{n}\right\rangle \phi_{n} \right\| \le | f| $$
Proof
$(a) \implies (b)$
Assume $(a)$. Then, by the lemma, $\sum \left\langle f, \phi_{n} \right\rangle\phi_{n}$ converges in the norm sense. Let’s define the difference of the series as $g$.
$$ g=f-\sum \limits_{n=1}^{\infty} \left\langle f, \phi_{n} \right\rangle\phi_{n} $$
Then, it can be shown that $g=0$.
$$ \begin{align*} \left\langle g,\phi_{m} \right\rangle &=\ \left\langle f,\phi_{m}\right\rangle - \sum \limits_{n=1}^{\infty}\left\langle f,\phi_{n} \right\rangle \left\langle \phi_{n}, \phi_{m} \right\rangle \\ &=\ \left\langle f,\phi_{m}\right\rangle - \left\langle f,\phi_{m}\right\rangle \\ &=\ 0 \end{align*} $$
Therefore, by assumption, $g=0$. Thus, $f= \sum_{n=1}^\infty \left\langle f, \phi_{n} \right\rangle\phi_{n}$
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$(b) \implies (c)$
Assume $(b)$. Then, since $f=\sum_{1}^\infty \left\langle f, \phi_{n}\right\rangle\phi_{n}$,
$$ \begin{align*} \| f \|^2 &=\ \left\| \sum \limits_{n=1}^{\infty} \left\langle f, \phi_{n} \right\rangle \phi_{n} \right\| ^2 \\ &= \left\| \lim \limits_{N \rightarrow \infty} \sum \limits_{n=1} ^{N} \left\langle f, \phi_{n} \right\rangle\phi_{n} \right\| ^2 \\ &= \lim \limits_{N \rightarrow \infty} \left\| \sum \limits_{n=1} ^{N} \left\langle f, \phi_{n} \right\rangle\phi_{n} \right\| ^ 2 \\ &= \lim \limits_{N \rightarrow \infty} \sum _{n=1}^{N} | \left\langle f,\phi_{n} \right\rangle |^2 \\ &= \sum \limits _{n=1} ^{\infty} | \left\langle f, \phi_{n} \right\rangle |^2 \end{align*} $$
The third equation holds because the series converges in the norm sense by assumption. The fourth equation is valid due to the Pythagorean theorem.
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$(c) \implies (a)$
Assume $(c)$. Then,
$$ \| f \|^2 =\sum \limits _{n=1} ^{\infty}\left| \left\langle f,\phi_{n} \right\rangle \right|^{2} $$
Therefore, for all $n$, if $\left\langle f, \phi_{n} \right\rangle=0$, then $f=0$ is true.
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