Definition:Complete Orthonormal Set of Real Functions

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Definition

Let $I$ be an indexing set.

Let $S := \family {\map {\phi_i} x}_{i \mathop \in I}$ be an orthonormal set of real functions over the interval $\openint a b$.


Let $S$ have the property that:

$\forall n \in I: \ds \int_a^b \map \psi x \map {\phi_n} x \rd x = 0 \implies \map \psi x \equiv 0$

for any real function $\psi$ integrable over the interval $\openint a b$.


Then $S$ is a complete orthonormal set of real functions.


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