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.
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 8$. Orthonormal Sets of Functions