Complete system of functions

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An orthonormal system of functions $\{\phi(x)\}$ in some Hilbert space $H$ such that in $H$ there is no function orthogonal to all the functions in that family. A system of functions that is complete in one space may be incomplete in another. For example, the system

$$\left\lbrace\sqrt\frac2\pi\cos nx\right\rbrace,\quad n=0,1,\ldots,$$

forms a complete system of functions in the space $L[0,\pi]$ but does not form a complete system in the space $L[-\pi,\pi]$.

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Complete system of functions. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article