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Complete (orthogonal system)

An orthogonal system with no nonzero vector orthogonal to every member of it.

An orthogonal system \{u_i\} in an inner product space is complete if no nonzero vector in the space is orthogonal to every u_i -- equivalently, the system leaves nothing out. Completeness of the trigonometric system \{1,\cos(nt),\sin(nt)\} is exactly the fact that makes Parseval's identity hold with no missing energy.

Examples

  • Dropping \sin(t) from the trigonometric system makes it incomplete: \sin(t) itself is then orthogonal to every remaining member.

See also

Hilbert space Energy (of a function)
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