Basic Formula
If B is an orthogonal basis of H, then every element x of H may be written as
When B is orthonormal, we have instead
and the norm of x can be given by
Even if B is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x, and the formula is usually known as Parseval's identity. See also Generalized Fourier series.
If B is an orthonormal basis of H, then H is isomorphic to ℓ 2(B) in the following sense: there exists a bijective linear map Φ : H → ℓ 2(B) such that
for all x and y in H.
Read more about this topic: Orthonormal Basis
Famous quotes containing the words basic and/or formula:
“Our basic ideas about how to parent are encrusted with deeply felt emotions and many myths. One of the myths of parenting is that it is always fun and games, joy and delight. Everyone who has been a parent will testify that it is also anxiety, strife, frustration, and even hostility. Thus most major parenting- education formats deal with parental emotions and attitudes and, to a greater or lesser extent, advocate that the emotional component is more important than the knowledge.”
—Bettye M. Caldwell (20th century)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)