proof of Riesz representation theorem for separable Hilbert spaces
Simplifying, . Hence converges to an element in .
For every basis element, . By linearity, it will also be true that
Any vector in the Hilbert space can be written as the limit of a sequence of finite superpositions of basis vectors hence, by continuity,
It is easy to see that is unique. Suppose there existed two vectors and such that . Then for all vectors . But then, which is only possible if , i.e. if .
|Title||proof of Riesz representation theorem for separable Hilbert spaces|
|Date of creation||2013-03-22 14:34:20|
|Last modified on||2013-03-22 14:34:20|
|Last modified by||asteroid (17536)|