proof of Riesz representation theorem
Existence - If we can just take and thereby have for all .
Suppose now , i.e. .
Recall that, since is continuous (http://planetmath.org/ContinuousMap), is a closed subspace of (continuity of implies that is closed in ). It then follows from the orthogonal decomposition theorem that
and as we can find such that .
It follows easily from the linearity of that for every we have
The theorem then follows by taking .
Uniqueness - Suppose there were such that for every
Then for every . Taking we obtain , which implies .
|Title||proof of Riesz representation theorem|
|Date of creation||2013-03-22 17:32:37|
|Last modified on||2013-03-22 17:32:37|
|Last modified by||asteroid (17536)|