proof of existence and uniqueness of best approximations
Existence : Without loss of generality we can suppose (we could simply translate by the set ).
As is convex, , and therefore
So we see that
which means that when , i.e. is a Cauchy sequence.
Since is complete (http://planetmath.org/Complete), for some .
As its norm must be . But also
which shows that . We have thus proven the existence of best approximations (http://planetmath.org/BestApproximationInInnerProductSpaces).
Uniqueness : Suppose there were such that . Then, by the parallelogram law
If then we would have , which is contradiction since ( is convex).
Therefore , which proves the uniqueness of the .
|Title||proof of existence and uniqueness of best approximations|
|Date of creation||2013-03-22 17:32:22|
|Last modified on||2013-03-22 17:32:22|
|Last modified by||asteroid (17536)|