User:Mct mht/some links/d3

Suppose  =  for all x, y in H.

Define &phi;x(y) = . For a fixed x, |&phi;x(y)| &le; ||Ax||&middot;||y|| by Cauchy-Schwarz. So each functional &phi;x is bounded.

If the set {x} is bounded, then for a fixed y, by symmetry |&phi;x(y)| = || &le; ||x||&middot;||Ay||. Therefore the family of functionals {&phi;x} is pointwise bounded.

The above shows the assumptions of the uniform boundedness principle are satisfied. So there exists some constant C s.t.
 * &phi;x(y)| = || &le; C||y|| for all x (and y). By the conjugate-isometry given by Riesz representation, the set {Ax} is bounded, i.e. A maps bounded sets to bounded sets, i.e. A is bounded.