Suppose <Ax , y> = <x, Ay> for all x, y in H.
Define φx(y) = <Ax, y>. For a fixed x, |φx(y)| ≤ ||Ax||·||y|| by Cauchy-Schwarz. So each functional φx is bounded.
If the set {x} is bounded, then for a fixed y, by symmetry |φx(y)| = |<x, Ay>| ≤ ||x||·||Ay||. Therefore the family of functionals {φx} is pointwise bounded.
The above shows the assumptions of the uniform boundedness principle are satisfied. So there exists some constant C s.t. |φx(y)| = |<Ax, y>| ≤ 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.