Let A be a linear operator on a Hilbert space X. Suppose that D(A) = X,
and that (Ax, y) = (x, Ay) for all x, y in H. Show that A is bounded.
The Attempt at a Solution
I've tried to prove it by using the fact that if A is continuous at
a point x implies that A is bounded.
Suppose that x_n converges to x.
|| Ax_n - Ax ||^2 = || A(x_n - x) ||^2 = ( A(x_n - x), A(x_n - x) ) =
(x_n - x, A^2 (x_n - x) ) <= ||x_n - x || || A^2(x_n - x) ||
But i don't think I can conclude from this that because ||x_n - x|| -> 0, this expression goes to zero, since || A^2(x_n - x) || may blow up.. Or doesn't it?
Please help me :)