Suppose $H$ is a Hilbert space (much of this still works if it's just a Banach space), $x\in H$, and $(x_n)$ a sequence in $H$. Does $x_n\to x$ strongly in H iff $x_n\to x$ as operators in the strong operator topology on $H^*$? (In terms of the inner product: does $\langle x_n,x_n\rangle \to \langle x,x\rangle$ iff $\langle x_n,y\rangle \to \langle x,y\rangle$ for all $y\in H$? )
It seems like using the Hahn-Banach Theorem will get the $\Leftarrow$ implication. Is there a nicer way to look at it?
The "strong operator topology on $H^*$" is just the same as the weak topology on $H$, so you're asking whether strong and weak convergence in $H$ is the same. No, it isn't. Standard example: an orthonormal sequence $u_n$ converges to $0$ weakly but does not converge strongly.