If $X$ and $Y$ are banach spaces and T map from $X$ to $Y$. If every sequence $x_n$ in weak topology in $X$ converges to $0$ , and $T(x_n)$ converges to $0$ in weak topology , does that imply that $T$ is bounded ?
From this question i would like to understand how convergence in weak topology differs from other topology. What would be the case if the topology defined was not weak ?
Thank you for your kind help . Keenly Looking forward to get some good ideas about weak topologies .