Let $X$, $Y$ be Banach spaces and $\{T_\alpha : X \to Y\}$ be a net of bounded linear map converging pointwise to a bounded linear operator $T : X \to Y$ and uniformly bounded in norm.
Then I remember vaguely that $T_\alpha$ converges to $T$ uniformly on bounded subsets of $X$. Is it correct? And how does one prove it? I think I have to use the linearity cleverly but cannot proceed.
Not true. If $H$ is Hilbert space with orthonormal basis $(e_n)$ then $T_n (\sum a_ie_i) = \sum\limits_{i=1}^{n} a_ie_i$ defines operators with $\|T_n|| \leq 1$ for all $n$. Also, $T_nx \to x$ for all $x$ but $\|T_n-I\|$ does not tend to $0$.
[Uniform convergence on bounded sets is same as convergence in operator norm. In this example $\|T_n-I\|=1$ for all $n$].