What are equivalent to a sequence of normed bounded linear operators? 
If $X$ and $Y$ are Banach spaces and $T_n\in B(X,Y)$, $n=1, 2, ...$, show that equivalent statements are:
(a) $(||T_n||)$ is bounded
(b) $(||T_nx||)$ is bounded for all $x\in X$
(c) $(|g(T_nx)|)$ is bounded for all $x\in X$ and all $g\in Y'$

All I have got is (b)$\rightarrow$(a) by the Uniform Boundedness Theorem. Any hints or solutions would be helpful.

(a)->(c): for each $g\in Y'$ and $x\in X$ we have $$\forall n\in \mathbb{N}: |g(T_n x)|\leq ||g||\cdot ||T_n x||\leq ||g||\cdot ||T_n||\cdot ||x||\leq ||g||\cdot C\cdot ||x||,$$ if $||T_n||\leq C$ for each $n$.
(c)->(b): this is in fact again the uniform boundedness theorem, applied to the family $\{\iota(T_n x): x\in X, n\in\mathbb{N}\}$, where $\iota$ denotes the canonical isometric embedding $Y\to Y''$, i.e. $\iota(y)(g)=g(y)$. (c) means that this family is pointwise bounded, therefore uniformly bounded. Since $\iota$ is isometric, we obtain (b). 
Hope that helps!
EDIT: My argument from (c)->(b) can be generalized to obtain that a set $A\subseteq Y$ is norm-bounded iff $g(A)$ is a bounded subset of the complex field for each $g\in Y'$. This usually is denoted by the phrase "strongly bounded iff weakly bounded". 
