Sequence of the linear operators on Banach spaces

Question

Let $X$ and $Y$ be Banah spaces and let $(T_n)$ be a sequence of bounded linear operator from $X$ to $Y$. I need to prove that following statements are equivalent:

(a) Sequence $(||T_n||)$ is bounded

(b) Sequence $(||T_n(x)||)$ is bounded for each $x\in X$

(c) Sequence $(|f(T_n(x))|)$ is bounded for each $x\in X$ and for each linear functionals define on $Y$.

Answer 1

Hint: check the Uniform Boundedness Principle in any Functional Analysis textbook.