Erwin Kreyzsig's Introduction to Functional Analysis , Section 4.8- Problem2:
Let $X,Y$ be normed spaces, $ T\in B(X,Y)$ and $(x_n)$ a sequence in X. Prove that if $ \forall f\in X' \quad f(x_n) \rightarrow f(x_0)$, then $ \forall g \in Y' \quad g(Tx_n) \rightarrow g(Tx_0).$
I attempted as follows: Consider $$ |g(Tx_n) - g(Tx_0) | \leq ||g||.||T||.|| x_n-x_0|| $$ I thought to replace $g$ with $T, $ which would give me
$$ |g(Tx_n) - g(Tx_0) | =| T(gx_n) - T(gx_0)| \leq ||T||.|g(x_n)-g(x_0)| $$ and conclusion directly follows. But I could not assure that what I did is correct.