Continuity of a map to a Frechet space

Question

Let $(A,\| \cdot \|)$ be a normed space and $B$ be a Frechet space equipped with a family $\{ p_k \}_{k \in \mathbb{N}}$ of seminorms.

Let $\phi: A \to B$ be a linear transformation satisfying the following: There is a $C > 0$ such that for all $k$ $$ p_k(\phi(u)) \leq C\| u \|.$$ Does it follow that $\phi$ is continuous on $A$? How about uniformly continuous on $A$?

Answer 1

The answers follow from the standard fact that $B$ has the metric $d(x,y) = \sum_{k=1}^\infty c_kp_k(x,y)/[1+p_k(x,y)]$ where $c_k$ is any sequence of scalars such that $\sum c_k < \infty$. E.g. $c_k=1/2^k$.