Let $V$ be a real Frechet space with directed seminorms $\{ \lVert \cdot \rVert_n \}$ and $W$ be a real Banach space with the norm $\lVert \cdot \rVert$.
Let $T : V \to W$ be a continuous linear mapping. Then, I know that there exists some $m \in \mathbb{N}$ and a constant $C>0$ such that
\begin{equation}
\lVert T(v) \rVert \leq C \lVert v \rVert_m
\end{equation}
for all $v \in V$. Now, let $C':=\sup_{\lVert v \rVert_m \neq 0 } \frac{\lVert T(v) \rVert}{\lVert v \rVert_m}$.
Then, I think for any "balanced" neighborhood $U$ of the origin in $V$, there exists a sequence $v_j$ such that $\frac{\lVert T(v_j) \rVert}{\lVert v_j \rVert_m} \to C'$ as $j \to \infty$.
But, I cannot find a way to justify my guess rigorously. Or do I need further conditions on $U$ in order to get such a sequence?
Could anyone pleae explain?
Best Answer
Just take any sequence $(w_j)$ such that $\frac{\lVert T(w_j) \rVert}{\lVert w_j \rVert_m} \to C'$ as $j \to \infty$ with $\|w_j\|_m \neq 0$. By continuity of scalar mulitplication there exist positive constants $c_j$ such that $v_j=c_jw_j \in U$ and this sequence $(v_j)$ does the job.