open and closed unit balls in topological vector space

Question

It is a famous fact that in any seminormed space $X$, the open unit ball $B$ and closed unit ball $C$, which are defined by $B=\{x\in X: \|x\|_{X}<1\}$ and $C=\{x\in X: \|x\|_{X}\leq 1\}$ respectively, satisfy that $\overline{B}={C}$, where $\overline{B}$ is the closure of $B$ in $X$.

On the other hand, assume that $X$ is a topological vector space that is not determined by a single seminorm only, so $X$ is not a seminormed space, still, a famous fact from the theory of topological vector spaces states that $X$ is determined by a family of seminorms $P=\{p_{i}: i\in I\}$. Now I am looking for some examples on such $X$ and $P$ for which

1) $\overline{B_{p_{i}}}=\overline{\{x\in X: p_{i}(x)<1\}}\nsubseteq C_{p_{i}}=\{x\in X: p_{i}(x)\leq 1\}$ for some $i\in I$.

2)$\overline{B_{p_{i}}}=\overline{\{x\in X: p_{i}(x)<1\}}\subset C_{p_{i}}=\{x\in X: p_{i}(x)\leq 1\}$ for some $i\in I$, where the inclusion is strict.

Can anyone here suggest any of those?

Answer 1

There are no examples of type (1), since $p_i$ is continuous, so $C_{p_i}$ is a closed set. There are no examples of type (2), since for any $x\in C_{p_i}$, $tx\in B_{p_i}$ for any $t\in [0,1)$, and $tx$ converges to $x$ as $t\to 1$.

Answer 2

You need the fact that the norm $p_{i}$ is continuous only in order to know that the set $C_{p_{i}}$ is closed - one uses the triangle inequality to find an open $p_{i}$-ball around every point of the complement to $C_{p_{i}}$.This proves the inclusion $\overline{B}_{p_{i}} \subseteq C_{p_{i}}$.

For the other inclusion, one does not in fact need it. For any $a \in C_{p_{i}}$, it is either in $B_{p_{i}}$ and we are finished, or we can assume $p_{i}(a) = 1$. Suppose that this is the case. Then simply pick any sequence $\{t_{n}\} \subseteq (0,1)$ such that $t_{n} \rightarrow 1$, and let $a_{n} := t_{n} \cdot a$. One has $p_{i}(a_{n}) = t_{n} \cdot p(a) < 1$, so $a_{n} \in B_{p_{i}}$.

As the scalar multiplication is a continuous map from $\mathbb{R} \times X$ to $X$, it follows that $\{ a_{n} \}_{n=1}^{\infty}$ converges to $a$. But this proves that $a \in \overline{B}_{p_{i}}$ and the converse inclusion $C_{p_{i}} \subseteq \overline{B}_{p_{i}}$ is proved.