If the kernel of a seminorm $p$ is closed, is $p$ then continuous?

Question

Let $V$ be an infinite-dimensional normed $\mathbb{C}$-vector space.
Let $p:V\to\mathbb{R}$ be a seminorm on $V$, and let $K=\ker p= \{ v \in V : p(v)=0\}$ be its kernel.

I would like to know if $$K\text{ is closed }\Longrightarrow\:p\text{ is continuous}$$ is true.

Possible duplicate of $T$ is continuous if and only if $\ker T$ is closed — Javi, Apr 20 '18 at 10:52
Which topology is $p$ supposed to be continuous with respect to? — Theo Bendit, Apr 20 '18 at 10:57
@Javi a seminorm is not a linear functional so my question isn't duplicate — Matey Math, Apr 20 '18 at 11:00
@TheoBendit $p$ is continuous as map from $V$ to $\mathbb{R}$ — Matey Math, Apr 20 '18 at 11:23
@MateyMath A vector space $V$, without some kind of topology/norm doesn't have a topology by default. So, saying $p$ is continuous as a map from $V$ to $\mathbb{R}$ doesn't work without extra information, as continuous maps only exist between topological spaces. What is the topology on $V$? Is it induced by $p$ (in which case $p$ is necessarily continuous)? Is it the discrete topology? The indiscrete topology? Is there another norm? — Theo Bendit, Apr 20 '18 at 11:52
@TheoBendit ok thanks, now i understand your question, the topology on $V$ is induced by norm ($V$ is a normed space) — Matey Math, Apr 20 '18 at 12:06

score 2 · Accepted Answer · answered Apr 20 '18 at 12:24

2

The answer is "no", even if you strengthen the question to norms, rather than seminorms. Note that each norm necessarily has a trivial kernel, which is always closed under the norm on $V$, but due to the fact that norms need not be equivalent in infinite dimensions, this implies that norms will not necessarily be continuous with respect to each other.

answered Apr 20 '18 at 12:24

Theo Bendit

50,900

thanks for your answer – Matey Math Apr 20 '18 at 12:30

If the kernel of a seminorm $p$ is closed, is $p$ then continuous?

1 Answers1