(Hahn-Banach) Is a subadditive and positively homogeneous function continuous at $0$?

Question

Let $E$ be topological vector space, i.e., a real vector space over $\mathbb{R}$ such that all points are closed sets and $+,\cdot\,$ are continuous.

Let $p: E \to \mathbb{R}$ be a subadditive ($p(x+y) \leq p(x) + p(y)$) and positively homogeneus ($\forall \lambda \geq 0, p(\lambda x) = \lambda p(x))$ function.

I want to know whether $p$ is always continuous at $0$.

(I'm trying to prove that the function you get with one of the Hahn-Banach theorems is continuous.)

What I got so far is that it suffices to prove that if $x_\ell \to 0$ then $p(x_\ell) \to L$ for some $L \in \mathbb{R}$. Then, using positive homogeneity, I can prove that $L = 0$.

However, I suspect it's false. In fact, I'm trying to devise a counterexample in $\mathbb{R}^2$.

Answer 1

I think this is false: Take some infinite-dimensional normed vector space $V$ and a discontinuous linear functional $p$ on $V$.

As a more concrete example you could choose $V = c_e(\mathbb{R})$ the space of real-valued sequences than eventually become zero. This space has a basis $e^1, e^2, \dots$, where $e^1 = (1, 0, \dots), e^2 = (0, 1, 0, \dots)$, etc. One could then choose define $g$ as the linear extension of $g(e^n) = n$.

Answer 2

The statement is false, but I doubt you can find a counterexample in finite dimensional space. One can find a counterexample in any infinite dimensional Hilbert space. If $(\phi_n)_{n \in \mathbb{N}}$ is any orthonormal set of vectors in a Hilbert space $H$, then $\phi_n\to 0$ in weak topology as one can see from Bessel's inequality. However, $\|\phi_n \| = 1 \not\to 0$ as $n\to \infty$. Of course, the norm $\|\cdot\|$ on $H$ is a subadditive, positively homogeneous functional.