Are these two definitions of sublinear functionals equivalent?

Question

Let $E$ be a t.v.s. In Brezis's book Functional Analysis, a sublinear functional $p:E \to \mathbb R$ is defined as a map that satisfies

• $p(\lambda x)=\lambda p(x)$ for all $\lambda > 0$ and $x\in E$.
• $p(x+y) \leq p(x)+p(y)$ for al $x, y\in E$.

On other sources, the nonnegative homogeneity is imposed also $\lambda = 0$, i.e.,

• $p(\lambda x)=\lambda p(x)$ for all $\lambda \ge 0$ and $x\in E$.
• $p(x+y) \leq p(x)+p(y)$ for al $x, y\in E$.

Are they equivalent, i.e., Brezis's definition implies $p(0)=0$?

From subadditivity, we have $p(x) \le p(x)+p(0)$ and thus $p(0) \ge 0$. However, I'm unable to prove the converse, i.e., $p(0) \le 0$.

Answer 1

In $p(\lambda x)=\lambda p(x)$ put $x=0$. You get $p(0)=\lambda p(0)$ for every $\lambda >0$. If $\lambda \neq 1$ this gives $p(0)=0$.