Unbounded convex not containing a ray - example without using a basis

Question

I prove here that an unbounded convex in a finite dimensional space contains a ray. At the same place, I give an example of an unbounded convex not containing a ray in the case of an infinite dimensional space. But this last example uses a basis.

It is possible to provide a similar example in a classical normed vector space but without using a basis?

Answer 1

In $L^p(\mathbb R)$ or $\ell^p$ for $1 \le p \le \infty$, $\{f: 0 < f(x) \le |x| \ \text{for}\ x \ne 0 \}$.