Vectors in Normed Space Must Have Finite Length?

Question

I have assumed this to be the case, and consequently this is why one looks at convergent sequences of vectors in normed, Banach, and Hilbert spaces.

But, I've never seen this listed explicitly as an axiom of a normed space.

(I thought it might perhaps be inferred from a requirement for homogeneity of the norm,, i.e. $||\alpha.v|| = |\alpha|.||v||$ where $\alpha $ is (finite) scalar: but this still works if $||v||$ is infinite.)

Any feedback would be appreciated.

Answer 1

Looking at the definition of a norm on wikipedia (which matches the definitions I encountered in textbooks) (link here)

if $V$ is a vector space, then a norm is a function $\rho: V\to \mathbb R$ that satisfies certain properties

so it is quite explicitly stated that the norm of every vector is a real number, thus finite.