Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?

Question

Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ? (Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is not normable ?)

I have now looked through several books on the subject but nowhere is something like this mentioned and I also wasn't able to find a way to construct such norm (or to find a counterexample).

Answer 1

Pick a basis $B$ (in the algebraic sense, also known as a Hamel basis), so any vector can be uniquely written as $\sum_{b\in B}\lambda_b b$, with only finitely many of the $\lambda_b$ being nonzero. Define for instance $$\left \|\sum_{b\in B}\lambda_b b\right \| := \max _{b\in B} |\lambda_b|$$ (another possibility would be $\sum_{b\in B} |\lambda_b|$ instead of taking the maximum).

Answer 2

Try books on the topic of "topological vector spaces": It is a theorem that every finite dimensional real or complex vector space has a norm, and that all norms are equivalent.

Correspondingly, there are infinite dimensional topological vector spaces that don't have a norm that induces the topology.

Canonical literature:

• François Treves: "Topological Vector Spaces, Distributions and Kernels"

• H.H. Schaefer, M.P. Wolff : "Topological Vector Spaces"