TITLE: Can all vector spaces be made into normed spaces?
QUESTION [5 upvotes]: Can all vector spaces be made into normed spaces (even trivial ones)? Vectorspace could be of infinite dimension. 
Update: I don't know how to make this question more specific. I am talking about a very general vectorspace. It can be any kind of vectorspace (finite dimensional, infinite dimensional) with any kind of underlying field ($\mathbb{R}$ or $\mathbb{C}$ or other). It seems like most sources I have simply start by discussing the definition of a normed vectorspace without discussing when it is possible for a vectorspace to have a norm. So I am thinking maybe there could be a vectorspace that we cannot define a norm on it? I couldn't think of an example myself.

REPLY [5 votes]: Sure. Take a (Hamel) basis $\{v_i\}_{i\in I}$ of your vector space $V$ (its existence is guaranteed by Zorn's lemma) and define an inner product by setting
$$\langle v_i,v_j\rangle = \delta_{ij}$$
and extending linearly over $V$. Now take the induced norm $\|w\| = \langle w,w\rangle$. However, this structure is not interesting at all, as it basically only depends on the "size" of $V$ (the cardinality of a basis), and not on any other information you have.

REPLY [3 votes]: Yes (as long as you have an absolute on the field, and you do not use non-common axioms schemes for your set-theory). 
Just recall that every vector space has a basis, fix one, and define the norm, e.g., as the $\infty$-norm  of the coordinates in this basis.