Does every real vectorspace have a symetric positive definite bilinear form?

Question

Does every real vectorspace $V$ (possibly not finite dimensional) have a symetric positive definite bilinear form? That is a map $s:V \times V \rightarrow \mathbb{R}$ such that: $$\forall v, w \in V:s(v,w)=s(w,v)$$ $$\forall u,v,w \in V\space\forall\lambda\in \mathbb{R}:s(\lambda u + v,w)=\lambda\space s(u,w)+s(v,w)$$ $$\forall v \in V:s(v,v)>0 \Leftrightarrow v \neq 0$$

Answer 1

Use the axiom of choice to create a Hamel basis $H$ for $V$. So every vector $v$ can be written as $\sum_{i=1}^n \alpha_i h_i$, where $n \ge 0$, $\alpha_i \in \mathbb R$, and $h_i\in H$.

Then for any $h,k \in H$, set $$ s(h,k) = \cases{ 1 & if $h = k$ \cr 0 & if $h \ne k$} $$ and extend bilinearly.