Does every vector space have an inner product?

Question

I have a question: Does every vector space have an inner product?

I think, yes. But I failed to find an essential reason. If it does not exist, then give me a counterexample. Thanks.

Answer 1

Inner products are defined for spaces over $\mathbb{R}$ or over $\mathbb{C}$ only.

The reason is that in the definition we require $$\langle u,v \rangle=\overline{\langle v,u\rangle}$$

and what is the meaning of $$\overline{\langle v,u\rangle}$$

when $\langle v,u \rangle$ is not complex ?

Answer 2

Take a vector space $X$ over $\mathbb R$ (or $\mathbb C$) and fix a basis $\{v_i\}_{i\in I}$.

If $x,y\in X$, then they can be uniquely expressed as $$ x=\sum_{i\in I}c_iv_i,\quad y=\sum_{i\in I}d_iv_i, $$ with each of the sum above finite.

Define $$\langle u,v\rangle=\sum_{i\in I}c_id_i. $$ This is an inner product in $X$.

In the case of $\mathbb C$, the definition is $$\langle u,v\rangle=\sum_{i\in I}c_i\overline{d_i}. $$