# Can we define inner product on every vector space? [duplicate]

Can we define inner product on every vector space?

I don't know any example of any vector space that do not have any inner product .

Help me

## marked as duplicate by Rhys Steele, Chappers, Gerry Myerson, José Carlos Santos, mechanodroidJul 25 '18 at 13:16

• $L^p$ spaces with $p$ different of two are the typical example . – Gustave Jul 25 '18 at 13:44
If you have a finite dimensional vector space just choose a basis and then define an inner product by assuming that basis is orthonormal. Another way to say the same thing: choosing a basis of a finite dimensional vector space over a field $K$ establishes an isomorphism with $K^n$ where there's a natural inner product.
• Even in an infinite-dimensional vector space (over $\mathbb R$ or $\mathbb C$), your method works fine. You need the axiom of choice to ensure that there is a basis, but then every basis gives rise to an inner product, in which that basis is orthonormal. – Andreas Blass Jul 25 '18 at 13:05