# Clarification on the definition of inner product

An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar.

More precisely, for a real vector space, an inner product $\langle \cdot,\cdot \rangle$ satisfies the following four properties. Let $u, v$, and $w$ be vectors and alpha be a scalar, then:

1. $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$.

2. $\langle\alpha v,w\rangle=\alpha\langle v,w\rangle$.

3. $\langle v,w\rangle=\langle w,v\rangle$.

4. $\langle v,v\rangle\geq0$ and equal if and only if $v=0$.

A vector space together with an inner product on it is called an inner product space. This definition also applies to an abstract vector space over any field.

How can one define a inner product when $V$ is a vector space over $\mathbb{Z}_{2}$ in order to satisfies $4$?

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4'. $\langle v,w \rangle=0$ for all $w$ implies that $v=0$.