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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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The fourth property is called "positive definiteness". Typically when moving over to other fields one replaces this with "non-degeneracy" (which is weaker than positive definiteness).

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

Note: If you move to working over the complex numbers then usually one replaces 3. with conjugate symmetry. And so this along with 2. imply that pulling scalars out of the second slot results in conjugation.

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