Inner Product with Vector is Linear Functional

Theorem
Let $\GF$ be a subfield of $\C$.

Let $\struct{ V, \innerprod \cdot \cdot }$ be an inner product space over $\GF$.

Let $v_0 \in V$.

Then the mapping $L: V \to \GF$ defined by:


 * $\map L v := \innerprod v {v_0}$

is a linear functional.

Proof
Let us directly check the definition of linear functional: