Riesz representation theorem (finite-dimensional)

Existence. Let $(e_1,\ldots,e_n)$ be an orthonormal basis of $E$. Define $u=\sum_i\varphi(e_i)e_i$. For any $v=\sum_j v_j e_j$:
$$\langle v,u\rangle=\sum_j v_j\langle e_j,u\rangle=\sum_j v_j\varphi(e_j)=\varphi\left(\sum_j v_j e_j\right)=\varphi(v).$$

Uniqueness. If $\langle v,u\rangle=\langle v,u'\rangle$ for all $v$, then $\langle v,u-u'\rangle=0$ for all $v$. Taking $v=u-u'$: $\|u-u'\|^2=0$, so $u=u'$.

Conclusion. Every linear form on a finite-dimensional inner product space is represented by a unique vector.

Matrix form. In matrix terms, a linear form $\varphi(v)=a^Tv$ is represented by the vector $u=a$ (in an orthonormal basis).