Assume a norm \( \| \cdot \| \) is defined by an inner product . Then
\[ \langle x, y \rangle = \tfrac{1}{4} (\|x + y\|^2 - \|x - y\|^2) \]in \( \mathbb{K} = \mathbb{R} \), and in \( \mathbb{K} = \mathbb{C} \):
\[ \langle x, y \rangle = \tfrac{1}{4} (\|x + y\|^2 - \|x - y\|^2 + i\|x + iy\|^2 - i\|x - iy\|^2). \]