Parseval Equality

For a vector $\vec{v}$ in $d$-dimensional space the Parseval Equality says that $\vec{v}$ is equivalent to every unit vector $\overrightarrow{u\_d}$ that make up the orthonormal basis of that $d$-dimensional space multiplied by the magnitude of $v$ in that specific direction. $|v|=\sqrt{(v\cdot u\_1{)}^2+...+(v\cdot u\_d{)}^2}$

The collective formula is

$$|v{|}^2=\sum \_{k=1}^d(\vec{v}\cdot \overrightarrow{u\_k}{)}^2$$

Such an equality also exists for Fourier Series as it does for vectors.