The Parallelogram Identity for Inner Product Spaces

# The Parallelogram Identity for Inner Product Spaces

We will now look at an important theorem. If $V$ is an inner product space and $u, v \in V$ then the sum of squares of the norms of the vectors $u + v$ and $u - v$ equals twice the sum of the squares of the norms of the vectors $u$ and $v$, that is:

(1)
\begin{align} \quad \| u + v \|^2 + \| u - v \|^2 = 2 \| u \|^2 + 2 \| v \|^2 \end{align}

This important identity is known as the Parallelogram Identity, and has a nice geometric interpretation is we're working on the vector space $\mathbb{R}^2$:

We will now prove this theorem.

 Theorem 1 (The Parallelogram Identity): Let $V$ be an inner product space. If $u, v \in V$ then $\| u + v \|^2 + \| u - v \|^2 = 2\| u \|^2 + 2 \| v \|^2$.
• Proof: Let $V$ be an inner product space and let $u, v \in V$. Noting that $\overline{-1} = -1$ and we have that:
(2)