The Quadratic Form of a Real Square Matrix is Bounded on the Closed Unit Ball

Let $n$ be a natural number with $1\le n$, let $[n]$ be the initial segment determined by $n$, and let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure and the absolute value $|\cdot|$. Let $A$ be a real $n\times n$ matrix, with entries $A_{ij}$.

On Euclidean space $\mathbb{R}^n$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, write $\xi\cdot\zeta$ for the dot product, $\lVert\xi\rVert$ for the Euclidean norm, and $A\xi$ for the matrix-vector product. All sums below are the finite sums of the field $\mathbb{R}$.

Then the set

$$Q_A=\bigl\{\,|\xi\cdot(A\xi)|\ :\ \xi\in\mathbb{R}^n,\ \lVert\xi\rVert\le1\,\bigr\}$$

is a nonempty subset of $\mathbb{R}$ admitting $\sum_{i=1}^{n}\sum_{j=1}^{n}|A_{ij}|$ as an upper bound, and therefore has a least upper bound in $\mathbb{R}$.