Let $m,n \in \mathbb{N}$, and let $\mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ denote the real [vector space](/page/Vector%20Space) of linear maps from $\mathbb{R}^m$ to $\mathbb{R}^n$. Let $\|\cdot\|_F: \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n) \to [0,\infty)$ denote the Frobenius norm defined by the Euclidean norm of the standard matrix, and let $(\cdot,\cdot)_F: \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n) \times \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n) \to \mathbb{R}$ denote the Frobenius [inner product](/page/Inner%20Product) defined by summing products of corresponding standard-matrix entries. Then for every $T \in \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$,

\begin{align*} \|T\|_F^2=(T,T)_F. \end{align*}