# A question on a special “metric”

Suppose we have a function $$F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$$ where $$\mathcal{M}_{n \times n }(\mathbb{R})$$ is the space of $$n \times n$$ real matrices, a compact set $$B \subset \mathbb{R}^n$$, a constant $$C \in \mathbb{R}$$. What kind of $$F$$ satisfies the following condition $$\Vert F(x_1) y_1 - F(x_2) y_2 \Vert_2 \le C \Vert y_1 - y_2 \Vert_2, \forall x_1, x_2 \in [a,b]^n, \forall y_1, y_2 \in B$$

If $$F$$ is constantly a contraction the problem becomes trivial, but what happens when $$F$$ varies? A simpler question I am interested in is, when $$A, B \in \mathcal{M_{n \times n}(\mathbb{R})}$$, what conditions must they satisfy such that $$\Vert A y_1 - B y_2 \Vert_2 \le C \Vert y_1 - y_2 \Vert_2, \forall y_1, y_2 \in B$$ Are there any previous work on problems like this? I would really appreciate some book/paper references!

Taking $$y_1=y_2=y$$, we see that $$F(x_1)y=F(x_2)y\,$$ for all $$x_1,x_2$$ in $$[a,b]^n$$ and for all $$y\in B$$, and hence for all $$y\in \text{span}\,B$$. In particular, if $$B$$ spans $$\mathbb R^n$$, then the matrix $$F(x)$$ does not depend on $$x$$, and in such a case the problem becomes "tirvial", as you said.