conformal maps between vector spaces?

Question

let $X,Y$ be two vector spaces furnished with inner products. Now consider a linear map $L:X\rightarrow Y$ with the following additional property: there exists a constant $C>0$ with

$$\langle Lv,Lw \rangle=C\cdot \langle v,w \rangle, \forall v,w\in X.$$

I am wondering if linear maps with this property do have a special name? This property does look so natural, but I could not find anything in the internet and in books about linear algebra.

Surely, one can call it a "conformal" map, since it is a conformal map in the sense of Riemannian geometry, when we think about the spaces $X,Y$ as Riemannian manifolds. But I would like to know if there is a name for it in the realm of linear algebra.

Thanks in advance

Answer 1

If $$\left< Lx , Ly \right> =C\left< x, y\right> $$ then the function $T(x) =\frac{1}{\sqrt{C}} L(x)$ satisfies $$\left< Tx , Ty \right> =\left< x, y\right> $$ thus $T$ is unitary map.