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In this article we consider the following equivalence relation on the classof all functions of two variables on a set $X$: we will say that $L,M: X\timesX\to \mathbb{C}$ are rescalings if there are non-vanishing functions $f,g$ on$X$ such that $M\left(x,y\right)=f\left(x\right)g\left(y\right)L\left(x,y\right)$, for any $x,y\in X$. We give criteria for being rescalingswhen $X$ is a topological space, and $L$ and $M$ are separately continuous, orwhen $X$ is a domain in $\mathbb{C}^{n}$ and $L$ and $M$ aresesqui-holomorphic.  A special case of interest is when $L$ and $M$ are symmetric, and $f=g$ onlyhas values $\pm 1$. This relation between $M$ and $L$ in the case when $X$ isfinite (and so $L$ and $M$ are square matrices) is known to be characterized bythe equality of the principal minors of these matrices. We extend this resultto the case when $X$ is infinite. As an application we characterizerestrictions of isometries of Hilbert spaces on weakly connected sets as themaps that preserve the volumes of parallelepipeds spanned by finite collectionsof vectors.

Principal Minor Assignment, Isometries of Hilbert Spaces, Volumes of Parallelepipeds and Rescalling of Sesqui-holomorphic Functions