Topological classification of systems of bilinear and sesquilinear forms

TL;DR

This paper establishes that the topological classification of systems of bilinear and sesquilinear forms is equivalent to their linear classification, simplifying the understanding of their structure.

Contribution

It proves that topological equivalence of such systems coincides with linear equivalence, providing a significant simplification in their classification.

Findings

Topological and linear classifications are equivalent for these systems.

Homeomorphisms correspond to linear bijections in this context.

The result applies to systems of bilinear and sesquilinear forms.

Abstract

Let $\cal A$ and $\cal B$ be two systems consisting of the same vector spaces $\mathbb C^{n_1},\dots,\mathbb C^{n_t}$ and bilinear or sesquilinear forms $A_i,B_i:\mathbb C^{n_{k(i)}}\times\mathbb C^{n_{l(i)}}\to\mathbb C$, for $i=1,\dots,s$. We prove that $\cal A$ is transformed to $\cal B$ by homeomorphisms within $\mathbb C^{n_1},\dots,\mathbb C^{n_t}$ if and only if $\cal A$ is transformed to $\cal B$ by linear bijections within $\mathbb C^{n_1},\dots,\mathbb C^{n_t}$.