Topological classification of systems of bilinear and sesquilinear forms

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 $\mathbb C^{n_1},\dots,\mathbb C^{n_t}$.

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