Lie Groups pp 31-37 | Cite as

# Lie Subgroups of \(\mathrm{GL}(n, \mathbb{C})\)

## Abstract

If *U* is an open subset of \({\mathbb{R}}^{n}\), we say that a map \(\phi: U\longrightarrow {\mathbb{R}}^{m}\) is *smooth* if it has continuous partial derivatives of all orders. More generally, if \(X \subset {\mathbb{R}}^{n}\) is not necessarily open, we say that a map \(\phi: X\longrightarrow {\mathbb{R}}^{n}\) is *smooth* if for each *x* ∈ *X* there exists an open set *U* of \({\mathbb{R}}^{n}\) containing \(x\) such that ϕ can be extended to a smooth map on *U*. A *diffeomorphism* of \(X \subseteq {\mathbb{R}}^{n}\) with \(Y \subseteq {\mathbb{R}}^{m}\) is a homeomorphism \(F: X\longrightarrow Y\) such that both *F* and *F* ^{−1} are smooth. We will assume as known the following useful criterion.