Banach–Lie group

Remarks

Example: let $E$ be a real Banach space and $\mathcal{L}(E)$ the Banach space of bounded linear operators with the operator norm. The group $GL(E)$ of invertible operators is an open subset of $\mathcal{L}(E)$, so it is a Banach manifold with one chart. Multiplication is the restriction of a continuous bilinear map, hence smooth, and inversion is smooth because near $A$ it is given by the convergent series $(A - X)^{-1} = \sum_{k \ge 0} (A^{-1} X)^k A^{-1}$.

Non-example: on $\R$ with its standard smooth structure, define $x * y = (x^{1/3} + y^{1/3})^3$. This is a group, isomorphic to $(\R, +)$ through $x \mapsto x^{1/3}$, but the partial derivative of $x * y$ in $x$ at the point $(0, 1)$ does not exist, so clause 1 fails.

The tangent space at the identity carries a Lie bracket and is a Banach–Lie algebra. Every finite-dimensional Lie group is a Banach–Lie group.