Quantum mechanics: The unitary group $U$ of a Hilbert space. If you use the norm topology, then it is a Banach Lie group with bounded skew hermitian operators as Lie algebra. But unbounded self-adjoint operators $A$ (Schr"odinger operators) lead to 1-parameter semigroups $e^{itA}$ in $U$, where now the strong operator topology plays a role.

Quantum mechanics: The unitary group $U$ of a Hilbert space. If you use the norm topology, then it is a Banach Lie group with bounded skew hermitian operators as Lie algebra. But unbounded self-adjoint operators $A$ (Schrödinger operators) lead to 1-parameter semigroups $e^{itA}$ in $U$, where now the strong operator topology plays a role.