Exponential of a bounded operator

Summary

The exponential of a bounded operator $T$ on a Banach space is the operator $e^T = \sum_{n \ge 0} T^n / n!$. It extends the matrix exponential to infinite dimensions. The family $t \mapsto e^{tT}$ solves the linear differential equation $x' = Tx$ in the Banach space and is the simplest example of an operator semigroup.