Then your linear operators are matrices. Then you have for example for a matrix $A$ $\mathrm{e}^{tA}$ defined as the power series $$\mathrm{e}^{tA} = \sum_{n = 0}^\infty \frac{t^n A^n}{n!}.$$
@BenjaminLim So now. We can have trouble on infinite-dimensional spaces where our operators are not continuous, then something as simple as a power series usually doesn't work.
As for unbounded operators. Are there any books on quantum mechanics which have a complete description of basics, using unbounded operators? As far as I know bounded are not fit for purposes of physics.
@Norbert if you like chatting, and you spend some time here, you might want to install the MathJax bookmarklet. It allows you to read things like $\int_0^\infty e^{-x}\mathrm{d}x$.
