Deduce that $E$ is a vector subspace of the vector space $\mathcal { C } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { R } \right)$ of continuous functions on $\mathbb { R } _ { + } ^ { * }$ with values in $\mathbb { R }$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.