> [!theorem] > > Let $E, F$ be [[Banach Space|Banach spaces]], $U \subset E$ be an [[Open Set|open set]], then the set of functions that can be [[Space of Continuously Differentiable Functions|continuously differentiated]] arbitrarily many times > $ > C^\infty(U, F) = \bigcap_{n \in \nat}C^n(U, F) > $ > is the space of **smooth functions**. > [!definition] > > Let $U \subset \real^d$ be an [[Open Set|open]] set, and $C^\infty(U) = C^\infty(U, \real)$ be the space of smooth real-valued functions. Let $\seq{U_n}$ be an [[Exhaustion by Compact Sets|exhaustion]] of $U$ by precompact open sets. For each $n \in \nat$ and [[Multi-Index|multi-index]] $\alpha$, define a [[Seminorm|seminorm]] > $ > \norm{f}_{n, \alpha} = \norm{\partial^\alpha f}_{u, \ol{U_n}} > $ > then > 1. The topology induced by $\bracsn{\norm{\cdot}_{n, \alpha}}$ makes $C^\infty(U)$ a [[Fréchet Space|Fréchet space]]. > 2. $f_n \to f$ if and only if $\partial^\alpha f_n \to \partial^\alpha f$ [[Uniform Convergence on Compact Sets|uniformly on compact sets]] for each multi-index $\alpha$.