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Let $M$ be a closed manifold and let $N$ be a connected manifold withoutboundary. For each $k\in\mathbb{N}$ the set of $k$ times continuouslydifferentiable maps between $M$ and $N$ has the structure of a smooth Banachmanifold where the underlying manifold topology is the compact-open $C^k$topology. We provide a detailed and rigorous proof for this important statementwhich is already partially covered by existing literature.

The Banach manifold $C^k(M,N)$