Topological vector spaces

Each space C N = {f ∈ C c (R) : sptf ⊂ [−N,N ]} ⊂ C[−N,N ] is strictly smaller than the space C[−N,N ] of all continuous functions on the interval [−N,N ], since functions in C N must vanish at the endpoints. Still, C o N is a closed subspace of the Banach space C [−N,N ] (with sup norm), since a sup-norm limit of functions vanishing at ±N must also vanish there. Thus, each individual C N is a Banach space. For 0 < M < N the space C M is a closed subspace of C o N (with sup norm), since the property of vanishing off [−M,M ] is preserved under sup-norm limits. But for 0 < M < N the space C M is nowhere dense in C o N , since an open ball of radius ε > 0 around any function in C N contains many functions with non-zero values off [−M,M ]. Thus, the set C c (R) is an ascending union of a countable collection of subspaces, each closed in its successor, but nowhere-dense there.