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Space of smooth functions on a compact interval ( $C^{\infty} ([0, 1])$ )

The space

C^{\infty} ([0, 1])

becomes a complete metric space under

d (f, g) = \sum_{r = 0}^{\infty} 2^{- r} min {1, ∥ f^{(r)} - g^{(r)} ∥_{\infty}} .

Convergence in this metric is exactly uniform convergence of every derivative separately.

Table of contents 187 3
- Completeness of the smooth-function metric Space of smooth functions on a compact interval 41
- Generic superfactorial derivative growth at rational points Space of smooth functions on a compact interval 123 1
  - Nowhere-analytic generic smooth function Generic superfactorial derivative growth at rational points 53

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