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Spaces of Continuous and Differentiable Functions

Learning Goals

  1. What is a Banach space?

  2. The spaces of continuous and differentiable functions.

  3. What do CkC^k norms measure?

Banach Spaces

Definition 1 (Banach Space)

A Banach space (X,∥⋅∥X)(X, \|\cdot\|_X) is a complete normed vector space.

Example 1 (Basic Banach Space)

Rn\mathbb{R}^n is a Banach space with any p-norm i.e. ∥x∥p=(∑i=1n∣xi∣p)1/p\|x\|_p = \left( \sum_{i=1}^{n}|x_i|^p \right)^{1/p}

Spaces of Continuous and Differentiable Functions

Definition 2 (C0C^0 — Continuous Functions)

Let Ω⊂Rn\Omega \subset \mathbb{R}^n be open. The space of continuous functions on Ω‾\overline{\Omega} is

C0(Ω‾)=C(Ω‾)={f:Ω‾→R:f is continuous}C^0(\overline{\Omega}) = C(\overline{\Omega}) = \{ f : \overline{\Omega} \to \mathbb{R} : f \text{ is continuous} \}
(1)

equipped with the sup-norm (or uniform norm)

∥f∥∞=sup⁡x∈Ω‾∣f(x)∣.\|f\|_\infty = \sup_{x \in \overline{\Omega}} |f(x)|.
(2)

When Ω‾\overline{\Omega} is compact, every continuous function is bounded and attains its supremum, so ∥f∥∞<∞\|f\|_\infty < \infty for all f∈C(Ω‾)f \in C(\overline{\Omega}).

Definition 3 (CkC^k — kk-times Continuously Differentiable Functions)

For k≥1k \geq 1, define

Ck(Ω‾)={f∈C(Ω‾):Dαf∈C(Ω‾) for all ∣α∣≤k}C^k(\overline{\Omega}) = \{ f \in C(\overline{\Omega}) : D^\alpha f \in C(\overline{\Omega}) \text{ for all } |\alpha| \leq k \}
(3)

where α\alpha is a multi-index and DαfD^\alpha f denotes the corresponding partial derivative. This space is equipped with the CkC^k norm

∥f∥Ck=∑∣α∣≤k∥Dαf∥∞=∥f∥∞+∑∣α∣=1∥Dαf∥∞+⋯+∑∣α∣=k∥Dαf∥∞.\|f\|_{C^k} = \sum_{|\alpha| \leq k} \|D^\alpha f\|_\infty = \|f\|_\infty + \sum_{|\alpha|=1} \|D^\alpha f\|_\infty + \cdots + \sum_{|\alpha|=k} \|D^\alpha f\|_\infty.
(4)

Definition 4 (C∞C^\infty — Smooth Functions)

The space of smooth (infinitely differentiable) functions is

C∞(Ω‾)=⋂k=0∞Ck(Ω‾).C^\infty(\overline{\Omega}) = \bigcap_{k=0}^{\infty} C^k(\overline{\Omega}).
(5)

This space does not carry a single norm but is topologized by the family of CkC^k seminorms: a sequence fn→ff_n \to f in C∞C^\infty if and only if ∥fn−f∥Ck→0\|f_n - f\|_{C^k} \to 0 for every kk.

C∞C^\infty is not a Banach space — it is a Fréchet space

C∞(Ω‾)C^\infty(\overline{\Omega}) is not a Banach space because no single norm can capture its topology. It is instead a Fréchet space: a complete, metrizable, locally convex topological vector space whose topology is defined by a countable family of seminorms (here, ∥⋅∥Ck\|\cdot\|_{C^k} for k=0,1,2,…k = 0, 1, 2, \dots).

Why can’t a single norm work? For any fixed kk, there exist sequences in C∞C^\infty that converge in CkC^k but diverge in Ck+1C^{k+1}. For example, on [0,1][0,1]:

fn(x)=sin⁡(nx)nf_n(x) = \frac{\sin(n x)}{n}
(6)

converges to 0 in C0C^0 (∥fn∥∞=1/n→0\|f_n\|_\infty = 1/n \to 0), but fn′(x)=cos⁡(nx)f_n'(x) = \cos(nx) does not converge at all — the C1C^1 norms satisfy ∥fn∥C1≥1\|f_n\|_{C^1} \geq 1 for all nn. No matter which CkC^k norm you pick, there is always a “higher” derivative direction that it fails to control.

A Banach space norm would have to simultaneously control all derivatives, but controlling the kk-th derivative imposes constraints that are strictly independent of controlling the (k−1)(k-1)-th. Formally, C∞C^\infty is not normable: by Kolmogorov’s criterion, a locally convex space admits a norm if and only if it has a bounded neighborhood of zero, and in C∞C^\infty every neighborhood of zero contains functions with arbitrarily large kk-th derivatives for sufficiently large kk.

Despite not being Banach, C∞C^\infty is complete in its Fréchet topology (a Cauchy sequence in every CkC^k norm converges in every CkC^k norm), and much of the Banach space theory — including versions of the Open Mapping Theorem and Closed Graph Theorem — extends to Fréchet spaces.

What Do CkC^k Norms Measure?

The CkC^k norm controls the function and its first kk derivatives uniformly. This has a concrete geometric meaning:

The key point: higher-order CkC^k norms are strictly stronger. A sequence can converge in C0C^0 (graphs converge) without converging in C1C^1 (slopes may oscillate wildly).

Example 2 (C1C^1 with the sup-norm is incomplete)

The space C1([0,1])C^1([0,1]) equipped with the wrong norm ∥⋅∥∞\|\cdot\|_\infty (instead of ∥⋅∥C1\|\cdot\|_{C^1}) is not a Banach space.

Consider the sequence fn(x)=x+1/nf_n(x) = \sqrt{x + 1/n} on [0,1][0,1]. Each fnf_n is C1C^1, and fn→f(x)=xf_n \to f(x) = \sqrt{x} uniformly. But f(x)=xf(x) = \sqrt{x} is not differentiable at x=0x = 0, so f∉C1([0,1])f \notin C^1([0,1]).

The sup-norm cannot detect that the derivatives fn′(x)=12x+1/nf_n'(x) = \frac{1}{2\sqrt{x + 1/n}} are blowing up near x=0x = 0. The C1C^1 norm would catch this: ∥fn∥C1→∞\|f_n\|_{C^1} \to \infty, so the sequence is not Cauchy in the C1C^1 norm.

This is a concrete instance of a general principle: a function space is only complete in a norm strong enough to detect all the structure the space requires. The sup-norm is too weak for C1C^1—it measures height but not slope, so Cauchy sequences can converge to functions that are continuous but not differentiable.

Proposition 1 (Ck(Ω‾)C^k(\overline{\Omega}) is a Banach space)

For Ω‾\overline{\Omega} compact and k≥0k \geq 0, the space (Ck(Ω‾),∥⋅∥Ck)(C^k(\overline{\Omega}), \|\cdot\|_{C^k}) is a Banach space.

Proof sketch:

Let {fn}\{f_n\} be Cauchy in CkC^k. Then for each ∣α∣≤k|\alpha| \leq k, the sequence {Dαfn}\{D^\alpha f_n\} is Cauchy in C0C^0 (sup-norm). Since (C0,∥⋅∥∞)(C^0, \|\cdot\|_\infty) is complete, each DαfnD^\alpha f_n converges uniformly to some continuous function gαg_\alpha. A standard argument (integrating and differentiating under the limit) shows gα=Dαg0g_\alpha = D^\alpha g_0, so g0∈Ckg_0 \in C^k and fn→g0f_n \to g_0 in CkC^k.

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