nLab
continuously differentiable map

Contents

Idea

Once beyond the realm of normed vector spaces, the various ways of defining differentiation diverge. This is particularly evident if one considers the slightly stronger notion of continuous differentiability wherein the assignment of the derivative must also be continuous.

One can make a reasonable start by saying that for a function f:EUFf \colon E \supseteq U \to F to be continuously differentiable then it must at least satisfy the notion of Gâteaux differentiability, and one can throw in the requirement that the assignment of the directional derivative be continuous and linear (this is known as Gâteaux–Lévy differentiability). Thus one obtains a map Df:U(E,F)D f \colon U \to \mathcal{L}(E,F). However, outside the realm of normed vector spaces there is not a unique topology on (E,F)\mathcal{L}(E,F) and thus one can come up with a variety of meanings for the phrase “DfD f is continuous”.

Finite Dimensions

Let us remind ourselves of the situation in finite dimensions.

Definition

A function f: mU nf \colon \mathbb{R}^m \supseteq U \to \mathbb{R}^n, where UU is open, is said to be continuously differentiable, or of class C 1C^1, if there is a continuous map Df:U( m, n)D f \colon U \to \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n) with the property that for each xUx \in U and h mh \in \mathbb{R}^m then

lim t0f(x+th)f(x)tDf(x)ht=0 \lim_{t \to 0} \frac{f(x + t h) - f(x) - t D f(x)h}{t} = 0

Note that for xUx \in U and h nh \in \mathbb{R}^n there is an open interval (ϵ,ϵ)(-\epsilon,\epsilon) with the property that for t(ϵ,ϵ)t \in (-\epsilon,\epsilon) then x+thUx + t h \in U and so the limit makes sense.

Infinite Dimensions

In infinite dimensions the difficulty with extending the standard definition is that of the topology on continuous linear maps. This becomes more evident with higher derivatives. Thus the definition depends on such a choice. In addition, one needn’t use a topology but can make sense of the definition with a convergence structure on the space of linear maps.

Definition

Let EE and FF be locally convex topological vector spaces. Let UEU \subseteq E be an open set. Let (E,F)\mathcal{L}(E,F) be the space of continuous linear maps from EE to FF. Let Λ\Lambda be a convergence structure on (E,F)\mathcal{L}(E,F). A continuous function f:UFf \colon U \to F is said to be differentiable of class C Λ 1C^1_\Lambda if there exists a continuous mapping Df:U Λ(E,F)D f \colon U \to \mathcal{L}_\Lambda(E,F), called the derivative of ff, such that for every x,hU×Ex,h \in U \times E then

lim t0f(x+th)f(x)t=Df(x)h \lim_{t \to 0} \frac{f(x + t h) - f(x)}{t} = D f(x) h

We define C Λ 1(X,F)C^1_\Lambda(X,F) to be the set of functions f:XFf \colon X \to F which are of class C Λ 1C^1_\Lambda.

Convergence Structures

There are a variety of convergence structures and topologies on (E,F)\mathcal{L}(E,F) that can be used. Some of them with particular properties are gathered in the list below. For these, the notation is condensed slightly as indicated.

In the following, given semi-norms α\alpha on EE and β\beta on FF we define a semi-norm ρ β,α\rho_{\beta,\alpha} on (E,F)\mathcal{L}(E,F) by

ρ β,α(u)sup{β(uh)|α(h)1} \rho_{\beta,\alpha}(u) \coloneqq \sup \{\beta(u h) | \alpha(h) \le 1\}
  1. C Θ 1C^1_\Theta: the translation-invariant convergence structure with filters:

    0\mathcal{F} \to 0 if there is a semi-norm α\alpha on EE such that for each semi-norm β\beta on FF and ϵ>0\epsilon \gt 0 there is some QQ \in \mathcal{F} with sup uQρ β,α(u)ϵ\sup_{u \in Q} \rho_{\beta,\alpha}(u) \le \epsilon.

  2. C Δ 1C^1_\Delta: (Marinescu’s convergence structure) the colimit in the category of convergence vector spaces of the following family of spaces. Let Φ\Phi denote the family of mappings from the set of continuous semi-norms on FF to that on EE. For ϕΦ\phi \in \Phi define:

ϕ(E,F){u(E,F)|ρ β,ϕ(β)<for allβ} \mathcal{L}_\phi(E,F) \coloneqq \{ u \in \mathcal{L}(E,F) | \rho_{\beta,\phi(\beta)} \lt \infty\; \text{for all}\; \beta\}
  1. C Π 1C^1_\Pi: the compatible convergence structure with filters:

    0\mathcal{F} \to 0 if for each semi-norm β\beta on FF there is a semi-norm α\alpha on EE such that for each ϵ>0\epsilon \gt 0 there is some QQ \in \mathcal{F} with sup{ρ β,α(u)|uQ}ϵ\sup\{\rho_{\beta,\alpha}(u) | u \in Q\} \le \epsilon.

  2. C qb 1C^1_{q b}: the quasi-bounded convergence structure. This has filters:

    0\mathcal{F} \to 0 if ( n)0\mathcal{F}(\mathcal{B}^n) \to 0 for every quasi-bounded filter \mathcal{B} on EE.

    Recall that a filter \mathcal{B} on EE is quasi-bounded if 𝒱0\mathcal{V} \cdot \mathcal{B} \to 0 where 𝒱\mathcal{V} is the neighbourhood filter of 00 in \mathbb{R}.

  3. C c 1C^1_c: the continuous convergence structure. This is the coarsest convergence structure on (E,F)\mathcal{L}(E,F) which makes the evaluation map (E,F)×EF\mathcal{L}(E,F) \times E \to F continuous.

  4. C 𝒮 1C^1_{\mathcal{S}}: the topology of uniform convergence on a family 𝒮\mathcal{S} of bounded subsets of EE, in particular:

    1. C b 1C^1_b: the topology of uniform convergence on bounded subsets of EE,
    2. C pk 1C^1_{p k}: the topology of uniform convergence on precompact subsets of EE,
    3. C k 1C^1_k: the topology of uniform convergence on compact subsets of EE,
    4. C s 1C^1_s: the topology of uniform convergence on finite subsets of EE.

The relationships between the various definitions of C ? 1C^1_? are displayed in the following diagram.

Layer 1 C Θ 1 C^1_\Theta C Δ 1 C^1_\Delta C Π 1 C^1_\Pi C q b 1 C^1_{q b} C c 1 C^1_c C b 1 C^1_b C p k 1 C^1_{p k} C k 1 C^1_k C s 1 C^1_s E metrisable and barrelled E complete or metrisable E Schwartz E metrisable or Schwartz E metrisable E or F normable

Let us extract some particular cases:

  1. EE normable, FF arbitrary:
    1. C Θ 1C^1_\Theta, C Δ 1C^1_\Delta, C Π 1C^1_\Pi, C qb 1C^1_{q b}, C b 1C^1_b are equivalent
    2. C c 1C^1_c, C pk 1C^1_{p k}, C k 1C^1_k are equivalent
  2. EE Banach space, FF arbitrary:
    1. C Θ 1C^1_\Theta, C Δ 1C^1_\Delta, C Π 1C^1_\Pi, C qb 1C^1_{q b}, C b 1C^1_b are equivalent
    2. C c 1C^1_c, C pk 1C^1_{p k}, C k 1C^1_k, C s 1C^1_s are equivalent
  3. EE Fréchet space, FF arbitrary:
    1. C qb 1C^1_{q b}, C b 1C^1_b are equivalent
    2. C c 1C^1_c, C pk 1C^1_{p k}, C k 1C^1_k, C s 1C^1_s are equivalent
  4. EE Fréchet–Schwartz, FF arbitrary: C Π 1C^1_\Pi, C qb 1C^1_{q b}, C c 1C^1_c, C b 1C^1_b, C pk 1C^1_{p k}, C k 1C^1_k, C s 1C^1_s are equivalent
  5. EE finite dimensional and FF arbitrary, or EE Fréchet–Schwartz and FF normable: All equivalent

Chain Rule

An important question to ask of the various definitions of continuously differentiable is whether they satisfy the chain rule. The following result provides the basis for this.

Lemma

Let EE, FF, GG be LCTVS, let UEU \subseteq E and VFV \subseteq F be open sets, let Λ\Lambda and Λ\Lambda' be convergence structures on (E,F)\mathcal{L}(E,F) and (E,G)\mathcal{L}(E,G) respectively.

Assume that the composition map:

k(F,G)× Λ(E,F) Λ(E,G) \mathcal{L}_k(F,G) \times \mathcal{L}_\Lambda(E,F) \to \mathcal{L}_{\Lambda'}(E,G)

is continuous.

Let f:UFf \colon U \to F and g:VGg \colon V \to G be functions with f(U)Vf(U) \subseteq V. Suppose that ff is of class C Λ 1C^1_\Lambda and gg of class C k 1C^1_k. Then gfg \circ f is of class C Λ 1C^1_{\Lambda'}.

Corollary

The chain rule holds for each of C c 1C^1_c, C qb 1C^1_{q b}, C Π 1C^1_\Pi, C Δ 1C^1_\Delta, C Θ 1C^1_\Theta.

The following partial chain rules also hold:

  1. C Π 1(F,G)×C 𝒮 1(E,F)C 𝒮 1(E,G)C^1_\Pi(F,G) \times C^1_{\mathcal{S}}(E,F) \to C^1_{\mathcal{S}}(E,G)
  2. If EE is metrisable, C k 1(F,G)×C s 1(E,F)C s 1(E,G)C^1_k(F,G) \times C^1_s(E,F) \to C^1_s(E,G)
  3. If EE is metrisable, C k 1(F,G)×C k 1(E,F)C k 1(E,G)C^1_k(F,G) \times C^1_k(E,F) \to C^1_k(E,G)

Higher Derivatives

A minor wriggle enters the story with higher derivatives due to the fact that the higher derivatives are multilinear maps E nFE^n \to F and so not only are there different notions of convergence to put on these spaces, there are also different possible meanings of the statement that these are continuous. When dealing with one of the topologies (defined by some family of bounded sets), we will end up with derivatives in 𝒮 n(E,F)\mathcal{H}^n_{\mathcal{S}}(E,F) rather than 𝒮 n(E,F)\mathcal{L}^n_{\mathcal{S}}(E,F) in the notation of continuous multilinear operator.

However, we can start with a very weak notion to get the ball rolling. Let L n(E,F)L^n(E,F) be the space of all (not necessarily continuous) nn-linear maps E nFE^n \to F. We equip it with the topology of simple convergence.

Definition

Let EE and FF be LCTVS, UEU \subseteq E an open subset, pp \in \mathbb{N}. A function f:UFf \colon U \to F is said to be weakly pp-times differentiable if there exist functions D kf:UL k(E,F)D^k f \colon U \to L^k(E,F) for k=0,1,,pk = 0, 1, \dots, p such that D 0f=fD^0 f = f and for each xUx \in U, hEh \in E, and k=0,1,,p1k = 0, 1, \dots, p-1 then

lim t0D kf(x+th)D kf(x)t=D k+1f(x)h. \lim_{t \to 0} \frac{D^k f(x + t h) - D^k f(x)}{t} = D^{k+1} f(x)h.

Note that we don’t assume that the D kfD^k f are continuous. If some are continuous then some nice properties ensue. For example, if D pfD^p f is continuous then each D kfD^k f for kpk \le p is totally symmetric.

From the definition of weakly pp-times differentiable we can define the various classes of continuously pp-times differentiable. For the definition of 𝒮 n(E,F)\mathcal{H}^n_{\mathcal{S}}(E,F) see the page on continuous multilinear operator.

Definition

Let EE and FF be LCTVS, UEU \subseteq E an open subset, pp \in \mathbb{N}. Let 𝒮\mathcal{S} be a family of bounded sets in EE which covers EE. A function f:UFf \colon U \to F is said to be differentiable of class C 𝒮 pC^p_{\mathcal{S}} if ff is weakly pp-times differentiable and such that for k=0,1,,pk = 0,1,\dots, p then:

  1. D kf(U) 𝒮 k(E,F)D^k f(U) \subseteq \mathcal{H}^k_{\mathcal{S}}(E,F), and
  2. D kf:U 𝒮 k(E,F)D^k f \colon U \to \mathcal{H}^k_{\mathcal{S}}(E,F) is continuous.

Using convergence structures, we have:

Definition

Let EE and FF be LCTVS, UEU \subseteq E an open subset, pp \in \mathbb{N}. Let Λ\Lambda be one of the convergence structures Λ c\Lambda_c, Λ qb\Lambda_{q b}, Π\Pi, Δ\Delta, or Θ\Theta on p(E,F)\mathcal{L}^p(E,F). A function f:UFf \colon U \to F is said to be differentiable of class C 𝒮 pC^p_{\mathcal{S}} if ff is weakly pp-times differentiable and such that for k=0,1,,pk = 0,1,\dots, p then:

  1. D kf(U) Λ k(E,F)D^k f(U) \subseteq \mathcal{L}^k_{\Lambda}(E,F), and
  2. D kf:U Λ k(E,F)D^k f \colon U \to \mathcal{L}^k_{\Lambda}(E,F) is continuous.

For fixed pp, the relationships between the various C ? pC^p_? are the same as for p=1p = 1. For varying pp we have:

Lemma

Let EE and FF be LCTVS, UEU \subseteq E an open set. If f:UEf \colon U \to E is of class C c p+1C^{p+1}_c then ff is of class C Π pC^p_\Pi. Whence we have:

C c p+1(X,F)C Π p(X,F)C qb p(X,F)C b p(X,F) C^{p+1}_c(X,F) \subseteq C^p_\Pi(X,F) \subseteq C^p_{q b}(X,F) \subseteq C^p_b(X,F)

If EE is metrisable (resp. Fréchet) and ff is of class C k p+1C^{p+1}_k (resp. C s p+1C^{p+1}_s) then ff is of class C Π pC^p_\Pi.

Smooth Functions

We make the obvious definition for a smooth function:

Definition

Let Λ\Lambda be one of the convergence structures Λ s\Lambda_s, Λ k\Lambda_k, Λ pk\Lambda_{p k}, Λ b\Lambda_b, Λ c\Lambda_c, Λ qb\Lambda_{q b}, Π\Pi, Δ\Delta, or Θ\Theta. A function f:UFf \colon U \to F is said to be differentiable of class C Λ C^\infty_\Lambda if it is of class C Λ pC^p_\Lambda for all pp \in \mathbb{N}.

We have the following table of relationships.

Layer 1 C Θ C^\infty_\Theta C Δ C^\infty_\Delta C Π C^\infty_\Pi C q b C^\infty_{q b} C c C^\infty_c C b C^\infty_b C k C^\infty_k C s C^\infty_s E or F normable E metrisable E metrisable E metrisable and barrelled

Therefore, for Banach spaces all the notions of “smooth function” collapse, whilst for Fréchet spaces all of the notions coarser than Π\Pi collapse.

References

  • Keller, H. H. (1974). Differential calculus in locally convex spaces. Lecture Notes in Mathematics, Vol. 417. Berlin: Springer-Verlag. MR0440592

Last revised on October 13, 2017 at 23:38:00. See the history of this page for a list of all contributions to it.