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Section 3.4 Differentiable functions

So far, we showed functional spaces of functions defined over an interval. This is just the simplest (and yet not trivial!). More generally, in order to define the \(L^p\) spaces it is enough to have a measure space; in order to define the space of continuous functions \(C^0\text{,}\) it is enough to have a topological space; in order to define the spaces \(C^k\text{,}\) it is enough to have a smooth manifold.

Subsection 3.4.1 Functions on Manifolds

We will not need in this textbook any knowledge of differential geometry. We just recall that a manifold \(M\) without boundary of dimension \(n<\infty\) is a topological space that is locally euclidean, namely there are open sets \(\Omega_i\) and homeomorphisms \(\psi_i:\Omega_i\to\bR^n\) (these homeomorphisms are called "charts" of the manifold), such that the \(\Omega_i\) cover the whole \(M\) and, in each non-empty intersection \(\Omega_i\cap\Omega_j, i\neq j\text{,}\) the change of coordinates \(\psi_i\circ\psi_j^{-1}\) is \(C^\infty\text{.}\) By definition, \(C^k(M)\) is the set of all \(f\in C^0(M)\) such that, for every \(i\text{,}\) the function \(f\circ\psi_i^{-1}:\bR^n\to\bR\) belongs to \(C^k(\bR^n)\) (the map \(f\circ\psi_i^{-1}\) is called the "representative" of \(f\) in the chart \(\psi_i\)). A manifold with boundary is defined quite similarly, with the difference that some of the charts will be homeomorphic not to \(\bR^n\) but to the upper half semispace \(\{x^n\geq0\) of \(\bR^n\text{.}\) All points that, in a chart, end up on the \(x^n=0\) hyperplane form the boudary of \(M\text{,}\) that is usually denoted by \(\partial M\text{.}\)

A fundamental theorem of differential geometry grants the existence of a smooth partition of unit subordinated to any system of charts \(\Omega_i\text{.}\) This is a set of functions \(\rho_i\in C^\infty(M)\) such that: (i) \(0\leq\rho_i\leq1\text{;}\) (ii) \(\supp\rho_i\subset\Omega_i\text{;}\) (iii) each \(p\in M\) has a nbhd that intersects only finitely many \(\Omega_i\text{;}\) (iv) \(\sum_i\rho_i=1\text{.}\) Hence, for every \(f\in C^\infty(M)\text{,}\) \(f=\sum_i\rho_i f\text{,}\) with \(\supp f\subset\Omega_i\text{,}\) namely every smooth function can be decomposed in the sum of smooth functions where each addend has support inside a single chart. Therefore, the study of the properties of \(C^\infty(M)\) can be reduced to the study of properties of \(C^\infty(\bR^n)\) and similarly in case of boundaries.
As we did in Section 3.1, we now show how to get several functional spaces from \(C^\infty(\Omega)\text{.}\) Because of the discussion above, we will mostly use in this book functions defined on an open set \(\Omega\subset\bR^n\text{.}\) When \(\Omega\) is not the whole space, we denote by \(\partial\Omega\) its boundary and by \(\overline{\Omega}\) its closure. We are mostly interested in two cases: (i) \(\Omega\) is bounded; (ii) \(\Omega=\bR^n\text{.}\) Hence, when we write \(\Omega\text{,}\) unless we write otherwise, we assume that \(\Omega\) is an open set with compact closure.

Subsection 3.4.2 \(C^0(K),\dots,C^\infty(K)\)

We consider here two cases of compact sets \(K\text{:}\) the closure \(\overline{\Omega}\) of a bounded open set \(\Omega\) and a compact manifold, with or without boundary, such as \(\bS^1\) or the \(n\)-torus \(\bT^n\text{.}\) In either case, all norms discussed in Section 3.1 are well-defined over \(C^\infty(K)\) and we can repeat almost verbatim the whole discussion. In more detail, the completion of \(C^\infty(K)\) under the norm
\begin{equation*} \|f\|_{C^0(K)} = \sup_{x\in K}|f(x)| \end{equation*}
gives the Banach space \((C^0(K),\|\cdot\|_{C^0(K)})\text{.}\) The \(C^1\) norm, since now functions of \(C^\infty(K)\) are in \(n\) variables, is given by
\begin{equation*} \|f\|_{C^1(K)} = \|f\|_{C^0(K)} + \max\{\|\partial_{x^1} f\|_{C^0(K)},\dots,\|\partial_{x^n} f\|_{C^0(K)}\} \end{equation*}
and the completion of \(C^\infty(K)\) under this norm gives the Banach space \((C^1(K),\|\cdot\|_{C^1(K)})\text{.}\) Similarly one gets the Banach space \((C^k(K),\|\cdot\|_{C^k(K)})\text{.}\) Finally, \(C^\infty(K)\) is a Frechet space under the distance function
\begin{equation*} d(f,g) = \sum_{k=1}^\infty \frac{1}{2^k} \frac{ \|f-g\|_{C^k(K)} }{ 1 + \|f-g\|_{C^k(K)}}. \end{equation*}

Subsection 3.4.3 \(C^0(\Omega)\)

Since \(\Omega\) is not compact, \(C^\infty\) functions are not necessarily bounded in \(\Omega\) and so \(\|f\|_{C^0(\Omega)} = \sup_{x\in\Omega}|f(x)|\) is not well defined on the whole \(C^\infty(\Omega)\text{.}\) I an analogous way as we did in case of \(C^\infty([0,1])\text{,}\) we can now use the fact that, for each compact set
\begin{equation*} Q_k=\{x\in\bR^n:|x^i|<k,i=1,\dots,n\}, \end{equation*}
the norm
\begin{equation*} \|f\|_{C^0(Q_k)} = \sup_{x\in Q_k}|f(x)| \end{equation*}
is well-defined and that
\begin{equation*} \bigcup_{k=1}^\infty Q_k = \bR^n. \end{equation*}
Hence, the function
\begin{equation*} d(f,g)=\sum_{k=1}^\infty\frac{1}{2^k}\frac{\|f-g\|_{C^0(Q_k)}}{1+\|f-g\|_{C^0(Q_k)}} \end{equation*}
is a well-defined distance function on \(C^\infty(\Omega)\text{.}\) Indeed, one can easily check that \(d(f,g)=0\) if and only if \(f=g\) and that \(d(f,g)=d(g,f)\text{.}\) The triangular inequality can be proven the same way we did in case of the \(C^\infty\) distance in Section 3.1.

Now, consider a Cauchy sequence \(f_n\) under this distance. Then \(f_n|_{Q_k}\) is Cauchy under \(|\cdot\|_{C^0(Q_k)}\) and so it converges to some \(g_k\in C^0(Q_k)\text{.}\) Moreover, if \(k'>k\text{,}\) then \(g_{k'}|_{Q_k}=g_k\text{.}\) Hence, the \(g_k\) define a function \(g\in C^0(\Omega)\) and \(f_n\to g\) under the \(d\) above.

Finally, let \(f\in C^0(\Omega)\text{.}\) We can build a sequence of \(f_n\in C^\infty(\Omega)\) converging to \(f\) under \(d\text{.}\) Let \(\eta\in C^\infty_c(\Omega)\) be a mollifier. Then \(f_h(x)=\int_\Omega \eta_h(x-y)f(y)dy\) is a smooth function that converges uniformly to \(f\) over every compact \(K\subset\Omega\) as \(h\to0\text{.}\)

Hence, the completion of \(C^\infty(\Omega)\) under \(d\) is \(C^0(\Omega)\text{.}\) This topology is often called compact-open topology or also Whitney weak \(C^0\) topology. In this topology, \(f_n\to f\) if and only if, for every \(k=1,2,\dots\text{,}\) \(f_n|_{Q_k}\to f|_{Q_k}\) in \(C^0(Q_k)\text{.}\) In turn, this is equivalent to the following: for every compact \(K\subset\Omega\text{,}\) \(f_n|_{K}\to f|{K}\) in \(C^0(K)\text{.}\) Equivalently, a "basic neighborhood" of a function \(f\) is given by all functions of the form \(f+U_{K,\eps}\text{,}\) where
\begin{equation*} U_{K,\eps} = \{g\in C^0(\Omega)\,:\,\sup_{x\in K}|g(x)|<\eps\}, \end{equation*}
namely all functions in such neighborhood differ from \(f\) only within some compact \(K\) and within it they differ by at most \(\eps\text{.}\)

In other words, the weak Withney topology does not care about what happens outside of some compact set. For instance, consider the sequence \(f_n(x)=e^x/n\in C^\infty(\bR)\text{.}\) Within each compact, \(f_n\to0\text{,}\) so that \(f_n\to0\) in \(C^0(\bR)\) under the weak Withney topology even though, for each \(n\text{,}\) \(f_n(x)\) diverges exponentially as \(x\to\infty\text{.}\)

Since we mentioned the weak Withney topology, we briefly mention here a second topology, used mostly in differential geometry, called the Whitney strong \(C^0\) topology. In this topology, "basic neighborhoods" \(f+U_\eta\) of a function \(f\) are parametrized by \(C^0\) functions \(\eta:M\to(0,\infty)\) so that
\begin{equation*} U_\eta = \{g\in C^0(\Omega)\,:\,|g(x)|<\eta(x)\text{ for all }x\in K\}. \end{equation*}
This topology is designed to control the behavior of functions at infinity, and it does a pretty good job with it. Indeed, \(f_n\to f\) in this topology if and only if almost all \(f_n\) coincide with \(f\) outside of some compact set \(K\subset\Omega\) and \(f_n|_K\to f_K\) in the Banach space \(C^0(K)\text{.}\) The only if direction is trivial. Now, assume that \(f_n\to f\) in the strong topology and suppose that they differ outside of every compact subset of \(\bR^n\text{.}\) Then there is a sequence \(x_k\in\bR^n\) going to infinity and a subsequence \(n_k\) such that \(|f_{n_k}(x_k)-f(x_k)|>0\) for every \(k\text{.}\) Now, let \(\eta(x)\) be a continuous positive function on \(\bR^n\) such that \(\eta(x_k)<|f_{n_k}(x_k)-f(x_k)|\) for every \(k\text{.}\) Then, the open neighborhood \(f+U_\eta\text{,}\) centered at \(f\text{,}\) does not contain any of the \(f_{n_k}\text{,}\) so that it cannot be that \(f_n\to f\) in this topology.

The Whitney strong topology is very useful in differential geometry because "preserves open relations". For instance, consider the simple case of positive functions. We saw above that, in the compact-open topology, the sequence \(e^x/n\) converges to 0. Each of those maps is everywhere positive but their limit is not. Hence, the set of positive continuous functions of \(\bR\) is not open in the compact-open topology of \(C^\infty(\bR)\text{.}\) On the other side, it is open in the strong Withney topology. Indeed, in this case a sequence \(f_n\) converges to \(f\) is and only if the \(f_n\) converge to \(f\) uniformly over some compact \(K\) and are identical to \(f\) outside of it. Over any \(K\text{,}\) each \(e^x/n\) has a positive minimum and therefore every function close enough to \(f\) will be positive in \(K\) and therefore will be positive over the whole \(\bR\text{.}\) Hence, \(f_n>0\) for all \(n\) implies \(f>0\text{.}\) Similarly, the space of immersions, of free maps and so on are all open in the strong Withney topology.

This topology, though, it is not that useful in PDE theory, because, with this topology, \(C^\infty(\Omega)\) is not a topological vector space. Indeed the multiplication by a scalar is not a continuous operation in the Whitney strong topology. If it were, then the map \(\lambda\to\lambda\cdot f\) would be continuous for every \(f\in C^\infty(\Omega)\) and so \(f/n\) would converge to the 0 function. This, though, cannot happen when \(f\) does not have compact support, as we showed above. This is not a problem for geometers since, in general, this topology is used for spaces $C^\infty(M,N)$ of smooth maps from a manifold $M$ to a manifold $N$ and these spaces, in general, are not linear. For this reasons, we will not consider any further this topology in this textbook.

The reason why there is no Banach space structure compatible with the continuity of the injections of \(C^0(\Omega)\) into the \(C^0(K)\) is a bit involved and will be discussed in ???.

The following subsets of \(C^0(\Omega)\) are important:
\begin{align*} C^0_c(\Omega) \amp = \{f\in C^0(\Omega)\,:\supp f\text{ is a compact subset of }\Omega\}\amp\text{ (test functions) }\\ C^0_0(\Omega) \amp = \{f\in C^0(\Omega)\,:\,\lim_{\|x\|\to\infty}|f(x)|=0\}\amp\text{ (functions that go to 0 at $\infty$)} \\ C^0_b(\Omega) \amp = \{f\in C^0(\Omega)\,:\|f\|_{C^0}<\infty\}\amp\text{ (bounded functions) } \end{align*}
The first is the set of continuous test functions, that play a fundamental role, for instance, as mollifiers and in the definition of weak derivatives and distributions. This space is not closed in the compact-open topology and its closure is the whole \(C^0(\Omega)\) space. It is enough to prove this result in the strong topology, since \(f_n\to f\) in the strong topology implies that \(f_n\to f\) in the weak topology. To see this, it is enough to verify that there are functions with compact support in each compact-open neghborhood of any function \(f\in C^0(\Omega)\text{.}\) So, let \(K\subset\Omega\) compact and \(\eps>0\text{.}\) By the Urysohn's Lemma, there is a \(C^0\) function \(\phi\) on \(\Omega\) that is 1 on \(K\) and 0 outside of some compact set \(K'\supset K\text{.}\) Hence, \((f-\phi\cdot f)|_K=0\) and \(\phi\cdot f\) has compact support, so \(C^0_c(\Omega)\cap (f+U_{K,\eps})\neq\emptyset\) for every \(\eps>0\text{.}\)

On the space of bounded continuous functions \(C^0_b(\Omega)\text{,}\) the \(C^0\) norm is well defined and one can easily prove that \((C^0_b(\Omega),\|\cdot\|_{C^0})\) is a Banach space. Its linear subspace \(C^0_0(\Omega)\) is closed in the \(C^0\) uniform topology. As above, we prove the claim in the simpler case \(\Omega=\bR^n\text{.}\) Assume that \(f_n\to f\in C^0(\Omega)\) in the uniform topology with \(f_n\in C^0_0(\Omega)\text{.}\) In other words: 1. for every \(\eps>0\) and for each \(n>0\text{,}\) there is a \(r_n>0\) such that \(|f_n(x)|<\eps\) for \(\|x\|> r_n\text{;}\) 2. for every \(\eps>0\) there is a \(N_\eps>0\) such that \(|f_n(x)-f(x)|<\eps\) for all \(n>N_\eps\) and for all \(x\in\bR^n\text{.}\) Hence, \(|f(x)|\leq|f(x)-f_n(x)|+|f_n(x)|<2\eps\) for all \(n> N_\eps\) and \(x\) such that \(\|x\|> r_n\text{,}\) namely \(\lim_{|x|\to\infty}f(x)=0\text{,}\) i.e. \(f\in C^0_0(\bR^n)\text{.}\)

It is important to notice the following difference between \(C^0_0(\Omega)\) and \(C^0_b(\Omega)\text{.}\) For any \(f\in C^0_0(\Omega)\text{,}\) the difference \(f(x+h)-f(x)\) converges to 0 uniformly, so that the translation operator \(T_h:f(x)\to f(x+h)\) on \(C^0_0(\Omega)\) is continuous. On the contrary, there are \(f\in C^0_b(\Omega)\) for which \(f(x+h)-f(x)\) does not converge to 0 uniformly. Hence, the translation operator \(T_h\) is not continuous on \(C^0_b(\Omega)\text{.}\) For instance, take \(f(x)=\sin x^2\text{.}\) Then take \(x_n=\sqrt{n\pi+\pi/2}\) and \(y_n=\sqrt{n\pi}\text{.}\) Then
\begin{equation*} |f(x_n) - f(y_n)| = |\sin x_n^2-\sin y_n^2| = |\sin(n\pi+\pi/2)|=1 \end{equation*}
although \(|x_n-y_n|\to0\text{.}\) Hence, \(f(x+h)-f(x)\) does not converge to 0 uniformly.

The space \(C^0_0(\Omega)\) is the ambient space of the PDEs with Dirichlet boundary conditions, where functions are required to go to zero at the boundary. As above, one can prove that the closure of \(C^0_c(\Omega)\) in the uniform norm is \(C^0_0(\Omega)\text{.}\)

Subsection 3.4.4 \(C^k(\Omega)\)

One can repeat the whole discussion above for the \(C^k\) spaces, with \(k=1,2,\dots\text{,}\) obtaining similar results: \(C^k(\Omega)\) is a Frechet space with the compact-open topology and is a finer, non-metrizable, topological space with the Whitney strong topology; \(C^k_0(\Omega)\text{,}\) the space of functions that vanish at infinity along with their derivatives up to order \(k\text{,}\) is a Banach space under the \(C^k\) norm. The space \(C^k_c(\Omega)\) is dense in \(C^k(\Omega)\) in both the compact-open topology and to \(C^k_0(\Omega)\) in the uniform topology.

Subsection 3.4.5 \(C^\infty(\Omega)\)

Using the cubes \(Q_k\text{,}\) one can define a distance function on \(C^\infty(\bR^n)\) that generalizes the one we showed in case of \(C^\infty([0,1])\text{,}\) namely
\begin{equation*} d(f,g) = \sum_{k=1}^\infty \frac{1}{2^k} \frac{ \|f-g\|_{C^k(Q_k)} }{ 1 + \|f-g\|_{C^k(Q_k)}}. \end{equation*}
A Cauchy sequence \(f_n\) under the distance above is a Cauchy sequence in \(C^k(Q_k)\) for each \(k=1,2,\dots\) and so it converges, in \(C^k(Q_k)\text{,}\) to a \(C^k\) function. Each of these functions coincides with its restriction to \(Q_{k'}\) for all \(k'< k\text{,}\) so ultimately this defines a \(f\in C^\infty(\Omega)\) such that \(f_n\to f\text{.}\)

The argument above shows that this distance function defines the compact-open topology on \(C^\infty(\Omega)\text{.}\) In this topology, the "basic neighborhoods" of a function \(f\in C^\infty(\Omega)\) are of the type
\begin{equation*} U_{K,\alpha,\eps} = \{g\in C^0(\Omega)\,:\,\sup_{|\alpha|\leq m}\sup_{x\in K}|\partial_\alpha g(x)|<\eps\}. \end{equation*}