[ notation ]

Smoothness

Linear spaces, over ringlets that have the positive reals as a sub-ringlet, are apt to have limit points for all sequences that do something suitably convergent, in a sense reasonably similar to Cauchy's. As linear spaces, they also equip us to construct chords of function from one of them into another and, from requiring these to do something similarly convergent as the chord's corners get closer to a given point, induce a notion of differentiability, initially at each point. When a function meets the conditions for that throughout some subset of the linear space in which there are no limit points of sequences not in the region, we can think of that subset of space as a volume, region or domain within which the function is ubiquitously differentiable.

Smoothness within the linear world

The very fact that the differentiability condition at a point discusses chords of all suitably small voluminous simplices, in whose interior the given point lies, ensures that its truth at nearby points says things about the simplices inside which they both are that inescapably imply a continuity of variation in the function that tends to constrain the function's derivative to vary similarly continuously and thus to have, at least nearly everywhere within our domain, a derivative of its own; which, likewise, is apt to vary continuously and so on. This leads to a scale of smoothness that starts with a continuity property that derivatives typically have (if the function has abrupt changes at all of the points on some sequence that has a limit, then the function also makes an abrupt change (albeit possibly of derivative, rather than of value) at the limit of that sequence).

A set is closed if every convergent sequence within it has a limit within it; it is open if no convergent sequence outside it has a limit within it. A function maps any sequence of inputs to a sequence of outputs; the inputs might converge yet the output might tend to some infinity; so the image of a convergent sequence needn't be convergent. However, when a continuous function's image of a sequence, convergent within the function's domain of continuity, does have a limit, then the continuous function can take as its value, at the sequence's limit point, the limit point of the function's values at the serives that converged there. That's as much as to say that all convergent sequences within the domain, that converge on any given point within the domain, have the same limit value. However, convergent sequences within the domain, that converge on the same point outside the domain, might have outputs of the function that converge on distinct values. This shall involve there being no continuous deformation of either sequence that turns it into the other. Manifold-preserving deformation can probably be stated in terms of convergent sequences (that may vary continuously with deformation's parameter) and the sense in which they join-or-break with the progress of the deformation. If two sequences are mutually convergent for some parameter values and each point of either varies continuously with the deformation, then the set of parameter values at which the limits agree is {closed, open} ? If a continuous function's smoothness, of some order N, changes abruptly everywhere on some sequence that has a limit, then the function's smoothness does change abruptly, at the limit, for some order no greater than N's successor. The function can be satisfy some continuity requirement at all points on a sequence that converges to a limit yet fail to do so at the limit.

Next on the scale is having a derivative at each point throughout the domain. As observed above, this entails the function's chords at simplices within the domain having gradients of the function compatible with the (continuity and continuous) differentiability constraints; indeed, each level of smoothness tends to limit the scope of departure from continuous at nearby levels, although there are always exceptions. Still, a function can be differentiable throughout an open region without its derivative varying continuously.

A function can (like square root on the complex plane, for any given choice of cut) be differentiable throughout a domain in which its derivatives on all mutualy convergent sequences do also mutually converge, even when the limit lies outside the domain (to be specific, in the cut); such a function's derivative can be completed to a function continuous even where the original function's variation isn't continuous.

So next we look at the function's derivative and ask whether it works continuously; that is the next level of smoothness. If the function meets that condition, we ask where it has a derivative, providing another level; and so on. Levels of smoothness thus follow a regular pattern: given a function, first ask whether it's continuous, then whether it has a derivative, throughout some domain; as long as it keeps saying yes, take its derivative and ask the same questions of that. A function is smooth in every domain within which it meets the conditions for all levels of smoothness; no matter how many times we differentiate it, we get again a continuous function with a derivative everywhere in the domain.

One last thing before I move on to general smoothness: if we have a function (V: f |U) with V and U linear, and f is differentiable at u in U, then its derivative at u is a limear map from (displacements in) U to (changes in f's output in) V; so each f'(u) is linear (V:|U). Thus, while f is a function (V:|U), f' is a function ({linear (V:|U)}: f' :U). Note the right-hand end of that, (… f' :U) only says that there are some u in U that f' accepts as input, i.e. at which f is differentiable; but ({linear (V:|U)}: f' …) says that it maps every such input to a linear map from U to V. The space of linear maps from U to V is itself a linear space (because V is; U's linearity is only relevant to selecting the linear subspace of {functions (V:|U)} that f' happens to be in), allowing us once again to ask whether f' is differentiable at points within a given domain.

General smoothness

The essential idea of smooth manifolds is to ask what's the most general domain in which we can coherently discuss a notion of smoothness compatible with the above characterisation, along with some of the more basic results about differentiable functions, between linear spaces, and their derivatives (differentiation is linear on the space of functions and obeys the product and chain rules). One can tweak which exact details of that characterisation one cares about, and the change may matter, but the theory of smooth manifolds is shaped by a particular range of reasonable choices on those details; I say range because a wide range of choices reasonably similar to one another all produce the same theory, for all that those making those choices may describe the theory differently, each using the terms that match up with their choices.

This, indeed, applies to all robust disciplines: one may formalise in various ways and get the same structure, albeit differently described by the diverse formalisations. Sometimes one finds differences of formalisation that do change what's described. Understanding why each such difference does lead to describing genuinely different systems – and why the other changes of formalisation don't – can reveal a great deal. Measure theory's collections of measurable sets have an interesting amount of similarity of form with topology's open, closed and compact sets; yet lead to weirder conclusions. Both are worth understanding.

So let's consider a smooth domain. I can't see how its notion of smoothness could be compatible with the above characterisation other than by there being: some mappings between (domains within) the smooth domain and (domains within) vector spaces; and some notion of how smoothness in such domains connects: smoothness of functions between such domains and linear spaces, when they're composed before and after a smooth function on or between such domains; to smoothness of the composite that results.

As long as we have smooth functions between smooth domains and linear spaces and smooth functions between smooth domains (or from one to itself), we can compose a smooth functions from a linear space to a smooth domain, optionally before a smooth function from that domain to another, before a smooth function from a smooth domain to a linear domain; the composite is a function from a linear space to a linear space, and we have adequately well-defined senses of continuity, differentiability and smoothness on those. When we compose functions between linear spaces, the composite always enjoys at least the level of smoothness of the least smooth function composed to make it; in particular, if all constituents were smooth, so is the composite. Consequently, the most obvious and natural first requirement that we can impose on generalised smoothness is that such composites are smooth whenever each function composed is smooth; and that this entails no disagreement with the usual notion of smoothness on linear spaces, when the composite is between two of them.

The next constraint that naturally arises is the chain rule: if we have a composite (V: f&on;g |U) of smooth functions, at least when U and V are linear spaces, and an input u in U at which it is allegedly smooth, then (f&on;g)'(u) = f'(g(u))&on;g'(u). In this, g'(u) needs to behave as if it's linear from U to something (which may depend on u), let's call it S: and f'(g(u)) needs to be linear from the same S to V; each needs to be linear to behave enough like a derivative; and they need to map to and from the same space in order to be composable. What that intermediate space is need not concern us, although I shall infer a space with which it's necessarily isomorphic, which shall let me re-cast the domains' notion of smoothness in terms of the theory of smooth manifolds, which provides a natural choice (U's space of tangents at u) of a linear space isomorphic to S; whatever we start with as description of the domains and their notion of smoothness, the theory of smooth manifolds shall deliver a way to transform that description into a standard form, that'll facilitate working out what smoothness means between such smooth domains and other smooth domains that likewise establish their compatibility with smoothness on linear spaces.

In particular, if we have smooth maps from some module U over a ringlet R to a smooth domain M and from M to some module V also over R, we can compose smooth maps from R to U before and from V to R after such maps; the resulting composite shall be from R to R. In particular, considering the first and last two maps of the chain composed, we get smooth maps (R: r :M) and (M: m :R) that we can compose to get an (R: r&on;m :R); and we can identify domains within M on which r is smooth and in R on which m is smooth; where m maps the latter into the former, the composite is also smooth, giving us domains in R on which (R: r&on;m :R) is smooth. In such domains we can ask for (r&on;m)', the derivative of the composite. For any given m, we can then consider the various (M: s :R) we could substitute for r – and, for given r, we can consider the various (R: n :M) we can substitute for m – without changing the composite's derivative.

Any R-linear (R: f |R) satisfies f(x.y) = x.f(y) for each x, y in R; in particular, each x is x.1 so f(x) = f(x.1) = x.f(1) determines f's value at every value of the ringlet. Thus, for each t in R at which r&on;m is smooth, although (r&on;m)'(t) is formally linear (R: |R), it is synonymous with (r&on;m)'(t, 1). The natural isomorhpism (R: f(1) ←f |{linear (R: |R)}) thus lets us treat ({linear (R: |R)}: (r&on;m)' :R) as a function (R: :R), for all practical purposes.

So first, for each value p in M, let H(p) = {(R: r :M) for which p is within a domain of M within which r is smooth} and S(p) = {pairs (: (M: m :R)←t :R) for which m(t) = p and p is within a domain of M within which m is smooth}. (Here, a pair is an atomic relation that relates one value to one value.) We can compose any h in H(p) after the s of any s←t in S(p) to get (R: h&on;s :R), which we can infer is smooth at t, so we can ask for (h&on;s)'(t) and hope to make sense of it as h'(p)&on;s'(t), since p = s(t). That equips us to define an equivalence relation on each of H(p) and S(p), namely:

• (H(p): ~ :H(p)) relates h to k precisely if, for every s←t in S(p), (h&on;s)'(t) = (k&on;s)'(t).
• (S(p): ~ :S(p)) relates s←t to p←q precisely if, for every h in H(p), (h&on;s)'(t) = (h&on;p)'(q).

When H(p) is considered modulo its share of this equivalence, it is known as the gradient space, G(p), of M at p; when S(p) is considered modulo its, it is known as the tangent space, T(p), of M at p. (Usually M is implicit. However, when we come to consider sub-manifolds, it may be necessary to distinguish G(M, p) and T(M, p) from G(N, p) and T(N, p) with N and M as sub-manifolds of some common manifold, which may be either of them.) When we understand an r in H(p) modulo this equivalence, it is written dr(p), which is in G(p); when we understand an m←p pair in S(p) modulo the equivalence, it is written m'(t), which is in T(p) thanks to m(t) being p; and the equivalence ensures we have a well-defined action of the former on the latter, dr(p)·m'(t) = (r&on;m)'(t).

Spaces of mappings {(R: :X)} are R-linear under the standard pointwise addition and (by composing before R's multiplication) scaling; the constraint H(p) imposes is compatible with that, hence H(p) is R-linear; and our equivalence trivially respects linearity on H(p). As a result, G(p) is R-linear. The action of G(p) on T(p) makes each m'(t) in T(p), so m(t) = p, a linear map from G(p) to R; thus this action embeds T(p) in the dual of G(p). The nature of the equivalence is such that no two members of G(p) can be distinct unless some member of T(p) distinguishes them, which leads to T(p) being the whole of G(p). Thus the action of G(p) on T(p) enables us to understand T(p) also as a linear space, the dual of G(p).

Now, regarding values of R as scalars, the members of H(p) are scalar functions of position; the members of S(p) are scalar-parameterised trajectories passing through p. (A trajectory might pass through p repeatedly: if so, the trajectory is paired with each of the parameter values at which it passes through p to provide a distinct member of S(p) representing each pass through p; these need not be (and typically aren't) equivalent in the sense defined here.) The smooth trajectories in S(p) can (between them all) be presumed to exhaustively explore M's domain close to p; I need to finish up that thought, as it's relevant to G and T being mutually dual.