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This is somewhat vague, but please bear with me. Complete metric spaces are supposed to take care of "gaps", they're understood as a natural extension of dense sets.

The convolution, defined as $f*g = \int_{-\infty}^{\infty}(f-\tau)g(\tau)d\tau$ does not have an identity that is a "function", despite that you can define convolutions over spaces of compactly supported continuous functions (which are complete), but it does have a "distribution" that satisfies $f*\delta = f$, which is the dirac delta distribution.

Is there a name for this dilemma or classification of this identity gap, that, while a transform such as convolution may be defined over a complete normed vector space such as rapidly decaying continuous functions over $\mathbb{R}$, that vector space might be missing an identity, and thus we need to "extend" from the space of functions to a space of distributions to find its identity?

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    $\begingroup$ This is an example (though I'd never thought of it as such, and it's more concrete than the usual abstract such process) of adjoining a unit to an algebra that doesn't have one, but I don't know any other name for that than "adjoining a unit". $\endgroup$
    – LSpice
    Commented Aug 28, 2022 at 23:52
  • $\begingroup$ As @QiaochuYuan points out below, this is nothng to do with "completeness" in the sense of metric spaces. The Banach space $L^1({\bf R})$ is complete in its usual norm and is a Banach algebra when equipped with the convolution product. However it does not have an identity element for convolution (this would be the Dirac delta, as you observe) $\endgroup$
    – Yemon Choi
    Commented Sep 12, 2022 at 15:39
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    $\begingroup$ @LSpice Banach algebraists often call this "unitization" (it does what it says on the tin) $\endgroup$
    – Yemon Choi
    Commented Sep 12, 2022 at 15:39

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This is related to something which has always annoyed me: some authors define, say, the group algebra $k[G]$ of a group $G$ as the set of functions $G \to k$ with finite support, equipped with convolution. In my opinion this is a bad definition, because it misleads you as to the functoriality of this construction with respect to $G$. Namely, functions pull back but the group algebra pushes forward: if $f : G \to H$ is a group homomorphism, it induces an algebra homomorphism $k[G] \to k[H]$ which is annoying to write down in terms of functions but, in terms of the free vector space on $G$, trivial to write down as

$$\sum k_g g \mapsto \sum k_g f(g).$$

One way to describe what is going on is that the free vector space on a set $X$ consists of "distributions with finite support" on $X$. Functions pull back but distributions push forward, and this suggests the correct functoriality of taking the free vector space, which is covariant rather than contravariant. Moreover the free vector space functor is monoidal: it converts direct products into tensor products, which is an abstract explanation of why it converts groups into algebras.

The relevance of this discussion to continuous convolution is that if $G$ is, say, a locally compact Hausdorff group, and we want to define some sort of group algebra of $G$, whatever that construction means it should be covariantly functorial, which as above naturally suggests we ought to consider distributions rather than functions; if distributions are defined as dual to a topological vector space of functions which pull back then they will naturally push forward. Then the identity will be the Dirac delta supported at the identity, exactly as in the discrete case; functoriality with respect to the inclusion $1 \to G$ of the identity forces this.

The assignment $X \mapsto C_c(X)$ of continuous functions with compact support, on the other hand, is not functorial at all, so it's not a good replacement for the free vector space functor. This isn't about completeness, really.

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    $\begingroup$ Just a modest comment on the last paragraph. I don't think that the phrase " not functorial at all" is very helpful or even correct. This assignment IS functorial on the class of locally compact spaces, but with the proper mappings rather than the continuous ones as morphisms. This is, of course, just a semi-category rather than a category but that doesn't mean that it is not a useful circle of ideas. Another context where it is appropriate is the extension of Gelfand duality to one between locally compact spaces and commutative $C^\ast$-algebras. $\endgroup$
    – terceira
    Commented Aug 29, 2022 at 13:10
  • $\begingroup$ I was thinking about mentioning that but it's not relevant to this discussion because it's contravariant anyway. We're looking for a covariant functor in this setting. $\endgroup$
    – Qiaochu Yuan
    Commented Aug 29, 2022 at 17:39
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    $\begingroup$ Counterargument for someone who does like categorical thinking in functional analysis (as you know) - when doing any actual harmonic analysis on locally compact groups $C_c(G)$ is incredibly useful as a convolution algebra, both for building general theory and for doing actual calculations / 3-epsilon arguments ... $\endgroup$
    – Yemon Choi
    Commented Sep 12, 2022 at 15:36

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