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Today in my analytic Calculus course, we discussed the Algebraic and Order limit theorems of sequences: they provide operations like $+$ on both sequences $\{a_n\},\{b_n\}$ and the values they converge to $a,b$.

We also made the connection that, for example, the additive structure ($+$) is preserved across the set of sequences and the set of real numbers.

I am seeking the name of such structure-preservations. My first guess was isomorphism, possibly natural, but that doesn't feel right: my understanding of isomorphisms is basically that there is a one-to-one correspondence between elements in two sets.

My background is in computer science, and I have only studied the terminology of maps and (homo/iso)morphisms enough to understand their connections to functional programming—even then, I would not say that I understand them beyond an intuitive level.

I have seen and read this question about maps, but I confess it is a bit beyond me. I'm looking for a simpler explanation.

Asaf Karagila
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D. Ben Knoble
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    This is simply a consequence of the fact that $f:(a,b)\mapsto f(a,b)=a+b$ is continuous, (hence $\lim_n (a_n+b_n)= (\lim_n a_n)+(\lim_n b_n)$), no? AFAIK there is no special name for that (beside continuity of the + application) – Picaud Vincent Jun 27 '19 at 17:53
  • @PicaudVincent the theorem is a simple consequence, yes (though we proved it using an $\epsilon$-definition, without limits). I am more interested in the idea that sequences and real numbers have similar additive structures. – D. Ben Knoble Jun 27 '19 at 17:57

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Structure-preserving maps are generally called homomorphisms. For example, maps that preserve the group structure are called group homomorphisms, maps that preserve order are called order homomorphisms, and so on.

However, there are a few cases that are important enough to get their own names. For example, vector space homomorphisms are called linear functions or linear operators. In particular, the real sequences form a vector space over the real numbers, and so do (trivially) the real numbers themselves. Also, the convergent sequences form form a subspace of the vector space of all sequences. The limit function preserves the vector space function, therefore it is a linear function from the space of convergent series to the real numbers.

However, the convergent series also form an algebra, with multiplication being element-wise and the constant sequence $1,1,1,\ldots$ as neutral element. The limit also preserves this structure, therefore it is an algebra homomorphism. Of course every algebra is a vector space, and every algebra homomorphism is a linear function.

If you ignore all sorts of multiplication and only look at the addition, you get an abelian group (as with every vector space). And of course the limit is also a group homomorphism for addition.

In particular:

  • A group homomorphism (for additive groups) has the property $$\phi(a+b)=\phi(a)+\phi(b).$$ The limit on convergent sequences clearly has this property: $$\lim_{n\to\infty} (a_n+b_n) = \lim_{n\to\infty} a_n + \lim_{n\to\infty} b_n$$

  • A linear function (vector space homomorphism) has the additional property that for a scalar (in this case, real number) $\alpha$, we have $$\phi(\alpha a)=\alpha \phi(a).$$ The limit also has that property: $$\lim_{n\to\infty} (\alpha a_n) = \alpha \lim_{n\to\infty} a_n$$

  • An algebra homomorphism in addition has the property that $$\phi(ab)=\phi(a)\phi(b).$$ Again this is true for the limit: $$\lim_{n\to\infty} (a_n b_n) = \lim_{n\to\infty} a_n \cdot \lim_{n\to\infty} b_n$$

celtschk
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