Can there be "polynomial spaces"?

Question

I don't know how to better frame this question. Thinking about vector spaces and their role in basically everything Calculus touched, I can understand why they are so central, especially in areas like differential geometry. But Taylor's Theorem got me thinking; my personal view of what Taylor's Theorem tells us is that the derivative is not a "one ocurrence" phenomenon, it is a part of a larger set of objects that approximate the function to a polynomial. That is, the derivative is not the most central object, in a fundamental way, just the most convenient I guess? Since it is easier to deal with linear functions than with, say, quadratic or cubic functions, and of course, it is easier to work with vector spaces than with other kinds of spaces.
Considering that vector spaces' structure is, in some sense, fine tuned, so that linear applications preserve their structure, what space would be such that quadratic applications preserve its structure? Or cubic applications? Are there such "polynomial" spaces? What properties might they have? Are they useful? My guess is that in principle they would be useful, because just as we can approximate functions to linear or quadratic functions and build powerful analysis from that, one might guess that we could approximate non-linear spaces with "sums" of these "polynomial spaces", perhaps even yielding a notion of "derivatives for a space"?
This might be a meaningless question but I would appreciate some insight.
EDIT:
For illustrating what I mean. Consider a vector space $V$. Then a linear map $T$ from $V$ to another vector space preserves the structure of $V$ a vector space, that is, $$T(\alpha u+\beta v)=\alpha T(u)+\beta T(v), u,v\in V$$ Or in other words, the image of the vector space $V$ is also a vector space. What I'm looking for is a space $Q$ such that quadratic applications would have the property: $$T(\alpha u+\beta v)=\alpha^2 u^2+2\alpha \beta uv+\beta^2v^2, u,v \in Q$$ Or in other words, the image of Q by T is also a "----" space, that is, it preserves its structure, whichever it might be.

There are algebraic varieties, defined as the set of solutions of a system of polynomial equations over a field. They are studied in algebraic geometry, modern number theory and many other fields. "Are they useful"? Yes, indeed. — Dietrich Burde, Feb 04 '22 at 17:49
You might also be interested in manifolds, which locally look like Euclidean vector spaces — Milten, Feb 04 '22 at 17:50
@Milten I've recently started studying differential geometry and analysis as preparation for that :) — Lourenco Entrudo, Feb 04 '22 at 17:55
Here is an exercise. Prove that the set of polynomials with coefficients in your favorite field form a vector space over that field. — John Douma, Feb 04 '22 at 18:08
@JohnDouma that is certainly a good exercise but it isn't really what I'm asking. I'm not asking if the set of polynomials constitutes a vector space over some field, but rather if there are spaces that preserve "quadratic structure" as the vector space preserves "linear structure" — Lourenco Entrudo, Feb 04 '22 at 18:13
How do you define quadratic structure? Do you mean $f(x^2)=(f(x))^2$? — John Douma, Feb 04 '22 at 18:15
Aspects of (abstract) approximation theory deal with finite dimensional subspaces for approximative purposes. See Encyclopedia of Mathematics's article Best approximation and the google search for "Kolmogorov n-width" (and the 1985 book n-Widths in Approximation Theory by Allan Pinkus), among other topics. (continued) — Dave L. Renfro, Mar 08 '22 at 14:55
I know almost nothing about these topics, but I'm sure there are others here who are knowledgeable about modern approximation theory and functional analysis, and who might be willing to write a brief low-level overview. The subject itself is very far from being obscure and little known. (This last sentence is intentional word-play, and not accidental, in case anyone was curious.) — Dave L. Renfro, Mar 08 '22 at 15:02
Should the right hand side of the equality in your edit read $\alpha^2 T(u)^2 + 2\alpha \beta T(u) T(v) + \beta^2 T(v)^2$? — Sambo, Mar 14 '22 at 20:32

score 2 · Answer 1 · answered Mar 08 '22 at 20:15

Operator algebras and Banach algebras are structures that allow linear operations and multiplication. If you add a conjugation operation, you obtain $C^*$-algebras. I especially recommend the book by Arveson cited below.

A different direction is explored in the beautiful book "Rings of continuous functions" by Gilman and Jerison.

REFERENCES

https://en.wikipedia.org/wiki/Banach_algebra

https://en.wikipedia.org/wiki/Operator_algebra

https://en.wikipedia.org/wiki/C*-algebra

Arveson, W. (1976), An Invitation to C*-Algebra, Springer-Verlag, ISBN 0-387-90176-0.

Can there be "polynomial spaces"?

1 Answers1