[graycat]

"Functions Are Vectors"?

Let's see: Let A be a non-empty set, N the set of positive whole numbers, and X the set of all functions f

f: A --> N

with usual notation.

Assume as is common, the scalars are the set of real numbers, but the set of complex numbers will also do.

So, is X a vector space and, thus, each f in X a vector?

No, since -f is not in X. Neither is (1/2)f.

Some references (with TeX markup):

Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition\/}

linear algebra treated as functional analysis.

Walter Rudin, {\it Real and Complex Analysis\/}

with Lebesgue integration and, then, Banach and Hilbert vector spaces.

Walter Rudin, {\it Functional Analysis\/}

with Fourier theory.

Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability\/}

with random variables, that is, functions from a probability space to, usually, the set of real numbers with convergence results, building on the work A. Kolmogorov building on the work of H. Lebesgue.

[Tainnor]

Functions with arbitrary codomains are not vectors, but when your codomain is some field, they are.

This is what the article is very explicitly about. I guess you can quibble that the title is imprecise, but it's just a title and the article makes it clear.

[graycat]

In all those books I listed, and more, e.g., on axiomatic set theory and other foundations, never saw definition or mention of codomain. So, the term is obscure. Readers are supposed to guess at the meaning of obscure terms?

What I did was follow, as in the references, long established convention, that for a function to be a vector at least it had to be in a vector space where (1) can multiply a function by a number (e.g., reals or complex) and (2) add two functions and still get a function in the vector space. To be general, I omitted metrics, inner products, topologies, convergence, probability spaces, and more.

Or, as in the references I gave, math talks about vector spaces and vectors, and each vector is in a vector space. The references are awash in definitions of vector spaces with (1) and (2) and much more.

Computing is awash in indexes for data, e.g., B-trees, SQL (structured query language) operations on relational data bases, addressing in central processors, collection classes in Microsoft's .NET, REDIS, and calling all such also functions confuses established material, conventions, and understanding.

[Tainnor]

Codomain is not an obscure term. One second of Googling would have helped.

[selimthegrim]

Codomain is different than just range; just think for a bit. In the older literature you are used to their usage is probably synonymous though.

[taktoa]

Not just when the codomain is a field, but more generally when the codomain is itself a vector space. The former is a special case of the latter where you construct a 1D vector space from a field.

[bmacho]

> Functions with arbitrary codomains are not vectors,

Well, not with the operations pulled from the codomain at least.