# What mathematical object accepts a sequence as input?

A function inputs a single number and outputs a single number. (e.g. $y = f(x)$)

What mathematical object inputs a sequence, or even just a vector or tuple?

Edit:

I understand that a function doesn't have to take a single number and output a single number. But what do you call the functions that don't? Can you give me some Google-able terms, please?

Example #1: the "derivative" (calculus) operation inputs a function and outputs a function. So it is clearly different than the basic/classic/algebraic "function" $y = f(x)$.

Example #2: the "union" (set-theory) operation inputs two sets and outputs a set.

Can you provide an example of any operation that inputs a sequence?

• Functions don't have to take numbers as arguments or return numbers. You can even have a function taking as an argument an apple and returning the person who wants to eat it the most. – Wojowu Oct 31 '15 at 20:59
• They are called sequences eaters and they are very dangerous to deal with. – Jack D'Aurizio Oct 31 '15 at 20:59
• One can even take the point of view that everything is a function. – André Nicolas Oct 31 '15 at 21:02

A function can map from any set to any other set. However there are some names commonly used for specific cases:

• A function that maps functions to numbers is often called a functional.
• A linear function between linear spaces is often called an operator.
• A linear function that maps vectors to scalars is called covector or $1$-form.
• A $k$-linear antisymmetric function from $k$-tuples of vectors to scalars is called a $k$-form.
• A function that preserves some structure is often called a map.
• A function from the natural numbers to some set is called a sequence.
• A function on pairs written in infix form is called an operation.
• Thank you, this is what I was looking for, additional terms to research. K-forms were especially helpful. – Sam Porch Nov 3 '15 at 8:25

Two examples:

The function $f$ which operates on a sequence by adding 3 to every member of the sequence; this gives another sequence. If we call $S$ the set of all sequences, $f:S\to S$.

The function $g$ which operates on a sequence and whose value is the limit of the sequence if the limit exists and whose value is 123.444 (or any value you want to imagine) if the limit does not exist. This is a function from $S$ to $\mathbb {R}$.

In modern usage, a "function" can input anything and output anything. But in more classical usage, "function" meant something that took numbers as input and returned numbers as output, and other words were used for more complicated functions. My impression is that it was common to use "operator" for something that took functions as input and returned functions as output. This usage still persists in terms like "differential operator."

A basic example of an operator acting on sequences is the finite difference operator, which takes as input a sequence $a_0, a_1, a_2, \dots$ and returns as output the sequence

$$a_1 - a_0, a_2 - a_1, a_3 - a_2, \dots$$

That might be the definition you used in calculus, but really when mathematicians use the word 'function' it's a much more general notion. We can define a function from any set to any set. Not just the reals.

To answer your request for further research terms, you want to study relations. Functions are actually a special case of relations.

To keep up with your edits, there are lots of examples in math. I can write one down right now. Let $A$ be the set of eventually zero integer sequences. Let $f:A\to \mathbb{N}$ by summing the sequences.

An example would be a summation operator, which could be said to take a summable sequence as "input" and yield the (finite) sum, or sum in the limiting sense, of the sequence as "output". Thus $\sum$ could be considered as a function mapping the set of (say) real summable sequences to the set of real numbers (although the idea of summation has such deep historical roots that it wouldn't normally be introduced in this way). The product operator $\prod$ is analogous. Other examples are $\sup$, $\inf$, $\limsup$, and $\liminf$.

Well, actually... the real numbers are equivalent to rational sequence classes which converge to that number so any real number function is equivalent to a function taking in a sequence as an input. (Although the domain the domain of one representative sequence from an infinite class of sequences, all converging to the same point might be whiffy.)

" So it is clearly different than the basic/classic/algebraic "function" y=f(x)"

Actually it's not as clear as you'd think. The only real (pun unintended) difference is those algebraic functions have a specific domain and range. So for as I know there aren't any terms to differentiate functions that use numbers vs. anything else. If anything, it's real valued functions that are the exception and distinguished by the term "real valued functions".

So for googling terms I don't have many suggestions other than "functions that take sequences as input".

Examples of functions with sequences are plentiful. F: sequence space -> extended reals, F({a_n} = lim {a_n} is obvious.