A vector-valued function is:

... a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the dimension of the domain is not defined by the dimension of the range.

Therefore, considering the function to be a map within the real numbers...

$$\underbrace{\mathbb R \times \mathbb R \times \dots \times \mathbb R}_{\text{1 to }\infty}=\underbrace{\; \mathbb R^n\;}_{1\leq n} \to \underbrace{\mathbb R \times \mathbb R \times \dots \times \mathbb R}_{\text{1 to }\infty}=\underbrace{\,\mathbb R^m\,}_{1< m}$$

The question is what would be the name for a function mapping vectors to scalars - like the typical temperature across a surface. These function would map

$$\underbrace{\mathbb R \times \mathbb R \times \dots \times \mathbb R}_{\text{1 to }\infty}=\underbrace{\,\mathbb R^n\,}_{1\leq n} \to \mathbb R$$

Naming them "scalar functions" doesn't seem to necessarily imply that the domain is a vector, focusing only on the codomain. Likewise, "multivariate" ignores the codomain.


From the point of view of set theory, a function $f$ from $X$ to $Y$ is a subset of the Cartesian product $X\times Y$ which satisfies the property that $$ (x,y) \in f \ \text{and}\ (x',y) \in f \implies x=x'. $$ In essence, a function is a subset of the Cartesian product which satisfies the "vertical line property".

However, we typically don't like to think of functions in this manner. Instead, we like to think of functions as "arrows" which take an input from some domain and give an output in some codomain, perhaps according to some formula or rule. From this point of view, a function is defined by three data:

  1. a domain $X$,
  2. a codomain $Y$, and
  3. a rule which assigns elements of $X$ to elements of $Y$.

In describing a function in plain English (rather than notation), we might want to emphasize either the domain or the codomain, or properties of either of these sets.

If we want to emphasize some property of the codomain, we say that $f$ is a "$[Y]$-valued function". This means that the codomain of $f$ is either $[Y]$ where $[Y]$ is some set, or that the codomain of $f$ has some property $[Y]$. For example, a "vector-valued function" is a function whose codomain is a vector space, while an "$\mathbb{R}$-valued function" is a function with the real numbers as its codomain (such a function may also be described as "scalar valued" in many contexts).

If we want to emphasize some property of the domain, we typically just say that $f$ is "a function on $[X]$", where $[X]$ is a set. If the actual domain is not that important, but we only care about some property of that domain, we might say that $f$ is "a function on [some kind of space]", e.g. $f : \mathbb{R}^n \to \mathbb{R}$ could be described as a function on a vector space, or "a function of a vector variable". That being said, we generally like to specify the domains of functions fairly exactly, so I would think that such a vague description of the domain of a function would be uncommon.

In the cases raised by the question,

  • a function $f : \mathbb{R}^m \to \mathbb{R}^n$ is a vector-valued function on $\mathbb{R}^m$, or perhaps a vector-valued function of a vector variable, or even a vector-valued function of multiple variables (if we don't care so much about the vector space structure of $\mathbb{R}^m$);
  • a function $f : \mathbb{R}^m \to \mathbb{R}$ is a scalar-valued function on $\mathbb{R}^m$, or maybe a scalar-valued (or real-valued) function of a vector variable; and
  • a function $f : \mathbb{R} \to \mathbb{R}^n$ is a vector-valued function on $\mathbb{R}$, or a vector-valued function of a scalar variable, or perhaps simply a scalar function of several variables.

The various phrases used above can be mixed-and-matched a little, and are all relatively informal. They should be easily understood by most readers (particularly in the kinds of contexts where they are used, which is probably a bit tautological, but... oh, well), but can and should be more explicitly defined if there is any potential for ambiguity.

| cite | improve this answer | |
  • $\begingroup$ Excellent answer. I was hoping you would say something about "maps." It would be great in the context of your background definition of "functions." $\endgroup$ – Antoni Parellada Aug 17 '19 at 20:27
  • $\begingroup$ @AntoniParellada "Map" and "function" are virtual synonyms. If there is any distinction at all, I suspect that it is related to the idea that a map should preserve some structure. That is, a "function" is just any ol' arbitrary subset of the Cartesian product which has the vertical line property, while a "map" is a function which is, for example continuous (i.e. a morphism in the category of topological spaces), or smooth (i.e. a morphism in the category of smooth manifolds), or something like that. $\endgroup$ – Xander Henderson Aug 17 '19 at 20:31

It's just a scalar-valued function. It can depend on one or more inputs but it has a scalar output. In my mind, I usually see a function mapping $ \mathbb{R^{\mathit{n}}} \to \mathbb{R}$ and call it an objective function, because it is most likely an optimization problem.

If you wanted to specifically describe functions which assign a scalar value, such as the temperature, to every point on a 2- or 3-dimensional space, you could use the term "scalar field".

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.