# Why Is the Proximal Mapping (Proximal Operator) a Mapping?

The proximal map is defined in the sense of an operator as:

$\text{prox}_{\lambda f}(x) = \arg \min_y f(y) + \frac{1}{2\lambda}\|x-y\|^2$

I don't see why it is called a mapping. I suppose it is because $\arg \min$ might "return" not only one value but a few of them. What is an example of this situation?

What is an intuitive explanation why the proximal mapping is called a mapping? Can you give an example?

• "mapping" is just a synonym for "function". If $f$ is convex and lower semicontinuous then the $\arg \min$ set in this case is guaranteed to be a singleton. – littleO Jul 7 '18 at 10:22
• So why not call it "proximal function"? – 今天春天 Jul 7 '18 at 10:26
• In math, it's very common to use the word "mapping" as a synonym for "function". It happens all the time, not just in this case. Often people call it "proximal operator", also. The name "proximal function" would be ok. – littleO Jul 7 '18 at 10:34
• Ok, I understand...thank you :) maybe add an answer? – 今天春天 Jul 7 '18 at 10:40

It's just because the word "mapping" is a synonym for "function". The proximal mapping is also called the "proximal operator". If $f$ is convex and lower semicontinuous then the $\arg \min$ set in question is guaranteed to be a singleton.
• @max_zorn Hmm, maybe some authors require that the codomain of a function is either $\mathbb R$ or $\mathbb C$. Wikipedia says this: "Some authors, such as Serge Lang, use 'function' only to refer to maps in which the codomain is a set of numbers (i.e. a subset of the fields $\mathbb R$ or $\mathbb C$) and the term mapping for more general functions." en.wikipedia.org/wiki/Map_(mathematics)#Maps_as_functions – littleO Jul 9 '18 at 4:54