# Alternative function notation?

I have a professor that uses the notation

$$\lambda (x_1,x_2,\dots ,x_n)\ . \ c_1x_1 +c_2x_n + \dots +c_nx_n \colon \ \mathbb{R}^n \longrightarrow \mathbb{R}$$

for a function

$$f \colon \ \mathbb{R}^n \longrightarrow \mathbb{R}, \quad f(x_1,x_2,\dots,x_n)=c_1x_1+c_2x_2+\dots +c_nx_n$$

This is a Linear Optimization course but I don't think that's relevant. Also, the professor says a map isn't the same as a function.

I mean, it is simple enough to be understood, but I don't think I've ever seen that notation anywhere else, so is this an alternative accepted notation, or not really?

• I'm guessing it's something to do with lambda calculus-esque notation. (edit: unless it really is a dot instead of a period) Commented Mar 12, 2013 at 1:58
• Absolutely sure. I know what the dot product is, but it's not the case in here. It's always a dot "bellow" like it would be on a regular sentnce (just like in the question). Commented Mar 12, 2013 at 2:00
• Ah, @TylerBailey has it. Definitely is lambda calculus. Not sure what he means by the distinction between a map and a function - that came up in another problem today, too. If it is a programming class, I suppose a "function" is actual code, and a "map" is what a mathematician considers a function? That is, two different functions can compute the same "map?" Commented Mar 12, 2013 at 2:01
• From wikipedia: "the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function" ... "Some authors, such as Serge Lang, use "map" as a general term for an association of an element in the range with each element in the domain, and "function" only to refer to maps in which the range is a field." In other words, everybody is correct because it's just a matter of the way you set up your definitions. Commented Mar 12, 2013 at 2:38

See lambda calculus ... http://en.wikipedia.org/wiki/Lambda_calculus

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• Hmm that's definetly it then.. This professor also teaches Mathematical Logic, and apparently lambda calculus has to do with it. Commented Mar 12, 2013 at 2:04

I've never seen that notation before but as long as you can follow it that's all that matters.

As for the statement "a map is not a function" it really depends on the setting and the definitions given. Maybe in this course you've defined "map" as a linear transformation?

• this doesn't really answer the question. Commented Mar 12, 2013 at 2:01
• Good point. I was really responding the the implied secondary question "the professor says a map isn't a function." I would have commented it but I don't have that privilege yet.
– mv3
Commented Mar 12, 2013 at 2:06
• I didn't realize new users weren't allowed to comment... seems pretty inane to me! I'll help you get there :p Commented Mar 12, 2013 at 2:07