I have read the wiki page (https://en.wikipedia.org/wiki/Functional_(mathematics) but it is not helping.

I understand what a regular function is

input > do something to that input > output.

but not what functional is,

Wikipedia says "from a vector space into its underlying field of scalars"

That wording is confusing.

So the input is a vector but the output is a scalar? The only way I know to get scalar from vector is a dot product.

And a vector is just written as , if you take that as an input isn't it just essentially a regular function of 2 variables? y=f(a,b)


1 Answer 1


There are many ways you can go from a vector space to a scalar, such as your example of a dot product. The wiki page also gives another example of a functional as with the arclength calculation for a vector f(t).

You're assumption is roughly correct that a functional is a function that maps from a vector space to a scalar space, and thus covers a broad number of concepts.


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