Editing Representable functor (section)

== Analogy: Representable functionals ==
Consider a linear functional on a complex [[Hilbert space]] ''H'', i.e. a linear function <math>F: H\to\mathbb C</math>. The [[Riesz representation theorem]] states that if ''F'' is continuous, then there exists a unique element <math>a\in H</math> which represents ''F'' in the sense that ''F'' is equal to the inner product functional <math>\langle a, -\rangle </math>, that is <math>F(v) = \langle a,v\rangle </math> for <math>v\in H</math>. 

For example, the continuous linear functionals on the [[Square-integrable function|square-integrable function space]] <math>H = L^2(\mathbb R)</math> are all representable in the form <math>\textstyle F(v) = \langle a,v\rangle =  \int_{\mathbb R} a(x)v(x)\,dx</math> for a unique function <math>a(x)\in H</math>. The theory of [[Distribution (mathematics)|distributions]] considers more general continuous functionals on the space of test functions <math>C=C^\infty_c(\mathbb R)</math>. Such a distribution functional is not necessarily representable by a function, but it may be considered intuitively as a generalized function. For instance, the [[Dirac delta function]] is the distribution defined by <math>F(v) = v(0)</math> for each test function <math>v(x)\in C</math>, and may be thought of as "represented" by an infinitely tall and thin bump function near <math>x=0</math>.  

Thus, a function <math>a(x)</math> may be determined not by its values, but by its effect on other functions via the inner product. Analogously, an object ''A'' in a category may be characterized not by its internal features, but by its [[Functor represented by a scheme|functor of points]], i.e. its relation to other objects via morphisms. Just as non-representable functionals are described by distributions, non-representable functors may be described by more complicated structures such as [[Stack (mathematics)|stacks]].