Editing Vector space (section)

===Function spaces===
{{Main|Function space}}
[[File:Example for addition of functions.svg|class=skin-invert-image|thumb|Addition of functions: the sum of the sine and the exponential function is <math>\sin+\exp:\R\to\R</math> with <math>(\sin+\exp)(x)=\sin(x)+\exp(x)</math>.]]

Functions from any fixed set {{math|Ω}} to a field {{math|''F''}} also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions {{math|''f''}} and {{math|''g''}} is the function <math>(f + g)</math> given by
<math display=block>(f + g)(w) = f(w) + g(w),</math>
and similarly for multiplication. Such function spaces occur in many geometric situations, when {{math|Ω}} is the [[real line]] or an [[interval (mathematics)|interval]], or other [[subset]]s of {{math|'''R'''}}. Many notions in topology and analysis, such as [[continuous function|continuity]], [[integral|integrability]] or [[differentiability]] are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.{{sfn|Lang|1993|loc = ch. XII.3., p. 335}} Therefore, the set of such functions are vector spaces, whose study belongs to [[functional analysis]].