Editing Integral (section)

===Linearity===
The collection of Riemann-integrable functions on a closed interval {{math|[''a'', ''b'']}} forms a [[vector space]] under the operations of [[pointwise addition]] and multiplication by a scalar, and the operation of integration

: <math> f \mapsto \int_a^b f(x) \; dx</math>

is a [[linear functional]] on this vector space. Thus, the collection of integrable functions is closed under taking [[linear combination]]s, and the integral of a linear combination is the linear combination of the integrals:<ref name=":0">{{Harvnb|Apostol|1967|p=80}}.</ref>

: <math> \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,</math>

Similarly, the set of [[Real number|real]]-valued Lebesgue-integrable functions on a given [[Measure (mathematics)|measure space]] {{mvar|E}} with measure {{mvar|μ}} is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral

: <math> f\mapsto \int_E f \, d\mu </math>

is a linear functional on this vector space, so that:<ref name=":3" />

: <math> \int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu. </math>

More generally, consider the vector space of all [[measurable function]]s on a measure space {{math|(''E'',''μ'')}}, taking values in a [[Locally compact space|locally compact]] [[Complete metric space|complete]] [[topological vector space]] {{mvar|V}} over a locally compact [[Topological ring|topological field]] {{math|''K'', ''f'' : ''E'' → ''V''}}. Then one may define an abstract integration map assigning to each function {{mvar|f}} an element of {{mvar|V}} or the symbol {{math|''∞''}},

: <math> f\mapsto\int_E f \,d\mu, \,</math>

that is compatible with linear combinations.<ref>{{Harvnb|Rudin|1987|p=54}}.</ref> In this situation, the linearity holds for the subspace of functions whose integral is an element of {{mvar|V}} (i.e. "finite"). The most important special cases arise when {{mvar|K}} is {{math|'''R'''}}, {{math|'''C'''}}, or a finite extension of the field {{math|'''Q'''<sub>''p''</sub>}} of [[p-adic number]]s, and {{mvar|V}} is a finite-dimensional vector space over {{mvar|K}}, and when {{math|''K'' {{=}} '''C'''}} and {{mvar|V}} is a complex [[Hilbert space]].

Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of [[Daniell integral|Daniell]] for the case of real-valued functions on a set {{mvar|X}}, generalized by [[Nicolas Bourbaki]] to functions with values in a locally compact topological vector space. See {{Harvnb|Hildebrandt|1953}} for an axiomatic characterization of the integral.