Examples of vector spaces: Difference between revisions

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==Function spaces==
:''See main article at [[Function space]], especially the functional analysis section.''
Let ''X'' be ana non-empty arbitrary set and ''V'' an arbitrary vector space over '''F'''. The space of all [[function (mathematics)|function]]s from ''X'' to ''V'' is a vector space over '''F''' under [[pointwise]] addition and multiplication. That is, let ''f'' : ''X'' → ''V'' and ''g'' : ''X'' → ''V'' denote two functions, and let ''α''∈'''F'''. We define
:<math>(f + g)(x) = f(x) + g(x) \,</math>
:<math>(\alpha f)(x) = \alpha f(x) \,</math>