Examples of vector spaces: Difference between revisions

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Perhaps the most important example of a vector space is the following. For any [[Positive number|positive]] [[integer]] ''n'', the space of all ''n''-tuples of elements of '''F''' forms an ''n''-dimensional vector space over '''F''' sometimes called ''[[coordinate space]]'' and denoted '''F'''^''n''. An element of '''F'''^''n'' is written

:<math>x = (x_1, x_2, \ldots, x_n) \,</math>

where each ''x''_''i'' is an element of '''F'''. The operations on '''F'''^''n'' are defined by

:<math>x + y = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n) \,</math>

:<math>\alpha x = (\alpha x_1, \alpha x_2, \ldots, \alpha x_n) \,</math>

:<math>0 = (0, 0, \ldots, 0) \,</math>

:<math>-x = (-x_1, -x_2, \ldots, -x_n) \,</math>

The most common cases are where '''F''' is the field of [[real number]]s giving the [[real coordinate space]] '''R'''^''n'', or the field of [[complex number]]s giving the [[complex coordinate space]] '''C'''^''n''.

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The vector space '''F'''^''n'' comes with a [[standard basis]]:

:<math>e_1 = (1, 0, \ldots, 0) \,</math>

:<math>e_2 = (0, 1, \ldots, 0) \,</math>

:<math>\vdots \,</math>

:<math>e_n = (0, 0, \ldots, 1) \,</math>

where 1 denotes the multiplicative identity in '''F'''.

==Infinite coordinate space==

Let '''F'''^∞ denote the space of [[infinite sequence]]s of elements from '''F''' such that only ''finitely'' many elements are nonzero. That is, if we write an element of '''F'''^∞ as

:<math>x = (x_1, x_2, x_3, \ldots) \,</math>

then only a finite number of the ''x''_''i'' are nonzero (i.e., the coordinates become all zero after a certain point). Addition and scalar multiplication are given as in finite coordinate space. The dimensionality of '''F'''^∞ is [[countably infinite]]. A standard basis consists of the vectors ''e''_''i'' which contain a 1 in the ''i''-th slot and zeros elsewhere. This vector space is the [[coproduct]] (or [[direct sum of modules|direct sum]]) of countably many copies of the vector space '''F'''.

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:''See main article at [[Function space]], especially the functional analysis section.''

Let ''X'' be a 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>

where the operations on the right hand side are those in ''V''. The zero vector is given by the constant function sending everything to the zero vector in ''V''. The space of all functions from ''X'' to ''V'' is commonly denoted ''V''^''X''.