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→Finite vector spaces: projective geometry is great, but anyone who does it knows what a finite vector space is already. Tag: references removed |
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Note the role of the finiteness condition here. One could consider arbitrary sequences of elements in '''F''', which also constitute a vector space with the same operations, often denoted by '''F'''<sup>'''N'''</sup> - see [[Examples of vector spaces#Function spaces|below]]. '''F'''<sup>'''N'''</sup> is the ''[[product (category theory)|product]]'' of countably many copies of '''F'''.
By Zorn's lemma, '''F'''<sup>'''N'''</sup> has a basis (there is no obvious basis). There are [[uncountably infinite]] elements in the basis. Since the dimensions are different, '''F'''<sup>'''N'''</sup> is ''not'' isomorphic to '''F'''<sup>∞</sup>. It is worth noting that '''F'''<sup>'''N'''</sup> is (isomorphic to) the [[dual space]] of '''F'''<sup>∞</sup>, because a linear map ''T'' from '''F'''<sup>∞</sup> to '''F''' is determined uniquely by its values ''T''(''e<sub>i</sub>'') on the basis elements of '''F'''<sup>∞</sup>, and these values can be arbitrary. Thus one sees that a vector space need not be isomorphic to its double dual if it is infinite dimensional, in contrast to the finite dimensional case.
==Product of vector spaces==
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