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By De Simone A., Mundici D.

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Clearly, it is a vector space with respect to operations of addition and multiplication by elements of K that are defined just as they are for rows of finite length. We say that a sequence is finitary if only a finite number of its entries are nonzero. Finitary sequences form a subspace in the space of all sequences. Denote it by K°°. ,0,1,0,... ), i = 1,2,... (the identity is in the ith place). Hence KO° is countable-dimensional. 66, we can prove that every countable-dimensio- nal space over K is isomorphic to K.

Of course, K" has other bases as well. 65. 56). The coordinates of a complex number in this basis are its real and imaginary parts. 8. 66. Any vector space V over afield K with a basis consisting of n vectors is isomorphic to the space Kn. Proof. Let {el, e2, ... , en} be a basis of V. Consider the map cp: V - Kn that assigns to each vector the row of its coordinates in the basis {e1, e2, ... , en}. Clearly, this map is bijective. +bnen, then a + b = (al + b1)el + (a2 + b2)e2 + - + (an + bn)en, Aa = (Aal)el + (Aa2)e2 + ...

Consider the map cp: V - Kn that assigns to each vector the row of its coordinates in the basis {e1, e2, ... , en}. Clearly, this map is bijective. +bnen, then a + b = (al + b1)el + (a2 + b2)e2 + - + (an + bn)en, Aa = (Aal)el + (Aa2)e2 + ... + (Aan)e0. Hence, p is an isomorphism. 67. The space E2 (respectively, E3) is isomorphic to R2 (respectively,1R3). 8. Algebras Because their structure is so simple, vector spaces are not very interesting on their own. However, the notion of a vector space is a part of many algebraic (and not just algebraic) theories.

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