By Percy Alexander MacMahon
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Then the R-topology on V is the weakest topology in which the mappings f : V .... K, that constitute R, are continuous. Proof. Assume V has a topology in which every member of R is continuous. The theorem will be proved if we show that the various loci in V are closed in the given topology, and this will follow if we establish that the typical principal locus CV(f), where fER, is closed. But this is clear because CV(f) is the inverse image of the finite set whose only member is the zero element of K.
We now see that 1/1: V-W is a K- morphism and 1/1* = w. Moreover by combining our observations we obtain the important Theorem 21. There is a natural bijection between the K-morphisms V - W and the homomorphisms K[W] - K[V] of K-algebras. /J*: K[W] .... K[V], where this is defined in the manner explained above. We shall make a fairly deep study of K-morphisms at a later stage. For the moment we shall content ourselves with some simple observations. First of all the identity mapping of V is a K- morphism and it is associated with the identity homomorphism of K[V].
L, and ~l' ~2' A, ••• , ~n IS L a base for Hom L (V , L). Accordingly, by Theorem 19, LA..... , ••• , ~n]· Let S be the L-algebra obtained by restricting the domain of the functions forming L[V L ] to V. The natural surjective homomorphism 53 of L-algebras which results is such that ~i 1-+ ~r Consequently ~ ] = n Theorem 32. L K[Y] . Let Y be an n-dimensional vector space over K. Then Y can be regarded as an affine set defined over K and yL as an ~ set defined over Remark. L. If the field K is infinit~ then yL = y~ Theorem 31 shows that the requirement that K be infinite cannot be left out.