Theta Functions from a Geometric Viewpoint
Let X denote a finite dimensional vector space over R and L a lattice in X. (We shall see that X will play the role of W= X × X* in Chapter I.) Then L spans X over R. Otherwise, let X′ denote the subspace of X spanned by L over R. Then the quotient group X/X′ is a vector space over R of a positive dimension. On the other hand, it is a quotient group of the compact group X/L; hence X/X′ is compact. But this is a contradiction.
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