By Paul R. Halmos

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**Sample text**

Prove that a subset of '0 is an equivalence class modulo ml: if and only if it is a coset of ml:. 4. (a) Suppose that ml: is a subspace of a vector space '0. , an element of '0') j the linear functional z is defined by z(x) = y(x + ml:). Prove that the correspondence Y -+ Z is an isomorphism I between ('O/ml:)' and ml:0 • (b) Suppose that ml: is a subspace of a vector space '0. , an element z of ml:') j the linear functional z is defined by z(x) = y(x). Prove that z is unambiguously determined by the coset JC (that is, it does not depend on the particular choice of y), and that the correspondence :Ie -+ z is an isomorphism between 'O'/ml:0 and ml:'.

It is, however, at this point that the greatest advantage of the notation [x, y] appears; by means of it, it is easy to discuss 'O and its relation to 'O". If we consider the symbol [x, y] for some fixed y = Yo, we obtain nothing new: [x, Yo] is merely another way of writing the value Yo(x) of the function Yo at the vector x. If, however, we consider the symbol [x, y] for some fixed x = Xo, then we observe that the function of the vectors in 'Of, whose value at y is [xo, y], is a scalar-valued function that happens to be linear SEC.

Warning: the assertion that a vector space is reflexive if and only if it is finitedimensional would shock most of the experts in the subject. The reason is that the customary and fruitful generalization of the concept of reflexivity to infinitedimensional spaces is not the simple-minded one given in (h). § 18. Direct SUll1S We shall study several important general methods of making new vector spaces out of old ones; in this section we begin by studying the easiest one. If 'U and '0 are vector spaces (over the same field), their direct sum is the vector space VI' (denoted by 'U E9 '0) whose elements are all the ordered pairs (x, y) with x in 'U and y in '0, with the linear operations defined by DEFINITION.