 By C. H. Edwards

ISBN-10: 0486131955

ISBN-13: 9780486131955

Smooth conceptual therapy of multivariable calculus, emphasizing the interaction of geometry and research through linear algebra and the approximation of nonlinear mappings by way of linear ones. while, plentiful recognition is paid to the classical functions and computational equipment. thousands of examples, difficulties and figures. 1973 edition.

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Additional info for Advanced Calculus of Several Variables (Dover Books on Mathematics)

Example text

As a consequence, it follows that the norm p is absolutely continuous if and only if L,*consists only of integrals. , if there is no subset of X of positive measure such that all f E Li vanish on this subset (cf. 6). in the sense that every G E (Li)* is of the form I- for some g E Lp,and all f E Lp, and such that IlGll = p ‘ ( g ) (cf. 7). As a final theorem we mention that the space L, is reflexive if and only if the norms p and p’ are absolutely continuous and p has the Fatou property (cf. 2).

V) We recall that a subset A of a topological space is called regularly open whenever A = A - " . Show that the ideal I in X satisfies I = Zdd if and only if { P } i is regularly open. We recall that the collection of all regularly CH. 1 , § 81 COMPACTNESS AND SEPARATION PROPERTIES 39 open subsets of 9 is an order complete Boolean algebra (cf. 16). Show that the lattice of all ideals Zin XsatisfyingZ = Iddis lattice isomorphic to the Boolean algebra of all regularly open subsets of 9. Hence, the lattice of all ideals Z in X satisfying Z = Zdd is also an order complete Boolean algebra.

Check the statement that = {M},, holds if and only if {x}dd= {y}ddholds, and show that {x}" = {y}dd is not the same as x = y. Finally, determine explicitly the Stone representation of X . , 7. The hull-kernel topology As in the preceding section, let 9 'be the set of all proper prime ideals in the distributive lattice X with smallest element 8, and let 9 be a non-empty subset of 9. Also, just as before, given x E X , let {R}xbe the set of all R E 92 such that x is no member of R. We recall that {R}#is empty and ( { R } , : x E X )= 9.