Download A first course in calculus, Fifth Edition by Serge Lang PDF

By Serge Lang

This 5th version of Lang's publication covers all of the issues typically taught within the first-year calculus series. Divided into 5 components, every one component to a primary direction IN CALCULUS comprises examples and functions with regards to the subject coated. additionally, the rear of the publication comprises designated suggestions to quite a few the workouts, permitting them to be used as worked-out examples -- one of many major advancements over prior variants.

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Example text

Dg is an isometry between C [0, 1]* and the set bv [0, 1] of functions 0 at 0, right-continuous except at 0, and of bounded variation on [0, 1]. From these examples we can now construct more normed spaces, which shall be called substitution spaces or product spaces. Definition 2. Let 5 be an index set and let X be a Banach space of real-valued functions on 5. If for each s in 5 a normed space N. is given, Jet PxN. be the space of all those functions x on 5 such that (i) x 8 is an element of N.

Iii) A set V is a neighborhood of 0 if and only if there is a compact set E in N such that E" ~ V. (iiic) A net is convergent to 0 if and only if it is uniformly convergent to zero on compact subsets of N. (iv) V is a neighborhood of 0 if and only if there is a sequence (y;, i E w) in N such that lim Y; = 0 and V ~ n {y;}". iEw iEw The corresponding equivalences for the bw-topology hold because the bw-topology in N is the relative topology imposed on Q (N) by the bw*-topology of N**; hence the bw- and bw*-topologies are locally convex.

Iq) lfq. (d) In l1(5)*, sEa 11111 = sup {itsl: s Ea}. From these calculations follows certain useful isometries, which help to clarify the structure of certain conjugate spaces; BANACH gives the case 5 = w. Theorem 1. l1 (5) is linearly isometric with c0 ( 5)* und er the mapping Ty =I if for every x in c0 (5) l(x) =}; y(s) x(s). sES Under the same sort of mapping m(5) is equivalent to l1 (5)*, and, if p > I and Ifp + Ifq =I, lq(5) is isometric to lP(5)*. Proof. For the first case, let a be a finite subset of 5 and define Pa, a projection of c0 (5), by Pax(s) = x(s) if s E0', = 0 if s 4 a.

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