Q1 solve real analysis by rudin chp 7 sequence and Series of functions SEQUENCES AND SERIES OF FUNCTIONS 165Theorem 7.32 does not hold for complex algebras. A counterexample isgiven in Exercise 21. However, the conclusion of the theorem does hold, evenfor complex algebras, if an extra condition is imposed on A, namely, that abe self-adjoint. This means that for every fesd its complex conjugate f mustalso belong to A; J is defined by (x) = f(x).7.33 Theorem Suppose A is a self-adjoint algebra of complex continuousfunctions on a compact set K, A separates points on K, and A vanishes at nopoint of K. Then the uniform closure 3 of sA consists of all complex continuousfunctions on K. In other words, A is dense 8(K).Proof Let AR be the set of all real functions on K which belong to d.If feA and f = u + iv, with u, v real, then 2u =f+7, and since Ais self-adjoint, we see that u e AR. If x, + x2, there exists fe A suchthat f(x,) = 1,f(x2) = 0; hence 0 = u(x2) + u(x,) = 1, which shows thatseparates points on K. If x e K, then g(x) #0 for some g eA, andthere is a complex number 2 such that ig(x) 0; if f = 2g, f = u + iv, itfollows that u(x) 0; hence A vanishes at no point of K.Thus AR satisfies the hypotheses of Theorem 7.32. It follows thatevery real continuous function on K lies in the uniform closure of AR,hence lies in . If f is a complex continuous function on K, f = u +iv,then u e 8, v e B, hence fe B. This completes the proof.%3D)0;EXET CIŞESProve that every uniformly convergent sequence of bounded functions is uni-formly bounded.2. If (f) and (g.) converge uniformly on a set E, prove that (f.+ g.) convergesuniformly on E. If, in addition, {f) and (g.) are sequences of bounded functions,prove that (fag,) converges uniformly on E.3. Construct sequences (fa), {g.) which converge uniformly on some set E, but suchthat {f9) does not converge uniformly on E (of course, {fag.) must converge onE).4. Considerf(x) =1+n*xFor what values of x does the series converge absolutely? On what intervals doesit converge uniformly? On what intervals does it fail to converge uniformly? Is fcontinuous wherever the series converges? Is f bounded?166 PRINCIPLES OF MATHEMATICAL ANALYSIS5. LetSa(x) = { sin.Show that {f.} converges to a continuous function, but not uniformly. Use theseries Ef, to show that absolute convergence, even for all x, does not imply uni-form convergence.6. Prove that the seriesconverges uniformly in every bounded interval, but does not converge absolutelyfor any value of x.7. For n-1, 2, 3, ..., x real, put

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Q1 solve real analysis by rudin chp 7 sequence and Series of functions SEQUENCES AND SERIES OF FUNCTIONS 165Theorem 7.32 does not hold for complex algebras. A counterexample isgiven in Exercise 21. However, the conclusion of the theorem does hold, evenfor complex algebras, if an extra condition is imposed on A, namely, that abe self-adjoint. This means that for every fesd its complex conjugate f mustalso belong to A; J is defined by (x) = f(x).7.33 Theorem Suppose A is a self-adjoint algebra of complex continuousfunctions on a compact set K, A separates points on K, and A vanishes at nopoint of K. Then the uniform closure 3 of sA consists of all complex continuousfunctions on K. In other words, A is dense 8(K).Proof Let AR be the set of all real functions on K which belong to d.If feA and f = u + iv, with u, v real, then 2u =f+7, and since Ais self-adjoint, we see that u e AR. If x, + x2, there exists fe A suchthat f(x,) = 1,f(x2) = 0; hence 0 = u(x2) + u(x,) = 1, which shows thatseparates points on K. If x e K, then g(x) #0 for some g eA, andthere is a complex number 2 such that ig(x) > 0; if f = 2g, f = u + iv, itfollows that u(x) > 0; hence A vanishes at no point of K.Thus AR satisfies the hypotheses of Theorem 7.32. It follows thatevery real continuous function on K lies in the uniform closure of AR,hence lies in . If f is a complex continuous function on K, f = u +iv,then u e 8, v e B, hence fe B. This completes the proof.%3D)>0;EXET CIŞESProve that every uniformly convergent sequence of bounded functions is uni-formly bounded.2. If (f) and (g.) converge uniformly on a set E, prove that (f.+ g.) convergesuniformly on E. If, in addition, {f) and (g.) are sequences of bounded functions,prove that (fag,) converges uniformly on E.3. Construct sequences (fa), {g.) which converge uniformly on some set E, but suchthat {f9) does not converge uniformly on E (of course, {fag.) must converge onE).4. Considerf(x) =1+n*xFor what values of x does the series converge absolutely? On what intervals doesit converge uniformly? On what intervals does it fail to converge uniformly? Is fcontinuous wherever the series converges? Is f bounded?166 PRINCIPLES OF MATHEMATICAL ANALYSIS5. LetSa(x) = { sin.Show that {f.} converges to a continuous function, but not uniformly. Use theseries Ef, to show that absolute convergence, even for all x, does not imply uni-form convergence.6. Prove that the seriesconverges uniformly in every bounded interval, but does not converge absolutelyfor any value of x.7. For n-1, 2, 3, ..., x real, put

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Use the definition of sequence of functions to solve this problem, as mentioned only first, in next…

EXE CISES Prove that every uniformly convergent sequence of bounded functions is uni- formly bounded.