limit_interchange-1

Interchange of limits and uniform convergence Lets→aandt→bbe two limiting processes. (For example, these could be a real variable tending to a number, or towards infinity, a natural number tending towards infinity, a partition tending towards ‘infinity’ in the sense of refinement, etc.) Let f(s, t) be a function (real- or complex-valued now, though this is not necessary). We will give a general theorem justifying the interchange of the order of these two limiting processes if a certain one of them occurs uniformly, in the sense of the following definition. Definition. LetL(t) be a function of t. We say that lim s→af(s, t) =L(t)uniformly in tif for every /epsilon1 >0 there is s0such that for any s≥s0, and for all t, we have/vextendsingle/vextendsinglef(s, t)−L(t)/vextendsingle/vextendsingle< /epsilon1. Theorem. Suppose that (i) For each s, lim t→bf(s, t) =R(s). (ii) For each t, lim s→af(s, t) =L(t). (iii) lim s→a/parenleftbig lim t→bf(s, t)/parenrightbig =C. (iv) lim s→af(s, t) =L(t) uniformly in t. Then lim t→b/parenleftbig lim s→af(s, t)/parenrightbig =C. Proof. Let/epsilon1 > 0. By (iv) we may choose s0so that if s≥s0then for all twe have/vextendsingle/vextendsinglef(s, t)−L(t)/vextendsingle/vextendsingle< /epsilon1/ 3. By (iii) we may choose s1so that if s≥s1then/vextendsingle/vextendsingleR(s)−C/vextendsingle/vextendsingle< /epsilon1/3. Choose s2≥max{s0, s1}. We may use (i) to choose t0so that if t≥t0then/vextendsingle/vextendsinglef(s2, t)−R(s2)/vextendsingle/vextendsingle< /epsilon1/ 3. Now for t≥t0we have /vextendsingle/vextendsingleL(t)−C| ≤/vextendsingle/vextendsingleL(t)−f(s2, t)/vextendsingle/vextendsingle+/vextendsingle/vextendsinglef(s2, t)−R(s2)/vextendsingle/vextendsingle+/vextendsingle/vextendsingleR(s2)−C/vextendsingle/vextendsingle < /epsilon1/ 3 +/epsilon1/3 +/epsilon1/3 =/epsilon1. The standard theorems stating that the (appropriately uniform) limit of continu- ous/integrable/differentiable functions is continuous/integrable/differentiable, and that the limit/integral/derivative of the limit is the limit of the limit/integral/derivative, can all be derived from the above theorem. We will use it to prove a result on double series. Theorem. Letaijbe complex numbers, for i,j∈ {1, 2, 3, . . .}. Suppose that (i) For all i,/summationtext j|aij|<∞. (ii)/summationtext i/parenleftbig/summationtext j|aij|/parenrightbig <∞. Then (iii) For all j,/summationtext i|aij|<∞. (iv)/summationtext j/parenleftbig/summationtext iaij/parenrightbig =/summationtext i/parenleftbig/summationtext jaij/parenrightbig . Proof. Fixj. For any iwe have that |aij| ≤/summationtext /lscript|ai/lscript|. Hence/summationtext i|aij|converges by (ii) and the comparison test. Therefore (iii) holds. In particular,/summationtext iaijconverges (absolutely). By (ii) and the comparison test again,/summationtext i/vextendsingle/vextendsingle/summationtext jaij/vextendsingle/vextendsingleconverges, so that/summationtext i/parenleftbig/summationtext jaij/parenrightbig converges (absolutely). We claim that/summationtext iaijconverges uniformly in j. Let /epsilon1 > 0. By (ii) there isi0such that/summationtext∞ i=i0/parenleftbig/summationtext /lscript|ai/lscript|/parenrightbig < /epsilon1. For any j, and for all i≥i0,|aij| ≤/summationtext /lscript|ai/lscript|. Therefore/summationtext∞ i=i0|aij| ≤/summationtext∞ i=i0/parenleftbig/summationtext /lscript|ai/lscript|/parenrightbig < /epsilon1. The theorem now follows by using the previous theorem.