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This is the Title PhL 1.11.15 This has been installed in Buck Chapter 4 notes at the start of Section 4.4. 4.4 Improper Integrals with a Parameter [ 204 ] (a) A Review of Continuity and Convergence In Section 2.3 Bucks discuss continuity and uniform continuity for f: En → Em . The two notions are very closely related and we say For any ε > 0 and for a specific p0 in D which lies in En, we can find δ(p0) > 0 such that | f(p) - f(p0)| < ε when |p-p0| < δ => f(p) is continuous at the point p0 For any ε > 0 and all p0 in D which lies in En, we can find δ > 0 such that | f(p) - f(p0)| < ε when |p-p0| < δ => f(p) is uniformly continuous on E In both the above we have the idea that limp→p0 f(p) = f(p0). In Theorem 1 [57] Bucks showed that if we have this kind of continuity at some point p0, then for any point sequence pn → p0 we have f(pn) → f(p0). When the word "sequence" appears in the discussion, so does the word "convergence". In the above, we are saying that if pn converges to p0 in the domain, then f(pn) converges to f(p0) in the range. The phrase is that "continuity preserves convergence". This convergence is f(pn) → f(p0) in both cases stated above, the difference is in the nature of the convergence with respect to whether δ depends on p0 or not. It is true that this type of convergence does involve functions (like f). Notation anomaly: The symbol p0 is here overloaded since it represents our point of interest, but it also represents the first point in the sequence {pn}. These are completely different meanings, so one must just keep this in mind. Now, in the above we have pn → p0 being a "sequence of points in En" f(p) is a "function" such that f: En → Em . En is of course a particular Hilbert Space in the language of Stakgold vol I p 110. Stakgold discusses another Hilbert Space he calls L2(r)(a,b) and L2(c)(a,b) where are the spaces of real or complex valued functions on the interval (a,b) which are L2 integrable over that interval. We could generalize this to talk about L2(m)(D) which would be functions taking values in Em which are defined on a domain D in Em, which again are square integrable over D. Another choice is L1(m)(D) which is functions for which ∫D dmx || f(x) || is integrable and then || f ||1 = ∫D dmx || f(x) || , whereas in the previous case we had || f ||2 = ( ∫D dmx || f(x) ||2 )1/2 and in general || f ||p = ( ∫D dmx || f(x) ||p )1/p . Note that E2 and complex are not quite the same, we let that pass. Another norm of interest is this one: || f ||∞ = max( ||f(x)||) on D. I might call this "the max norm" but wiki seems to like the "infinity norm" because it is like the 1 and 2 norms except p → ∞ (though I have not proven that fact). Officially it is the "supremum" norm, but I will call it the max norm. This max norm is the one the Bucks use for functions, and they write it as || f ||E over set E. Back in Section 4.2 we have the following new concepts. The players are now functions fn(p) which form a sequence of functions {fn(p)} which perhaps converge to some function f(p) For any ε > 0 and for a specific p in D which lies in En, we can find δ(p) > 0 such that | fn(p) - f(p)|En < ε when n > N. // n is overloaded! => fn(p) converges pointwise to f(p) on D Notice that the norm used here is the geometric norm of En , it is not any of the fancy function space norms listed above. As we have learned, this pointwise convergence is not very "interesting" because it doesn't lead to nice facts like order interchange of things. What is really much more interesting is the following notion: For any ε > 0 we can find δ > 0 such that || fn - f ||∞ < ε when n > N. => sequence fn converges uniformly to f over D Now we are for the first time using this max norm, and that norm knows about the nature of fn(p) - f(p) for all values of p in D, because it needs to know that in order to compute the "max". So it seems best not to display the function argument p, since the norm here is for the function itself! The main idea here is that if you have || f1 - f2 ||∞ very small, then the two functions are close together for ALL values of their arguments. If || f1 - f2 ||∞ < ε then each function lies within the 2ε band of the other function: Recap. In the above, we have discussed two totally different notions of convergence: We have first a convergence of points pn or f(pn) which involves the single function f. And we have second a convergence of functions fn to some function f. In more detail: 1. In the discussion of continuity and uniform continuity, we had limp→p0 f(p) = f(p0) using the Em norm saying that that | f(p) - f(p0)|Em < ε. We could then bring into the discussion a sequence of points pn and we could then say limpn→p0 f(pn) = f(p0) and we then have convergence of the sequence {f(pn)} which is a sequence of points in Em. This convergence is associated with the continuity of the function f(p). In the case that the same δ works for all p0, we had uniform continuity over E, and you could say that we have uniform convergence of f(pn) → f(p0) where pn → p0 where p0 is any point in E. In other words, we are saying that f(pn) → f(p0) converges to p0 sort of "uniformly" for any p0 in E. This use of the phrase "uniform convergence" is not used by the Bucks. You might call it (uniform convergence)p since it involves a sequence of points. Note that, although f(p) is a function, here we are really talking about sequences of points, { pn } and { f(pn) }. We are not talking sequences of function, only one function f is involved in the discussion. 2. In the above subsequent discussion of sequences of functions fn(p), we talked above about fn(p) converging pointwise to f(p) on D, which involved the En norm, and we talked about fn converging uniformly to f using the max norm over D. We say that fn converges uniformly to f over D, and this is the official definition of the phrase "uniform convergence". Maybe call this (uniform convergence)f since it involves a sequence of functions. (b) Application to a bystander parameter. Imagine that p lies in some set E, and that t lies in some set T. Consider a function of both variables f(p,t). Then consider this ε δ statement: 1. For any ε > 0 and for a specific p in E and for a specific t0 in T, we can find δ(t0,p) > 0 such that | f(t,p) - f(t0,p)| < ε when |t-t0| < δ. I would describe this as "continuity at a point t0 in T with a bystander parameter p in E". I would also say that f(t,p) "converges" to f(t0,p) as t → t0. This means there are sequences tn so that f(tn,p) "converges" to f(t0,p) as tn → t0. Our first "upgrade" of this concept would be: 2. For any ε > 0 and for all p in E and for a specific t0 in T, we can find δ(t0) > 0 such that | f(t,p) - f(t0,p)| < ε when |t-t0| < δ. I would describe this as "continuity at a point t0 in T which is uniform over E". I would also say that f(tn,p) (converges uniformly)p over E" to f(t0,p) as tn → t0. Notice that there is only one function here, it is called f. A different upgrade of item 1 might be this: 3. For any ε > 0 and for a specific p in E and for all t0 in T, we can find δ(p) > 0 such that | f(t,p) - f(t0,p)| < ε when |t-t0| < δ. I would describe this as "uniform continuity" over T with a bystander parameter p". One could also say we have (uniform convergence)p of f(tn,p) → f(t0,p) as tn → t0. The sequence involved is a sequence of points {tn} or { f(tn,p) }; only function is f. Now in the next item we combine both the above upgrades: 4. For any ε > 0 and for all p in E and for all t0 in T, we can find δ > 0 such thatc when |t-t0| < δ. I would describe this as "uniform continuity over T which is also uniform over P". This latter is a strange new use of the word "uniform". The statement involves being uniform in two different spaces at the same time, E and T. The "continuity" aspect only involves space T. One could also say we have (uniform convergence)p of f(tn,p) → f(t0,p) as tn → t0. The sequence involved is a sequence of points {tn} or { f(tn,p) }; only function is f. Notice that we still have the Em norm, not the max norm. We still have only one function. So even this double upgrade thing has nothing to do with (uniform convergence)f of a set of functions fn → f in the Hilbert Space of the max norm. On page 204 the Bucks use the above double-upgrade concept, which involves to different underlying sets E and T. They want to say that the result is uniform convergence of f(tn,p) → f(t0,p) since it is uniform in both the T and E sense. I will call this (uniform convergence)p2 to indicate that we are still using the point norm for f(tn,p) → f(t0,p), but that we are uniform in 2 spaces, T and E. So this really is a brand new definition of the term "uniform convergence". Then f(t,p) is continuous at t = t0 and p = p. We say there is convergence, limt→t0 f(t,p) = f(t0,p).