MT2002 Analysis

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## Convergence in infinite dimensional spaces

We will consider some more examples of convergence in metrics.

First, we look at some examples of convergence in spaces of sequences.

1. Let X be the set of all bounded real sequences with the metric d defined earlier.
Take the sequence of "points" in X (that is, a sequence of sequences) given by:

x1 = (1/1 , 1/2 , 1/3 , ... ), x2 = (1/12 , 1/22 , 1/32 , ... ), x3 = (1/13 , 1/23 , 1/33 , ... ) , ...
Then this sequence of sequences converges to the sequence y = (1 , 0 , 0 , 0 , ... ) in d.

Proof
d(xn, y) = lub { 0 , 1/2n, 1/3n, ... } = 1/2nand this is small when n is large. 2. This next example shows one of the differences between the finite and infinite-dimensional cases.

Take the sequence in X given by:

x1 = (1 , 0 , 0 , ... ), x2 = (0 , 1 , 0 , ... ), x3 = (0 , 0 , 1 , 0 , ... ), ...
Then although the "sequence of first components", the "sequence of second components", the "sequence of third components", ... all converge to 0, the sequence xn does not converge to the zero sequence: 0, since d(xn, 0) = 1 for all n.

So the theorem proved for finite dimensional spaces in the last section does not hold for this infinite dimensional space. 3. Define a sequence of functions in C[0, 1] by f1(t) = t , f2(t) = t/2 , f3(t) = t/3 , ...
We claim that this sequence converges to the 0-function in the metric d.

Proof
The maximum of the function |fn(x) - 0| is at x = 1 and is 1/n. Thus the real sequence (d(fn, 0))→ 0 and so the 0-function is the limit. Remarks We can get a "picture" of what convergence in the metric d "Looks like".
Given a function g on (say) [0, 1], we can draw "An ε-band" around its graph.

Then a function is within ε of g provided its graph lies in this ε band.

Thus a sequence (fn)→ g in d if the graphs of every function "far enough down" the sequence lie in an arbitrarily small ε band.

4. This time we consider converence in the d1 metric given by d1(f, g) = |f (x) - g(x)|dx. (i.e. the "distance between the functions is the area between their graphs").

1. The picture above shows that the sequence of functions (x/n) converges to the 0-function. 2. Here is a curious sequence of "spike functions"

Define fn(x) by:

Then (fn)→ the 0-function in the metric d1.

Proof
d1(fn, 0) = the area under the curve = 1/n and so (d1(fn, 0))→ 0 and the sequence converges. Note however, that this sequence does not converge to the 0-function in d since the graph of fn "sticks out" of any small ε band of the 0-function.

Thus it can happen that a sequence which converges in one metric may fail to converge in another.

Remarks
1. Convergence of sequences of functions in the metric d is particularly important. It is called uniform convergence and it turns out that this convergence has particularly nice properties, particularly in relation to differentiation and integration.

2. Convergence of sequences of functions in the metric d1is also important. It is called convergence in the mean (because in some sense it "Averages out" the distance between functions). It is important in some applications. For example, the partial sums of the Fourier series of a function converges to the function in the mean.

3. The most obvious way of considering convergence of a sequence of functions is pointwise convergence. That is, a sequence (fn) of functions in (say) C[0, 1] is pointwise convergent to f if at each x ∈ [0. 1], the real sequence (fn(x)) converges in R to f (x).
For example, the pointwise limit of the sequence of "spike functions" above is the function defined by f (1/2) = 1 and f (x) = 0 otherwise.

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JOC September 2001