0Introduction
IB Analysis II
0 Introduction
Analysis II, is, unsurprisingly, a continuation of IA Analysis I. The key idea in
the course is to generalize what we did in Analysis I. The first thing we studied
in Analysis I was the convergence of sequences of numbers. Here, we would like
to study what it means for a sequence of functions to converge (this is technically
a generalization of what we did before, since a sequence of numbers is just a
sequence of functions
f
n
:
{
0
} → R
, but this is not necessarily a helpful way
to think about it). It turns out this is non-trivial, and there are many ways
in which we can define the convergence of functions, and different notions are
useful in different circumstances.
The next thing is the idea of uniform continuity. This is a stronger notion
than just continuity. Despite being stronger, we will prove an important theorem
saying any continuous function on [0
,
1] (and in general a closed, bounded subset
of
R
) is uniform continuous. This does not mean that uniform continuity is a
useless notion, even if we are just looking at functions on [0
,
1]. The definition
of uniform continuity is much stronger than just continuity, so we now know
continuous functions on [0
,
1] are really nice, and this allows us to prove many
things with ease.
We can also generalize in other directions. Instead of looking at functions, we
might want to define convergence for arbitrary sets. Of course, if we are given a
set of, say, apples, oranges and pears, we cannot define convergence in a natural
way. Instead, we need to give the set some additional structure, such as a norm
or metric. We can then define convergence in a very general setting.
Finally, we will extend the notion of differentiation from functions
R → R
to general vector functions
R
n
→ R
m
. This might sound easy — we have been
doing this in IA Vector Calculus all the time. We just need to formalize it a bit,
just like what we did in IA Analysis I, right? It turns out differentiation from
R
n
to
R
m
is much more subtle, and we have to be really careful when we do so, and it
takes quite a long while before we can prove that, say,
f
(
x, y, z
) =
x
2
e
3z
sin
(2
xy
)
is differentiable.