Recommended Syllabus
This is the recommended syllabus for the module detailed below. The module should
contain all the topics listed below in some form, but be aware that there may be
additional material covered that can also be examined.
MA244 Analysis III
1. Reminder of key points from MA131 Analysis (convergence, continuity, a continuous function on a closed interval is bounded). f : R → R is continuous iff
U ⊂ R open implies f −1 (U ) open. Uniform or supremum norm on bounded
functions [a, b] → R. A continuous function f : [a, b] → R is uniformly continuous, hence is a uniform limit of step functions.
2. The integral for step functions. The vector space of step functions [a,R b] →
w
R. The definition of the integral is independent of the partition, u =
Rv Rw
Rb
Rb
+ v , ϕ 7→ a ϕ is linear and | a ϕ| ≤ (b − a)kϕk∞ .
u
3. f : [a, b] → R is regulated if it is the uniform limit of a sequence of step
Rb
Rb
functions (ϕn ). a f := limn a ϕn converges and is independent of (ϕn ). It
Rw Rv Rw
Rb
Rb
satisfies u = u + v , f 7→ a f is linear and | a f | ≤ (b − a)kf k∞ .
Rx
4. The indefinite integral F (x) := a f is uniformly continuous; if f is continuous
then F is differentiable and F 0 = f . Fundamental theorem of calculus for a
regulated function f : if g : [a, b] → R is differentiable with g 0 = f then
Rb
f = g(b) − g(a). Application to integration by parts and by substitution.
a
5. Normed vector spaces. Definition and examples of a norm. All norms on Rn
are equivalent. Continuity using ε, δ and by open sets. The operator norm.
6. Closed, convergent, Cauchy, completeness and contractions. These concepts
in a normed vector space. The Contraction Mapping Theorem for a closed
subset of a complete normed space.
R1
R1
7. Pointwise convergence and its disadvantages. Examples of lim 0 f 6= 0 limn fn
or where the pointwise limit of continuous functions is not continuous.
8. Uniform convergence: its Radvantages
R b for continuity and integrals. Theorem:
b
if fn → f uniformly then a fn → a f and, if each fn is continuous, then f is
continuous.
9. Uniform convergence: its advantages for differentiability. A sequence (fn ) of
C 1 functions with (fn0 ) converging uniformly. Examples with fn → f uniformly
but f not differentiable or fn0 6→ f 0 .
10. A space-filling curve. Construction with image a triangle in the plane.
11. Series of functions. The integral, continuity or differentiability of the sum of
a series of functions. Weierstrass M -test. Uniform convergence of a Fourier
series (very briefly). Riemann zeta function. A nowhere differentiable function.
12. Power series. Continuity and differentiability of a power series in the open
interval of convergence. exp, cosh, cos, periodicity of sin.
13. A solution for an ODE. Application of the Contraction Mapping Theorem to
the existence of a solution g for g 0 (x) = F (x, g(x)).
Last updated 12th November 2008