The University of British Columbia
Final Examination - April 12 , 2006
Mathematics 421/510, Real Analysis II, Term 2
Instructor: Dr. Brydges
Closed book examination Time: 2.5 hours
Special Instructions:
- This exam has five questions
1. Let Ybe a topological space and let Abe a set. Let YA={f:A→Y}be the
product space Qα∈AYwith the product topology.
(a) The product topology on YAis the weakest topology such that .. . ?
(b) Describe a neighbourhood base for a point f∈YA.
(c) Show that pointwise convergence, fn(α)→f(α) for each α∈A, implies
fn→f.
(d) Does every sequence {fn}with fn∈ {0,1}[0,1) have a convergent subse-
quence? (Yes/No plus very brief comment in either case).
2. Let Xbe a normed vector space over the complex numbers and let X∗be the
space of continuous linear functionals on X.
(a) Define the norm kfkof f∈ X ∗.
(b) State the complex version of the Hahn Banach theorem.
(c) Let x0∈ X . Show that there is a linear functional f∈ X ∗such that
f(x0) = kx0kand kfk= 1.
(d) Suppose that xn→xweakly. Prove that kxk ≤ lim inf kxnk.
(e) Suppose that Xis a Hilbert space, that xn→xweakly and kxk= lim kxnk.
Prove that xn→xin norm.
(f) Is it possible for xn→xweakly and kxk<lim inf kxnk? Hint: Bessel
inequality.
3. (a) Are continuous functions dense in L∞([0,1], dx)? (Yes/No plus brief ex-
planation in either case).
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