Indian Statistical Institute (ISI) 2005 M.Sc Mathematics Second Semester Mid-Sem -2006 - Question Paper

Time: three hrs Max. Marks : 100

Remarks: every ques. carries 20 marks.

1. Let (X, d) be a metric space. Let Cb(X) denote the normed linear space
consisting of all (real or complex valued) continuous functions on X,
with usual operations and supremum norm.
(a) Show that Cb(X) is a Banach space.
(b) Fix a point x0 two X. For any x two X, let x : X ! IR be described by
x(y) = d(x, y) - d(x0, y), y two X. Show that x 7! x is an isometric
embedding of X in Cb(X).
(c) show that every metric space occurs as a dense subspace of a complete metric space.

2. Let X be a complex Banach space. Let XIR denote the identical space,
viewed as a real Banach space. Show that f 7! Re (f) is an isometry
from X onto XIR.

3. (a) Prove that every non-empty closed and convex subset of a Hilbert
space has a unique element of smallest norm.
(b) Let C be the Banach space of all continuous function on [0,1] into
C, with supremum norm. Let M = {fEC :1R/20
f(t)dt-R11/2f(t)dt = 1}.
Show that M is a closed and convex non-empty subset of C containing
no element of smallest norm.

4. Let K : [0, 1) × [0, 1) ! IR be described by K(x, y) = min(x, y).
(a) Prove that K is an n.n.d. kernel. Let H denote the Hilbert space
with reproducing kernel K.

(b) Show that every element f of H is a continuous function with
f(0) = 0.

(c) Let 0 = x0 < x1 < x2 < . . . < xn and m1,m2, . . . ,mn be real
numbers. Let f be the unique continuous function on [0,1) such that
f(0) = 0, f(x) = constant for x > xn, and f |[xi-1,xi] is a linear function
of slope mi, one  i  n. Show that f two H and calculate its norm.

5. Let U : L2(T) ! L2(T) (T = unit circle with normalised arc-length
measure) be described by (Uf)(z) = zf(z), z two T, f two L2(T). Prove
that U is a unitary and calculate its spectrum.

Earning:   Approval pending.