# 2005 Indian Statistical Institute MSc Mathematics Second Semester Mid Sem Examination 2005 2006 University Question paper

** University Question Papers **

**
**2005 Indian Statistical Institute MSc Mathematics Second Semester Mid Sem Examination 2005 2006 University Question paper

Time: 3 hrs Max. Marks : 100

Remarks: Each question 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 2 X. For any x 2 X, let x : X ! IR be defined by

x(y) = d(x, y) - d(x0, y), y 2 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 same 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 defined 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, 1 i n. Show that f 2 H and compute its norm.

5. Let U : L2(T) ! L2(T) (T = unit circle with normalised arc-length

measure) be defined by (Uf)(z) = zf(z), z 2 T, f 2 L2(T). Prove

that U is a unitary and compute its spectrum.