Nalanda Open University 2009 B.Sc Mathematics Bachelor of Science ( Hors), Part-III Term End , - V (Mathematcs) - Question Paper
Bachelor of Science (Mathematics Honours), Part-III Term End Examination, 2009
Paper- V (Mathematcs)
Bachelor of Science (Mathemalirs Honoius), Part-IH Term End Examination, 2009 Pap hi -V (Matliema tic s)
lime: 3.00 His. Eull Marks: 75
Answer any Six Questions, selecting at least one question from each group.
Group-A
1. (at) Define an open sphere. Let X be a non-empty set and let d be a metric on X. Let a: X > R be difmed as g (x, y) = d(x, y) if d(x, y) < 1.,
Show that a is a metric on X.
(b) Show that the intersection of a finite number of open sets in (X, d) a metric space is open.
2. (a) Let (X, d) be a metric space. Let F cX. Then prove that F is closed if and only if F contains each of its accumulation points.
(b) Show that the metric space (C, d) of the complex plane where' d(zj, z2) = | Zj--l zl z2 e 13 complete.
1. (a) Let (X, dj) and (Y, d2) be metric spaces and- -'X > Y. Prove that f is continuous at x if and only if xn > x => f(xn) > f(x).
(b) Let (X, d), (Y, d2), (Z, d3) be metric spaces and f: X > Y, g : Y > Z be continuous mappings. Then prove that g f is a continuous mapping of X into Z.
4. (a) Let X be a non-empty set arid JT-} .be.-a family of topologies on X. Show that Tj is a topology on X.
(b) Give an example of a one-to-one continuous mapping of a topological space onto another such that the mapping is not a homeomorp.hisni
Group-B
5. (a) Show that a necessary any sufficient condition for the Riemann integrability of a
bounded function f over an interval [a, b] is that for every e > 0 there exists a lower
sum L(Pj) and an upper sum U(P2) such that U(P2) - L(Pj) < e.
(b) Define f on [0, 1 ] such that f(x) = 2rx for < x < . Then show that f e R[Q, 1 ] and pi t r+1 r
5. (a) If f is R - integrable over [a, b] and f possesses a primitive F over [a, b], then prove that J f(x)dx = F(b) - F(a).
a rb
(b) Iff is continuous and non-negative on [a, b] and J f(x)dx = 0, then prove that f(x) = 0
3.
for all x in [a, b].
Group- C1
State, -and prove Pringsheim's theorem on rearrangement of series.
3. (a) State and Drove Dinchlet's test.
Group-D
?. (Ja)-.Ll F be the'set of all sequences (x) such that at most a finite number of the terms
} .
-is continuous-'
; F' reive that' ~k nor me d hnear spacrV Banach space if any ;rm,;ty if ..evr absolutely = summable serie3: is Bummable-(b) Lt fi, a Banach space be such that B = L M where L and M are linear subspaces. Let fz = x+y be the unique re present at io n\o f a.ne'citctt' % :ih B as the sum ofvectors x and y'iii VL and M respectively.'Sh&W:that a new norm can be defined on the linear space B by
l|z|f= INI + llyll.
. the new normed linear space, prove frnf .iif.lL"and M are closed in B then B'is a
Banach space.
FI ,.(a). If M is a closed linear subspace of a harmed linear space N and if T is the mapping of N onto N/M defined by T(x) = x+M. Show that T is a continuous linear transformation for which ||T|| < 1.
(b) Define a Hilbert space. Let Lbe an inner product space over a field F. Define a norm on L by ||x|| = + V(x;x) for all x in L. Show that L is a normed linear space..
12.(a) Prove that an inner product space X is uniformly convex.
(b) Consider the hnear space P[0 1] af all real-valued polynomials ;bn [0, 1] wif-OB-her product given by
= ji(t)g(t)dt
where f, g are elements of P[0, 1]. Show that it is :anlhneT. .product space but not a Hilbert space.
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