*****STUDY MATERIAL FOR JRF MATHEMATICS****SYLLABUS*****neral Topology : Topological s
paces, Continuou
s functions,
Connectedness, Compactness, Separ
ation Axioms. Product spaces.
Complete
metric spaces. Uniform conti
nuity. Baire category theorem.****nctional Analysis : Normed
linear spaces, B
anach spaces, Hi
lbert
spaces, Compact operators. Knowledge of some standard examples
like C[0,
1],L
p
[0, 1]. Continuous linear maps (li
near operators).
 Hahn-Banach
Theorem,
Open mapping theorem, Closed graph theorem and the uniform boun
dedness
principle. *****al analysis : Sequences and se
ries, Continuity and different
iability of
real valued functions of one and two real variables and applica
tions, uniform
convergence, Riemann integration. *****ear algebra : Vector spaces
, linear transformations, chara
cteristic
roots and characteristic vecto
rs, systems of linear equations,
inner product
spaces, diagonalization of symmetric and Hermitian matrices, qu
adratic forms. *****mentary number theory and Co
mbinatorics: Divisibility, con
gruences,
standard arithmetic functions, permutations and combinations, a
nd combinatorial
probability. *****besgue integration
 : Lebesgue measure
on the line, measurab
le
functions, Lebesgue integral, conv
ergence almost everywhere, mo
notone and
dominated convergence theorems.******