Complete metric space|in complete metric, its subspace is complete iff it is closed|BSc3rdyr|L26

Описание: topic cover

complete metric space
if (X,d) is complete metric space and Y is a subspace of X then if Y is complete iff Y is closed.

topic cover in lecture 25
   • Every CAUCHY sequence in a metric space is...  
every cauchy sequence in a metric space is bounded.
E is a subset of metric space (X,d). E is bounded iff diameter of E is finite real number.

topic cover in lecture 24
   • Every convergent sequence in a metric spac...  
cauchy sequence
convergent sequence
Every convergent sequence in a metric space is cauchy. but its converse is not true.

topic cover in lecture 23
   • Every convergent sequence in a metric spac...  
definition : closure set
interior of set
bounded set
Every convergent sequence in a metric space is bounded.

Topic cover in lecture 22
   • set is Closed if it contain its Derived se...  
in a metric space, union of finite number of closed set is closed.
in a metric space, a set is closed if and only if it contains its set of limit point.

topic cover in lecture 21
   • closed SPHERE is a closed SET|intersection...  
in metric space, every closed sphere is a closed set.
in metric space, the intersection of an arbitrary collection of closed set is closed.

topic cover in lecture 20
   • Intersection of finite no of open set is o...  
In metric space, the intersection of a finite number of open sets is open.

topic cover in lecture 19
   • In Metric Space|Open sphere are open set|B...  
In a metric space, every open sphere is an open set.

topic cover in lecture 18
   • Part 2|Metric Space|practice ques.|B.Sc 3r...  
some more examples : to solve metric

topic cover in lecture 17
   • Metric Space|practice ques.|B.Sc 3rd year|...  
Metric space
question to check mapping is metric

topic cover in lecture 16
   • Fourier series of function of period 2pie|...  
Fourier Series of function with period 2 pie

Topic cover in lecture 15
   • Abel's|Drichlete|Test for convergence of 2...  
Test for the convergence of Beta function
Abel's test for 2nd kind of improper integral(prac. que. )
Drichlete test for 2nd kind of improper integral(prac. que.)

topic cover in lecture 14
   • Test for convergence of 2nd improper integ...  
Test for the convergence of 2nd kind of improper integral
Comparison test
Mue test

topic cover in lecture 13
   • Abel's|Drichlet|comparisionIMue test|Test ...  
test of convergence of first kind of improper integral
necessary and sufficient condition
comparison test
miu test
abel's test
drichlete test
absolute convergence
practice questions

topic cover in in lecture 12
   • Test for convergence of improper integral|...  

improper integral
types of improper integral
convergence of 1st kind improper integral

topic cover in lecture 11
   • Differentiability of function of two varia...  
test for differentiability of function of two variable

topic cover in lecture 10
   • Function of several variable|limit|example...  
continuity of function of several variable
with examples

topic cover in lecture 9
   • Function of several variable|niehbourhood|...  
fuction of several variable
niehbourhood of a point of several variable
limit of function of two variable

topic cover in lecture 8
   • Fundamental theorem of integral calculus|B...  
FUNDAMENTAL theorem of INTEGRAL CALCULUS

topic cover in lecture 7
   • Monotonic function is R[a,b]|if f is R[a,b...  
every monotonic function is riemann integrable

if f is riemann integrable then |f| is also riemann integrable

topic cover in lecture 6
   • every continuous function is riemann integ...  
every continuous function is riemann integrable

topic cover in lecture 5
   • Que to show that function is riemann integ...  
que to show that function id riemann integrable

topic cover in lecture 4
   • Upper & lower riemann integration|Dorboux ...  
definition: upper riemann integral
lower riemann integral
Dorboux the0rem

topic cover in lecture 3
   • U(P,f)|L(P,f) are bounded|theorem on Daurb...  
theorem on daurbox sum
L(P,f) and U(P,f) are bounded



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lectures on real analysis part II (with proofs) beneficial for students target B.Sc, M.Sc., UPSC Optional.


lectures on REAL ANALYSIS (target NET, JAM, GATE, SET, NBHM, TIFFER.... etc.
   • Real analysis (CSIR NET,JAM,GATE,PhD,MSC e...  

CSIR NET 2020maths, solved question paper:    • CSIR NET  2020maths, solved question paper  

lectures on Group theory:    • Lectures on group theory  

Lectures on Discrete Mathematics:    • Lectures on Discrete Mathematics  



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