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Real Analysis
Prove that an open interval in R is an open set and a closed intervals is a closed set
Real Analysis
Using the ε −δ definition, show that lim 3( x )5 1
Real Analysis
Show that T1-space X is regular iff for each point' a' element of X and each open set U containing 'a' ,there is an open set W containing 'a' whose closure is contained in U
Real Analysis
Prove or disprove
a. Every topological space is metrizible
b. Any metric defined on X(#0) induces a topology on x
a. Every topological space is metrizible
b. Any metric defined on X(#0) induces a topology on x
Real Analysis
Give an example of a regular space that is not normal
Real Analysis
Show that the set of rational numbers with the subspace topology of R is disconnected
Real Analysis
Prove that Hilbert space is seperable
Real Analysis
Let f element R(alpha) on [a,b] where alpha is of bounded variation on [a,b] and let v(x) denote the total variation of f on [a,x] for each x in [a,b] and let v(a) =0 , show that | integral a to b f d alpha |less than or equal to integral a to b |f| dv less than or equal to M.v(b) where M is an upper bound for |f| on [a,b]
Real Analysis
Use Euler summation formula,or integration by parts in a Reimann stieltjies to show that
Summation k from one to infinity 1/k= log n- integral one to n x-[x]/x^2 dx+1
Summation k from one to infinity 1/k= log n- integral one to n x-[x]/x^2 dx+1
Real Analysis
let {an} be a decreasing sequence of positive terms .prove that the series summation an sin (nX) converges uniformily on R if and only if nan tends to 0 as n tends to infinity
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