## Recent **Others** Questions & Answers

###### Prove Theorem 6.11....

##### Prove Theorem 6.11....

###### Let c:'(R) be the space of infinitely differentiable funcdons 01i
R with compact support and let C:(E) ...

##### Let c:'(R) be the space of infinitely differentiable funcdons 01i
R with compact support and let C:(E) ...

###### A continuous function/: R -+ R is said to have compactsupport
ifthere is a compact set KcR suchthat/(x) =0 wheneverx~K.
The class of all such function...

##### A continuous function/: R -+ R is said to have compactsupport
ifthere is a compact set KcR suchthat/(x) =0 wheneverx~K.
The class of all such functions on R is denoted by Cc(R).
Further, if Eis a measurable set in R, define the class CJ..E)
=f.ftE:/ECc(R)}. Prove that (a) Cc(R)cL'(R) and C11(E)CC(E)...

###### Let 11/.-/11,,-+ 0 as n-+ oo, where{/.} is a sequence in LP,
and/ELP, (1 ~ p < oo). Suppose {g.} is a sequence of measur...

##### Let 11/.-/11,,-+ 0 as n-+ oo, where{/.} is a sequence in LP,
and/ELP, (1 ~ p < oo). Suppose {g.} is a sequence of measur...

###### Let {/n} be a sequence of functions in LP, I < p < oo, which
converge a.e. to a function/ in LP and suppose there is a constant
K such that 11/n 11,,,...

##### Let {/n} be a sequence of functions in LP, I < p < oo, which
converge a.e. to a function/ in LP and suppose there is a constant
K such that 11/n 11,,, ::;;;;; K for all n. Prove that for each function
gin L'
ffg =I~...

###### Let f be a real-valued measurable function on [a, b], 1 ~ p~ oo
and (1/p)+(l/q)= I. Then show that
11/11,,,=supJ: fg,
where supremum is taken over all...

##### Let f be a real-valued measurable function on [a, b], 1 ~ p~ oo
and (1/p)+(l/q)= I. Then show that
11/11,,,=supJ: fg,
where supremum is taken over all real-valued functions g with
II gli11 ~ 1 and J:!g < oo....

###### A sequence {/.}in LP is said to converge weakly tofin L" if, for
every function gEL' with I/p+ 1/q= l.
Prove that if {/n} converges in the mean of ord...

##### A sequence {/.}in LP is said to converge weakly tofin L" if, for
every function gEL' with I/p+ 1/q= l.
Prove that if {/n} converges in the mean of order p tof in LP,
1 ~ p ~ oo, it also converges weakly to fin LP....

###### Let/ :;;;i: obe such that/ ELP.p > 0, and let /.=min (/, n).
Show thatf,.EL" and lim 11/-f,.1~=0.
~...

##### Let/ :;;;i: obe such that/ ELP.p > 0, and let /.=min (/, n).
Show thatf,.EL" and lim 11/-f,.1~=0.
~...

###### Prove Corollary 6.10 [Hint: Apply the method of contradiction
and use Theorem 6.9]...

##### Prove Corollary 6.10 [Hint: Apply the method of contradiction
and use Theorem 6.9]...

###### Prove Theorems 6.4 and 6.6....

##### Prove Theorems 6.4 and 6.6....