Dear MHF members,
I have two questions on Banach spaces,
and I would really appreciate if you help me with a complete answer.
Let be a interval and .
- Denote by the set of all real valued bounded functions on ,
which may have jump type discontinuity from the left-hand side at the point ,
i.e., and .
Is a Banach space with the norm?- Denote by the set of all real valued bounded functions on ,
which may have removable type discontinuity at the point .
Is a Banach space with the norm?
Thanks.
bkarpuz
Here are my arguments for the first question.
I will try to give an affirmative answer.
Proof of 1. It is clear that is a norm on .
To complete the proof, we need to show that any Cauchy sequence
converges a function in this space.
Then, for every, , there exists such that for all with .
In particular, for each , we have for all with ,
i.e., is a Cauchy sequence, and thus for each ,
the limit exists, to complete the proof, we have to now show that
converges uniformly to and .
We have for all and all with ,
which yields by letting that for all and all with ,
and proves that converges uniformly to .
Let , then for every , there exists such that for all .
Let . As the functions are continuous at , there exists a neighborhood of
such that for all .
Then, for any , we have
.........................
.........................
for all proving that is continuous at .
Let , then we follow similar steps above with a right neighborhood of ,
and show that is right-continuous at .
This shows that , and thus it is complete.
Hence, is a Banach space.
It would be very appreciated if you confirm that my arguments are correct.