Let be the space of sequences such that , with inner product and norm .
I need to show that is complete with respect to the inner product i.e. a Hilbert Space. I am assuming that is complete.
I know that I need to show that every Cauchy sequence in H is convergent. I can see that this follows relatively easily from the definition of H.
So I have taken a sequence but for this to be Cauchy . But I can't see how to show that, given that the inner product on has a below it. Any help would be great. Thanks
Thanks for the reply, okay so I was already assuming that was Cauchy, the bit I was confused about was the inequality
Which is need to show that is also Cauchy, for each j.
My question is: If you have the in the inequality, does that still satisfy the definition of Cauchy?