I'm trying to prove some properties of sequence spaces. I already know that the space l

Feinsn

Feinsn

Answered question

2022-06-16

I'm trying to prove some properties of sequence spaces. I already know that the space l of all bounded sequences isn't an inner product space, isn't separable but it is complete with sup norm.
The space c of all convergent sequences is also complete, as a closed subspace of l , and so is c 0 - the space off all sequences with entries equal to zero from some point on.
I also know that both c and c 0 are separable.
My question is - are c and c 0 inner product spaces. Should I look for an example of sequences which don't satisfy the parallelogram law.

Answer & Explanation

candelo6a

candelo6a

Beginner2022-06-17Added 24 answers

Should I look for an example of sequences which don't satisfy the parallelogram law.

Yes you should. Take the sequences x = ( 1 , 0 , 0 , 0 , ) and y = ( 0 , 1 , 0 , 0 , ), write down
x + y 2 + x y 2 = ? 2 ( x 2 + y 2 )
and watch it fail.

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