How would I show that the vector space with the 2 norm is not a Banach space?
I would perhaps have to show that it doesn't converge in the appropriate norm..? I am not sure how..!
You need to show that the space is not complete in that norm. To see that, start by taking a sequence that is in but not in . For example, you could take the element with . Then consider the sequence of elements in defined by
Then is Cauchy for the 2-norm, but it does not have a 2-norm limit in the space (because it is trying to converge to an element that isn't in ).