I have just read that every Banach space with a countable schauder basis will have the approximation property and so every compact operator will be the limit of some $T_n's$ finite rank operators. Now is there a relatively "normal" example of a Normed vector space with a schauder basis that doesn't have the approximation property?
And also can we have a space with the approximation property but that doesn't have a schauder basis ?
Thanks in advance.