Is the closure of the span in a Banach space complete?

As indicated by the title os the post, I am interested in the following question

Is the closure of the linear span in a Banach space again a Banach space?

If it is true, where I could find the reference, and if it is false, is there a counter example? Thanks!

Solution 1:

As Theo Bendit nicely stated in the comments, any closed set (and the closure of any set is indeed a closed set) in a complete metric space is complete. Then Theo showed you a nice hint to prove the proposition that I totally endorse. (Note that it is also true that a complete subset of a metric space is closed).

I just wanted to add a remark about the fact that you can use the sequential characterization of closed sets (the definition Theo used) since you are in a Banach space (hence in a metric space). Having your space to be metrizable is a sufficient (though not necessary (!) ) condition to ensure that a closed set (defined as “the complement is open”) is sequentially closed and viceversa.