Originally Posted by

**surjective** Hey,

Thanks for the reply. Yes I know that a vector space is a nonempty set V that has been equipped with two operations, namely addition and multiplication. The thing is that in order for a set to qualify as a vector space, the mentioned operations must adhere to certain axioms. They are quite many. Therefore I would, as far as possible, like to avoid showing all of them.

But as far as I have understood from your post it would be sufficient to choose two arbitrary bounded sequences in $\displaystyle l^{\infty}(\mathbb{N})$ and show that their sum is bounded as well as multiplying a scalar with an arbitrary bounded sequence in $\displaystyle l^{\infty}(\mathbb{N})$ and show that it is bounded. Would that really be sufficient to show that the space mentioned is a vector-space.

Appreciate the help.