Are these sets subspaces of a vectorspace?

Hi,

I have three sets and I should proof as they are subspaces of a vectorspace.

I write down the task and than I write my ideas and problems.

Let k be a field, V a k-vectorspace and U,W subspaces of V. Are this sets subspaces of V?

Okay to the first set. I think I have to show three things for every set which are valid if we have a subvectorspace.

1.)

2.)

3.)

Okay, first I have to show that the set is not empty. This means there must exist minimum one element which is not the empty set itself. Well:

Is this one line enough? If it is not, how to complete the argument that the first set is not empty?

Another idea was that A vectorspace has a abelian group which have a neutral element and if I found here a neutral element I have shown it but the neutral element of addition is here the empty set and I am not allowd to use it or?

thanks for help

greetings