2 proofs regarding linear transformations and operators and subspaces

1.) Prove that $\displaystyle l^1(\mathbb{R}^w) = \{(\beta\subscript_1, \beta\subscript_2, \beta\subscript_3.......) \in \mathbb{R}^w : \sum\limits_{n = 0}^\infty |\beta\subscript_n| < \infty \}$ is a subspace of $\displaystyle \mathbb{R}^w$

2.) Prove that if $\displaystyle \phi: V -> W$ is a linear transformation, the the set $\displaystyle Null(\phi) = \{v \in V: \phi(v) = 0\}$ is a subspace of V.

Okay, so I really know very little about these things. (I'm also currently enrolled in a non proof base linear algebra class)

I know (see read) that W is a subspace of V iff:

1.) W is nonempty

2.) $\displaystyle \alpha, \beta \in \mathbb{R} $ and $\displaystyle w\subscript_1, w\subscript_2 \in W$ always implies $\displaystyle \alpha w\subscript_1 \oplus \beta w\subscript_2 \in W$

I also get the definition of a linear operator (but latex takes along time for me) I can try to type it out if it helps anyone.

Is the Null function significant here? I've never seen it. Any other hints or words of advice will be greatly appreciated. I hope the class goes back to differential equations soon, although I'm guessing this all will be significant.

Thanks!