Is there a norm on ${\Bbb R}^{\Bbb N}$ Let $E={\Bbb R}^{\Bbb N}$ be the real vector space of real sequences. 

1) Is there a norm on $E$?

2) Is there a norm $N$ on $E$ such that the restriction of $N$ to $\ell^2$ is finer than the standard norm $N_2$ of $\ell^2$ ?
Pick a basis, and define the norm of a vector to be the sum of the absolute values of the coefficients of that vector in the basis.

For your extra requirement: Find a subspace $U$ of $\mathbb R^\mathbb N$ such that $U\oplus\ell^2=\mathbb R^\mathbb N$. Put on $\mathbb R^\mathbb N$ the norm which is the sup of the norm of $\ell^2$ and any norm (constructed as in my first paragraph) on $U$. This restricts to the norm of $\ell^2$, of course.
There is no norm on the entire $E=\Bbb R^{\Bbb N}$ (with the property that $||x||=0 \implies x=0$).

Suppose such norm exists. Let's denote the elements of $E$ as $a=(a_1,...,a_n,...)$ and suppose $||a_i||=\alpha_i&gt;0$. Then form a vector $b=(b_1,...,b_n,...)$ such that $||b||=\infty$. 

Here's how you do that: start with $b_1=2^1\frac{a_1}{||a_1||}$, then choose between $b_2=\pm 2^2\frac{a_2}{||a_2||}$, depending for which $b_2$ the norm is greater, etc. This would make the norm of $b$ unbounded.