I can not find an open set which does not belong to Uniform Topology of $R^w$ but belongs to $l^2$ topology of $R^w $.
I know that $l^2$ topology of $R^w $ contains Uniform Topology of $R^w$. But I can not find those sets which make $l^2$ topology of $R^w $ bigger than Uniform Topology of $R^w$.
Can anyone please help me?
As Nate Eldredge(https://math.stackexchange.com/users/822/nate-eldredge) mentioned , the unit ball i.e the set of all sequences $x$ satisfying $∑|x(n) |^2<1$ can be an example. There are sequences of arbitrarily small uniform norm which are not contained in this ball.
So this is an open ball around the zero sequence in $l^2$ which is not so in the Uniform Topology. As open ball around zero sequence of radius $1$ will look like $B(0,1)=⋃_{δ<1 }U(0,δ)$ where $U(0,δ) = (-δ , δ) \times (-δ , δ) \times...........$ .