Is't a correct observation that No norm on $B[0,1]$ can be found to make $C[0,1]$ open in it? There's a problem in my text which reads as:

Show that $C[0,1]$ is not an open subset of $(B[0,1],\|.\|_\infty).$

I've already shown in a previous example that for any open subspace $Y$ of a normed linear space $(X,\|.\|),~Y=X.$ Even though using this result this problem turns out to be immediate the sup-norm is becoming immaterial. 
And I can't believe what I'm left with:
No norm on $B[0,1]$ can be found to make $C[0,1]$ open in it.

Is this a correct observation?

This is true.  Indeed, you can show more:

Let $X$ be a normed vector space.  If $E \subseteq X$ is a linear subspace which is open, then $E = X$.

My guess is this follows from the same argument you already have.
