Does there exist a linear subspace of $\mathbb C ^{\mathbb N}$ that can be endowed a Banach space topology that is not finer than the locally convex topology of pointwise convergence?
Best, Martin
Does there exist a linear subspace of $\mathbb C ^{\mathbb N}$ that can be endowed a Banach space topology that is not finer than the locally convex topology of pointwise convergence? Best, Martin 


Take any of the usual Banach spaces X of sequences. Let f be any unbounded linear functional on X such that $f({1,0,0,...})\neq 0$. Now consider the mapping $\Phi$ between sequences defined as follows: $$\Phi({x_1,x_2,x_3,...})=({f(x),x_2,x_3,...}).$$ Let $Y=\Phi(X)$, endowed with the topology $\y\=\\Phi^{1}(y)\_X$. By construction, Y is a Banach space. However, since f was assumed unbounded, convergence in the norm of Y does not imply convergence of the first component. 


Martin, your question is at the level of a good homework problem in a course on functional analysis, so I will give only hints.


