Norms and pointwise convergence

It is known that

1. There is no norm $\|.\|$ on the space $E$ of continuous real-valued functions on an interval, say $[0,1]$ such that $f_n \to f$ for $\|.\|$ if and only if $f_n$ converges pointwise to $f$, cf Norm for pointwise convergence

2. For any finite dimensional subspace $F$ of $E$ there is such a norm (for polynomials, on can use the Lagrange interpolating polynomials).

Hence my question: is there an infinite dimensional subspace of $E$ having such a norm.

Note that the problem concerns only sequences: it is not about inducing the pointwise convergence topology

Thanks