Continuous linear functionals on the space $C[0,1]$ with topology of pointwise convergence

If $f: V \to \Bbb R$ is a (linear) continuous functional, we know by continuity at $0$ that for some basic open subset of $\mathcal{T}_p$ (presumably the topology generated by the semi-norms $p_t, t \in [0,1]$ and topologists would simply say that $f: C_p([0,1]) \to \Bbb R$ is continuous and linear) then there are $t_i \in [0,1], i=1, \ldots,n$ and $r_i>0$ so that $$f \in \bigcap \pi_{t_i}^{-1}[(-r_i, r_i)] \subseteq (-1,1)$$ (this are the basic sets from the standard subbase; I use that the weak topology induced by the seminorms $p_t$ is the same as the one induced by the evaluations/projections $\pi_t$). From this inclusion it follows quite easily that $\bigcap_{i=1}^n \text{ker}(\pi_{t_i}) \subseteq \text{ker}(f)$ (a point common to all kernels is in the intersection and thus in the kernel of $f$).

By a standard result on functionals (e.g. from here, we get that $f$ must be a linear combination of the $\pi_{t_i}, i = 1, \ldots n$ which is what is claimed.