Consider the following: Let $(x_n)_{n \in \mathbb{N}}$ be a sequence of real valued functions $[0,1] \rightarrow \mathbb{R}$. And $\mathcal{P}:={p_t:t \in [0,1]}$ a family of seminorms such that $p_t(x)=0$ for any function $x$ with $x(t)=0$. Given that, we can talk about the pointwise convergence of a sequence of functions. $x_n \rightarrow x$ pointwise iff $\forall t \in [0,1]: p_t(x_n-x) \underset{n \rightarrow \infty}{\rightarrow} 0 $.
I was told that one can rewrite the condition on the right in the sense of net convergence. I did look up the definitions of nets, but since I am not very familiar with them, I do not know how to rewrite the statement $\forall t \in [0,1]: p_t(x_n-x) \underset{n \rightarrow \infty}{\rightarrow} 0 $ in the sense of nets.
Definition: A directed set $(I,\leq)$ consists of a set $I$ togheter with a relation $\leq$ on $I$ such that:
- $\forall i \in I: i \leq i$
- $\forall i,j,k \in I: i \leq j$ and $j \leq k \Rightarrow i \leq k$
- $\forall i,j \in I: \exists k \in I$ s.t. $i \leq k$ and $j \leq k$.
Then a net on some set $S$ is a map $\tilde{x}:I \rightarrow S$.
As far as I see, in the example I mentioned $I=[0,1] \times \mathbb{N}$, but how do I find the correct relation $\leq$ on $I$?