What is the topology on a vector space generated by a family of seminorms? I have a question about topological vector spaces. I'm getting so confused about what we mean by saying a topology generated by a family of seminorms.
At first, I thought it just means the initial topology generated by the family of seminorm functions. But then, I read about this sample, Example 2. 
Let X be a set, then $\mathbb{C}^X = \{ f: X \to \mathbb{C} \}$ is a complex vector space. To each $x \in X$, consider $\| \cdot \|_x : \mathbb{C}^X \rightarrow \mathbb{R}$, ${\| f \|}_x = | f(x) |$. This is a seminorm on $\mathbb{C}^X$. The topology generated by this set of seminorms as $x$ varies over $X$ is the product topology on $\mathbb{C}$. 
Why is this the same topology as the product topology on $\mathbb{C}^X$? If this is true, the property of initial topology would imply $f_n$ converges to $f$ in $\mathbb{C}^X$ iff $|f_n(x)|$ converges to $| f(x)|$, for all $x\in X$. However, the property of the product topology would imply $f_n$ converges to $f$ in $\mathbb{C}^X$ iff $f_n(x)$ converges to $f(x)$, for all $x\in X$. Consequently, $f_n(x)$ converges to $f(x)$, for all $x\in X$ iff $|f_n(x)|$ converges to $| f(x)|$, for all $x\in X$, which I think is not true.
Why do I get a contradiction here?
Thanks in advance for any help!
 A: The topology generated by a set of seminorms is not just the initial topology they generate.  Rather, it is the initial topology that they and all of their translates generate (or equivalently, the initial translation-invariant topology they generate).  That is, for all $f$ in the space and each seminorm $p$, we require not just that $p$ is continuous with respect to our topology but also that $g\mapsto p(g-f)$ is continuous.  The idea is that for each $f$, we should think of $p(g-f)$ as a way of measuring the distance from $g$ to $f$, and so the set of $g$ such that $p(g-f)$ is small should be a neighborhood of $f$.
So in the example of the product topology, if $f_n$ converges to $f$, that means not just that $|f_n(x)|$ converges to $|f(x)|$ for all $x$, but that for any $f'$, $|f_n(x)-f'(x)|$ converges to $|f(x)-f'(x)|$.  Taking $f'=f$, this implies $|f_n(x)-f(x)|$ converges to $0$, so $f_n(x)$ converges to $f(x)$.
More generally, it is typical to describe a topology on a vector space by just describing the neighborhoods of $0$, and then obtaining neighborhoods of other points by translation.  That is what's going on here: when you say that a set of seminorms "generates" the topology, that actually just means they generate the neighborhoods of $0$, and neighborhoods of all other points are then obtained by translation.
