I have tried to formulate the notion of products myself and this is what I came up with:
Let $(X_i, |*|_i), i\in I$ be a collection of normed linear spaces and $f_i:Y\to X_i$ a collection of bounded linear maps, all index by the set $I$.
As long as $I$ is a finit set we can factor all maps in the following way:
$Y\overset{\triangle}{\to}\prod_{I}Y\overset{\prod_{i\in I} f_i}{\to}\prod_{i\in I}X_i\overset{p_i}{\to}X_i$
where the norm on the products is given by:
$|x|=\underset{i\in I}{sup}(|x_i|_i)$
As can be seen this is a close relative to $l^\infty$. We can formulate the notion of coproducts in the dual way and get something similiar to $l^1$, but only with finit sums.
$|x|=\sum_{i\in I}|x_i|_i$
I originally choose the supremum norm (on the product) so that it would work with infinite products, I am, however, no longer sure.
I wanted to use the definiton above to illuminate the difference between the weak topology and weak convergence.
To start I imagine we are in the following position:
$X\overset{\triangle}{\to}\prod_{X'}X\overset{\prod_{\lambda\in X'} \lambda}{\to}\prod_{X'}\mathbb{R}\overset{p_i}{\to}\mathbb{R}$
The weak topology on $Y$ can be obtained as the coarsest topology where $(\prod_{\lambda\in X'}\lambda)\triangle$ is continuous with respect to the product topology in $\prod_{X'}\mathbb{R}$ while weak convergence correspondes to the product norm defined above, which correspondes to the box topology.
Now for the objects this seem to be working out fairly well but it occurs to me that $(\prod_{\lambda\in X'}\lambda)\triangle$ need not be bounded, infact as long as X' isn't uniformly bounded it won't.
So now I'm hoping that I did something wrong. I really want the category of normed vectorspaces to have infinit products and it makes me a bit sad to think it might not.
Alternatively this is why the ideas of uniform boundedness are so important and I need to incorporate these somehow. I read about them but I honestly didnt get thier significanse at the time and I can't really see thier place in the big picture. Any help in that regard would also be very appreaciated.
I apologice if the question is to vague to be a proper StackExchange question.