Why do my resource caps continuously increase slowly?

Seems like a bunch of fonts were removed from the base operating system.

• Open FontBook
• Find the font you want to get
• Click "Download"
• Voilà!

The product norm is indeed a norm. The key observation is the following: If $g_1, g_2\in \mathbb{F}_p((Y))$ have the same norm, then there is $c\in \mathbb{F}_p$ such that $|g_1-cg_2|_2<|g_1|_2$. Hence, if one has an expression $f_1\otimes g_1+f_2\otimes g_2$ with $|g_1|_2=|g_2|_2$, one can also express it as $f_1\otimes g_1+f_2\otimes g_2=f_1\otimes (g_1-cg_2)+f_1\otimes cg_2+f_2\otimes g_2 =f_1\otimes (g_1-cg_2)+(cf_1+f_2) \otimes g_2$

and

$\max\{ |f_1|\cdot |g_1|, |f_2|\cdot |g_2| \} =\max\{ |f_1|, |f_2|\} \cdot |g_2| \geq \max\{ |f_1|\cdot |g_1-cg_2|, |cf_1+f_2|\cdot |g_2| \}$

This tells us that for computing the infimum, one only has to take into account sums of tensors $\sum_i f_i\otimes g_i$ where all $g_i$ have different norms.

In the next step, one shows that all such expressions $c=\sum_{i=1}^n f_i\otimes g_i$ with $g_i$ having different norms, have the same value $\max_i \{ |f_i|\cdot |g_i|\}$.

This is done as follows (a bit sketchy here):\ Consider two expressions $c=\sum_{i=1}^n f_i\otimes g_i$, $c=\sum_{j=1}^m f'_j\otimes g'_j$, with $|g_1|<\ldots<|g_n|$ and $|g'_1|<\ldots<|g'_m|$ of the same element $c$ (all $f_i$ and $f'_j$ non-zero). Then the set $\{ g_1,\ldots, g_n, g'_1,\ldots, g'_m\}$ has to be $\mathbb{F}_p$-linearly dependent. In particular, $|g_n|=|g'_m|$. Using the same trick as above, we can replace the second expression by one with $g'_m=g_n$, and the norm of the new expression does not exceed the old one. Then, $\sum_{i=1}^{n-1} f_i\otimes g_i- \sum_{j=1}^{m-1} f'_j\otimes g'_j=(f'_m-f_n)\otimes g_n$, and either we have $f'_m-f_n=0$ or $g_n$ is linear dependent of the other $g$'s. Due to the norms of the $g$'s, the latter can't be, and we conclude $f'_m=f_n$. Going on inductively in the same way, one can transform one expression into the other, and obtains that $\max_i \{ |f_i|\cdot |g_i|\} \leq \max_j \{ |f'_j|\cdot |g'_j|\}$. By switching the roles, we obtain equality.