I found a proof in a book and can't see how an inequality holds.
From (J. Lindenstrauss, L. Tzafriri; Classical Banach spaces 1: sequence spaces; page 31 proposition 1.e.3):
Let $X,Y$ be Banach spaces and put on $L(X,Y)$ the topology $\tau$ on uniform convergence on compact sets in $X$. (Topology generated by the seminorms $||T||_K := sup\{||Tx|| , x \in K\}$). Then, the continuous linear functionals on $(L(X,Y),\tau)$ all consist of the form $$\varphi(T) = \sum_{i \in \mathbb{N}} y'_i(Tx_i), \quad (x_i) \subset X \quad y'_i \subset Y' \quad \sum\limits_{i \in \mathbb{N}} ||x_i||||y'_i|| < \infty$$ Proof. Assume that $\varphi$ has such a representation. We may clearly assume that $x_i \neq 0$. Let {$\eta_i$} be a sequence of positive scalars tending to $\infty$ so that $\sum\limits_{i \in \mathbb{N}} \eta_i ||x_i||||y'_i|| = C < \infty$. Put $K=\{ \frac{x_i}{||x_i|| \eta_i}\} \cup \{0\}$. Then K is compact and $$|\varphi(T)| \leq \sum\limits_{i \in \mathbb{N}} ||y'_i|| ||x_i|| c_i ||T\left(\frac{x_i}{||x_i||c_i}\right)|| \leq C ||T||_K$$
note that this is just the first half of the poof.
My problem now is, that I can't see how the last inequality holds. It's clear that $$ \sum\limits_{i \in \mathbb{N}} ||y'_i|| ||x_i|| c_i ||T\left(\frac{x_i}{||x_i||c_i}\right)|| = C\sum\limits_{i \in \mathbb{N}} ||T\left(\frac{x_i}{||x_i||c_i}\right)|| $$ but at this point $\sum\limits_{i \in \mathbb{N}} ||T\left(\frac{x_i}{||x_i||c_i}\right)||$ needs to be smaller than $||T||_K$ . Which I guess is not the case as the supremum could be any of the summands. I can't even see how this is always convergent. Like if $c_i = i$ and $T(\frac{x_i}{||x_i||}) \geq 1$ .
I was too wondering if the proof ( in case its correct ) could be done using the operator norm instead of these seminorms. Then could we just prove it without mentioning the topology?
You are ignoring one of the assumptions and going for too big an estimate.
You have $$ \sum\limits_{n \in \mathbb{N}} \|y'_i\| \,\|x_i\| c_i\, \left\|T\left(\frac{x_i}{||x_i||c_i}\right)\right\| \leq\sum\limits_{n \in \mathbb{N}} \|y'_i\| \,\|x_i\| c_i\, \|T\|_K=C\,\|T\|_K. $$ And yes, the same argument would work for the operator norm, since it is the case where $K$ is the unit ball. Being continuous in operator norm is easier, so I don't see what you expect from "not mentioning" the topology. And the converse would not work, as not every bounded functional is of the form mentioned; that's where they likely use that $K$ is compact.