I'm reading the book "Theory of operator algebras" by Takesaki. In this book, the $\sigma$-strong$^*$ topology on the space $B(H)$ (bounded operators on the Hilbert space $H$) is defined (see def 2.3 p68) to be the locally convex topology generated by the seminorms

$$B(H)\ni x \mapsto \left(\sum_{n=1}^\infty \|x\xi_n\|^2 + \|x^*\xi_n\|^2\right)^{1/2}.$$I am trying to understand the proof of lemma 2.4:

Question:

Why does it suffice to use only one seminorm? Shouldn't we have something like $$\sum_{k=1}^m \left(\sum_{n=1}^\infty (\|x(\xi_k)_n\|^2+ \|x^*(\xi_k)_n\|^2\right)^{1/2} \le 1 \implies |\omega(x)| \le 1$$ where $\xi_1, \dots, \xi_m$ are sequences in $H$.