Hello everyone I have the following question:
Let $\{x_k\} \in \mathbb{R}^n$ be a sequence such that $\|x_k\|^q_q \leq M$ for every $k \in \mathbb{N}$, where $\|x\|^q_q = \sum^n_{i=1}|x_i|^q$ and $M>0$. I know that $\|\cdot||_q$ it's not a norm and it doesn't even induces a metric as mentioned here (L1 convergence and Lp bounded implies Lq convergence). But I was reading an article which says if $\|x_k\|^q_q \leq M$ then $\{x_k\} $ is bounded and it has a subsequence which is convergent. But we say that a set $X$ in a metric space is bounded when for every $x,y \in X$ we have that $d(x,y) \leq M, \, M>0 $. How can a pseudo norm bound a sequence?
Thank you
In this argument boundedness is not w.r.t the given pseudo-metric. Since $|x_i|^{q} \leq M$ for each $i$ the sequence is bounded in the usual sense and hence it has a convergent subsquence. You can see that this convergence takes place w.r.t. the given pseudo-metric also.