In my lectures, it was stated but not proved that
$$\bigoplus _{i\geq 1} l_q \cong l_q$$
meaning that the infinite direct sum of $l_q$ sequnce spaces is isometrically isomorphic to $l_q$. But I don't see how this can be the case. In particular, all elements $x \in l_q$ have $\Sigma_{i=1}^\infty |x_i|^q = a < \infty$. But now having a direct sum of these spaces, we have no guarantee that the infinite sum $\Sigma |a_i| $ is $< infty$.
Am I missing something here?
Split $\mathbb N$ into countably many infinite sets $\Gamma_n$, say. The space
$$\ell_p(\Gamma_n)= \{f\in \ell_p(\mathbb N)\colon f|_{\mathbb N\setminus \Gamma_n} = 0\}$$
is isometrically isomorphic to $\ell_p$. For each $n$ let $T_n\colon \ell_p \to \ell_p(\Gamma_n)$ that arises from a bijection between $\mathbb N$ and $\Gamma_n$. The map $T\colon \ell_p \to (\bigoplus_n \ell_p)_{\ell_p}$ given by
$$Tf = (T_n (f|_{\Gamma_n}))_{n=1}^\infty$$
is the sought isometry. (Requires checking.)