We have from book Bases in Banach spaces II by ivan singer "Proposition" Let $\{G_n\}$ be a sequence of closed linear subspaces of a Banach space $E$ and let $D_1$ be the linear space of sequences of elements $$D_1 = \Bigl\{\{y_n\} \subset E \big| y_n \in G_n \text{ }(n=1,2,...),\text{ }\displaystyle\sum_{i=1}^\infty y_i \text{ }converges\Big \},$$ endowed with the norm $$\|\{y_n\}\| = \displaystyle \sup_{1 \leq n <\infty}\|\displaystyle\sum_{i=1}^n y_i\|.$$ Then $D_1$ is a Banach space.
and "Proposition" Let $\{G_n\}$ be a decomposition of a Banach space $E$, such that each $G_n (n = 1,2, ...)$ is closed, and let $\{v_n\}$ be the associated sequence of coordinate projections to $\{G_n\}$.
Then a) The Banach space $D_1$ introduced in the last proposition is isomorphic to $E$, by the mapping $$w: \{y_n\} \rightarrow \displaystyle\sum_{i=1}^\infty y_i.$$
b) The numbers $$|||x||| = \displaystyle\sup_{1 \leq n < \infty} \|\sum_{i=1}^n v_i(x)\| (x \in E)$$ define a norm on the space $E,$ equivalent to the initial norm of $E.$
On the other hand, If we have a sequence of Banach spaces $(X_n)_{n=1}^\infty$ and a number $1 \leq p <\infty$, we define the $l_p$-sum of $X_1, X_2,...$ to be the space of all sequences $(x_n)_{n=1}^\infty$, with $x_n \in X_n$ for $n=1, 2, ...$, for which $\sum_{n=1}^\infty \|x_n\|_{X_n}^p < \infty,$ in case $p < \infty,$ or $\|(x_n)\|_\infty = \sup_n \|x_n\|_{X_n} < \infty,$ in case $p = \infty,$ and we use also the shorthand $(X_1 \oplus X_2 \oplus ...)_p$ to denote this new space. In brief, for any $1 \leq p \leq \infty,$ we have $$(X_1 \oplus X_2 \oplus ...)_p = \{(x_n): x_n \in X_n \text{ and } (\|x_n\|)_{n=1}^\infty \in l_p \}.$$
Also, The $c_0$-sum of spaces is defined in an entirely analogous fashion. In this case we write $$(X_1 \oplus X_2 \oplus ...)_0 = \{(x_n): x_n \in X_n \text{ and } (\|x_n\|)_{n=1}^\infty \in c_0\}.$$
We also always have $$(l_p \oplus l_p \oplus ...)_p = l_p \text{ and } (c_0 \oplus c_0 \oplus ...) = c_0,$$
So say in $(l_p \oplus l_p \oplus ...)_p = l_p$ space if we take $x = (v_i(x))_{i=1}^\infty \in l_p$ space then so $|||x|||= \displaystyle\sup_{1 \leq n < \infty} \|\sum_{i=1}^n v_i(x)\|$ is equivalent to $(\sum_{i=1}^\infty \|v_i(x)\|^p)^{1/p}$
Question So, if $x \in l_p$ space, as the two norms are equivalent does $$|||x||| = \sup_{1 \leq n < \infty} \|\sum_{i=1}^n (0,0,...,0,v_i(x),0,...)\| \leq c (\sum_{i=1}^\infty \|(0,0,...,0,v_i(x),0,...)\|_{l_p}^p)^{1/p} = c (\sum_{i=1}^\infty ((\|v_i(x)\|^p)^{1/p})^p)^{1/p} = c (\sum_{i=1}^\infty \|v_i(x)\|^p)^{1/p} = c \|x\|??$$