Definition 14.3.1 ($l^{p}$-Direct Sum).label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty)$, then the $l^{p}$-direct sum of $\seqi{X}$ is the space
\[[l^{p}(I); X_{i}] = \bracs{x \in \prod_{i \in I}X_i \bigg | \sum_{i \in I}\norm{x_i}_{X_i}^p < \infty}\]
equipped with the norm
\[\norm{x}_{[l^p(I); X_i]}= \braks{\sum_{i \in I}\norm{x_i}_{X_i}^{p}}^{1/p}\]
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