Definition 15.5.1 ($c_{0}$-Direct Sum).label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$. For any $x \in \prod_{i \in I}X_{i}$, $x$ vanishes at infinity if for each $\eps > 0$, $\bracs{i \in I| \norm{x_i}_{X_i} \ge \eps}$ is finite. The space
\[[c_{0}(I); X_{i}] = \bracs{x \in \prod_{i \in I}X_i \bigg | x \text{ vanishes at infinity}}\]
equipped with the uniform norm
\[\norm{x}_{[c_0(I); X_i]}= \sup_{i \in I}\norm{x_i}_{X_i}\]
is the $c_{0}$-direct sum of $\seqi{X}$.
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