Definition 10.11.2 (Direct Sum).label Let $\seqi{E}$ be TVSs over $K \in \RC$, then there exists $(E, \seqi{\iota})$ such that:
- (1)
$E$ is a TVS over $K$.
- (2)
For each $i \in I$, $\iota_{i} \in L(E_{i}; E)$.
- (U)
For each $(F, \seqi{T})$ satisfying (1) and (2), there exists a unique $T \in L(E; F)$ such that the following diagram commutes:
\[\xymatrix{ E \ar@{->}[r]^{T} & F \\ E_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & }\]
The space $E = \bigoplus_{i \in I}E_{i}$ is the direct sum of $\seqi{E}$.
Proof. Let $(E, \seqi{\iota})$ be the direct sum of $\seqi{E}$ as vector spaces, and equip it with the inductive topology induced by $\seqi{\iota}$, then $(E, \seqi{\iota})$ satisfies (1) and (2).
(U): By (U) of the direct sum, there exists a unique $T \in \hom(E; F)$ such that the diagram commutes. In which case, by (4) of Definition 10.11.1, $T \in L(E; F)$.$\square$
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