Proposition 11.7.3.label Let $\seqf{E_j}$ be TVSs over $K \in \RC$, then the following spaces coincide:
- (1)
The product $\prod_{j = 1}^{n} E_{j}$.
- (2)
The direct sum of $\seqf{E_j}$ as topological vector spaces.
- (3)
The direct sum of $\seqf{E_j}$ as locally convex spaces.
Proof. By Proposition 10.11.3, it is sufficient to show that (1) and (3) coincide. The proof is exactly the same as Proposition 10.11.3, but included here for completeness.
Let $1 \le k \le n$, then for each $1 \le k, l \le n$, $\pi_{l} \circ \iota_{k} \in L(E_{k}, E_{l})$, so by (U) of the product, $\iota_{k} \in L(E_{k}; \prod_{j = 1}^{n} E_{j})$. Thus $\prod_{j = 1}^{n} E_{j}$ satisfies (1) and (2) of the direct sum.
For any locally convex space $F$ over $K$ and $\seqf{T_j}$ with $T_{j} \in L(E_{j}; F)$ for each $1 \le j \le n$, let
then $T \in L(\prod_{j = 1}^{n} E_{j}; F)$ is the unique continuous linear map such that the following diagram commutes:
Hence $\prod_{j = 1}^{n} E_{j}$ satisfies (U) of the direct sum, so the spaces coincide.$\square$
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