Proposition 10.11.3.label Let $\seqf{E_j}$ be TVSs over $K \in \RC$, then
\[\prod_{j = 1}^{n} E_{j} = \bigoplus_{j = 1}^{n} E_{j}\]
Proof. 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 TVS $F$ over $K$ and $\seqf{T_j}$ with $T_{j} \in L(E_{j}; F)$ for each $1 \le j \le n$, let
\[T: \prod_{j = 1}^{n} E_{j} \to F \quad (x_{1}, \cdots, x_{n}) \mapsto \sum_{j = 1}^{n} T_{j}x_{j}\]
then $T \in L(\prod_{j = 1}^{n} E_{j}; F)$ is the unique continuous linear map such that the following diagram commutes:
\[\xymatrix{
E \ar@{->}[r]^{T} & F \\
E_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} &
}\]
Hence $\prod_{j = 1}^{n} E_{j}$ satisfies (U) of the direct sum, so the spaces coincide.$\square$
Post a Comment