10.11 Inductive Limits
Definition 10.11.1 (Inductive Topology).label Let $\seqi{E}$ be TVSs over $K \in \RC$, $\seqi{T}$ such that $T_{i} \in \hom(E_{i}; E)$ for all $i \in I$, and $E$ be a vector space over $K$, then there exists a topology $\topo$ on $E$ such that:
- (1)
$(E, \topo)$ is a TVS over $K$.
- (2)
For each $i \in I$, $T_{i} \in L(E_{i}; E)$.
- (U)
For any topology $\mathcal{S}$ on $E$ satisfying (1) and (2), $\mathcal{S}\subset T$.
- (4)
For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T_{i} \in L(E_{i}; F)$ for all $i \in I$.
The topology $\topo$ is the inductive topology on $E$ induced by $\seqi{T}$.
Proof. (1): Let
To see that $\mathcal{B}$ is a fundamental system of neighbourhoods at $0$ for a vector space topology on $E$, it is sufficient to verify the following and apply Proposition 10.1.12.
- (TVB1)
Every set in $\mathcal{B}$ is radial and circled by definition.
- (TVB2)
For any $U \in \mathcal{B}$, $U$ is circled, so $\frac{1}{2}U + \frac{1}{2}U \subset U$. Since $\frac{1}{2}U$ is also circled and radial, $\frac{1}{2}U \in \mathcal{B}$.
Let $\topo$ be the vector space topology such that $\mathcal{B}$ is a fundamental system of neighbourhoods at $0$, then $(E, \topo)$ is a TVS.
(2): For each $i \in I$ and $U \in \mathcal{B}$, $T_{i}^{-1}(U) \in \cn_{E_i}(0)$, so $T_{i} \in L(E_{i}; E)$.
(U): Let $U \in \cn_{(E, \mathcal{S})}(0)$ be circled and radial, then by (2), $T_{i}^{-1}(U) \in \cn_{E_i}(0)$ for all $i \in I$. Thus the circled and radial neighbourhoods of $0$ in $(E, \mathcal{S})$ is a subset of $\mathcal{B}$.
(4): Let $U \in \cn_{F}(0)$ be circled and radial and $i \in I$. Since $T \circ E_{i} \in L(E_{i}; F)$, $T_{i}^{-1}(T^{-1}(U)) \in \cn_{E_i}(0)$, so $T^{-1}(U) \in \mathcal{B}\subset \cn_{E}(0)$.$\square$
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, $T \in L(E; F)$.$\square$
Proposition 10.11.3.label Let $\seqf{E_j}$ be TVSs over $K \in \RC$, then
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
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.$\square$
Definition 10.11.4 (Inductive Limit).label Let $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of TVSs over $K \in \RC$, then there exists $(E, \bracsn{T^i_E}_{i \in I})$ such that:
- (1)
$E$ is a TVS over $K$.
- (2)
For each $i \in I$, $T^{i}_{E} \in L({E_i, E})$.
- (3)
For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
\[\xymatrix{ E_i \ar@{->}[rd]_{T^i_E} \ar@{->}[r]^{T^i_j} & E_j \ar@{->}[d]^{T^j_E} \\ & E }\] - (U)
For any pair $(F, \bracsn{S^i_F}_{i \in I})$ satisfying (1), (2), and (3), there exists a unique $S \in L({E, F})$ such that the following diagram commutes
\[\xymatrix{ E_i \ar@{->}[d]_{T^i_E} \ar@{->}[rd]^{S^i_F} & \\ E \ar@{->}[r]_{S} & F }\]for all $i \in I$.
- (5)
For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T^{i}_{E} \in L(E_{i}; F)$ for all $i \in I$.
The pair $(E, \bracsn{T^i_E}_{i \in I})$ is the inductive limit of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$.
Proof. Let $(E, \bracsn{T^i_E}_{i \in I})$ be the direct limit of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ as vector spaces over $K$ (Proposition 1.3.3). Equip $E$ with the inductive topology induced by $\bracsn{T^i_E}_{i \in I}$, then $(E, \bracsn{T^i_E}_{i \in I})$ satisfies (1), (2), and (3).
(U): By (U) of Proposition 1.3.3, there exists a unique $S \in \hom(E; F)$ such that the given diagram commutes. By (4) of Definition 10.11.1, $S \in L(E; F)$.
(5): By (5) of Definition 10.11.1.$\square$
Definition 10.11.5 (Strict).label Let $(\seqi{E}, \bracsn{\iota^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of TVSs over $K \in \RC$, then the system is strict if:
- (1)
For each $i, j \in I$ with $i \lesssim j$, $\iota^{i}_{j}: E_{i} \to E_{j}$ is injective.
- (2)
For each $i, j \in I$ with $i \lesssim j$, the topology of $E_{i}$ is induced by $\iota^{i}_{j}$.
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