11.8 Inductive Limits
Definition 11.8.1 (Inductive Locally Convex Topology).label Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_{i} \in \hom(E_{i}; E)$ for all $i \in I$, then there exists a topology $\topo$ on $E$ such that:
- (1)
$(E, \topo)$ is a locally convex space 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 locally convex space $F$ 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 $i \in I$.
- (5)
The family
\[\mathcal{B}= \bracs{U \subset E|U \text{ convex, circled, radial}, T_i^{-1}(U) \in \cn_{E_i}(0) \forall i \in I}\]is a fundamental system of neighbourhoods for $E$ at $0$.
- (6)
If $E$ is spanned by $\bigcup_{i \in I}T_{i}(E_{i})$, then
\[\fB = \bracs{\Gamma\paren{\bigcup_{i \in I}T_i(U_i)} \bigg | U_i \in \cn_{E_i}(0)}\]is a fundamental system of neighbourhoods for $E$ at $0$.
The topology $\topo$ is the inductive locally convex topology on $E$ induced by $\seqi{T}$.
Proof. (1), (5): 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 locally convex space.
(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 convex, circled, and radial. By (2), $T_{i}^{-1}(U) \in \cn_{E_i}(0)$ for all $i \in I$. Thus the convex, circled, and radial neighbourhoods of $0$ in $(E, \mathcal{S})$ is a subset of $\mathcal{B}$.
(5): Let $i \in I$ and $U \in \cn_{F}(0)$ be convex, circled, and radial. Since $T \circ E_{i} \in L(E_{i}; F)$, $T_{i}^{-1}(T^{-1}(U)) \in \cn_{E_i}(0)$, so $T^{-1}(U) \in \mathcal{B}\subset \cn_{E}(0)$.
(6): If $E$ is spanned by $\bigcup_{i \in I}T_{i}(E_{i})$, then each set in $\fB$ is radial. Hence $\fB$ is a family of neighbourhoods of $E$ at $0$.
Let $U \in \cn_{E}(0)$ be convex, circled, and radial, then for each $i \in I$, $T_{i}^{-1}(U) \in \cn_{E_i}(0)$, so $U \supset \bigcup_{i \in I}T_{i}[T_{i}^{-1}(U)]$. Since $U$ is convex and circled, $U \supset \Gamma\paren{\bigcup_{i \in I}T_i[T_i^{-1}(U)]}\in \fB$. Therefore $\fB$ forms a fundamental system of neighbourhoods for $E$ at $0$.$\square$
Definition 11.8.2 (Locally Convex Direct Sum).label Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, then there exists $(E, \seqi{\iota})$ such that:
- (1)
$E$ is a locally convex space 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:
\[\xymatrix{ E \ar@{->}[r]^{T} & F \\ E_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & }\] - (4)
The family
\[\fB = \bracs{\Gamma\paren{\bigcup_{i \in I}\iota_i(U_i)} \bigg | U_i \in \cn_{E_i}(0)}\]is a fundamental system of neighbourhoods for $E$ at $0$.
The space $E = \bigoplus_{i \in I}E_{i}$ is the locally convex 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 locally convex 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 11.8.1, $T \in L(E; F)$.
(4): By (6) of Definition 11.8.1.$\square$
Proposition 11.8.3.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.$\square$
Definition 11.8.4 (Inductive Limit).label Let $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of locally convex spaces over $K \in \RC$, then there exists $(E, \bracsn{T^i_E}_{i \in I})$ such that:
- (1)
$E$ is a locally convex space 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 locally convex space $F$ 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 $i \in I$.
- (6)
The family
\[\mathcal{B}= \bracs{U \subset E|U \text{ convex, radial, circled}, (T^i_E)^{-1}(U) \in \cn_{E_i}(0) \forall i \in I}\]is a fundamental system of neighbourhoods for $E$ at $0$.
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 11.8.1, $S \in L(E; F)$.
(5): By (5) of Definition 11.8.1.$\square$
Remark 11.8.1.label The projective topology behaves well across the constraints of topological vector spaces and locally convex spaces: the preimage of a vector space/locally convex topology is also a vector space/locally convex topology.
On the inductive side, the story is not as simple: In principle, the locally convex inductive topology is smaller than the vector space inductive topology, which is smaller than the inductive topology. As such, the same construction must be performed three separate times, each time restricting to a smaller collection of sets.
In addition to the neighbourhood construction given above, the inductive topology may also be constructed as the weak topology generated by all topologies satisfying certain properties. While this more non-constructive method is simpler, it does not directly provide an explicit fundamental system of neighbourhoods at $0$.
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