9.9 The Projective Tensor Product
Definition 9.9.1 (Projective Tensor Product). Let $E, F$ be locally convex spaces over $K \in \RC$, then there exists a pair $(E \otimes_{\pi} F, \iota)$ such that:
$E \otimes_{\pi} F$ is a locally convex space over $K$.
$\iota \in L^{2}(E, F; E \otimes_{\pi} F)$ is a continuous bilinear map.
For any $(G, \lambda)$ satisfying (1) and (2), there exists a unique $\Lambda \in L(E \otimes_{\pi} F; G)$ such that the following diagram commutes:
\[\xymatrix{ E \times F \ar@{->}[rd]_{\lambda} \ar@{->}[r]^{\iota} & E \otimes F \ar@{->}[d]^{\Lambda} \\ & G }\]For any topology $\topo$ on $E \otimes_{\pi} F$ satisfying (1) and (2), $\topo$ is coarser than the topology on $E \otimes_{\pi} F$.
$E \otimes_{\pi} F$ is the linear span of $\iota(E \times F)$.
For any $U \subset E$ and $V \subset F$, let $U \otimes V = \bracs{u \otimes v|u \in U, v \in V}$, then the convex circled hulls
\[\fB = \bracsn{\Gamma(U \otimes V)| U \in \cn_E(0), V \in \cn_F(0)}\]is a fundamental system of neighbourhoods at $0$ for $E \otimes_{\pi} F$.
The space $E \otimes_{\pi} F$ is the projective tensor product of $E$ and $F$, and the mapping $\iota \in L^{2}(E, F; E \otimes_{\pi} F)$ is the canonical embedding.
The space $E \widetilde{\otimes}_{\pi} F$ denotes the Hausdorff completion of $E \otimes_{\pi} F$.
Proof. Let $E \otimes_{\pi} F = E \otimes F$ be the tensor product of $E$ and $F$ as vector spaces. Let $\mathscr{T}\subset 2^{2^X}$ be the collection of all locally convex topologies satisfying (1) and (2), and let $\mathcal{S}$ be the projective topology on $E \otimes_{\pi} F$ generated by $\mathscr{T}$.
(1): By Proposition 9.5.1, $\mathcal{S}$ is a locally convex topology on $E \otimes_{\tau} F$.
(2): Since $\iota: E \times F \to E \otimes_{\pi} F$ is continuous with respect to every topology in $\mathscr{T}$, it is also continuous with respect to $\mathcal{S}$.
(U2): Since $\mathcal{T}\in \mathscr{T}$, $\mathcal{S}\supset \mathcal{T}$.
(U1): By (U) of the tensor product, there exists a unique $\Lambda \in \hom(E \otimes_{\pi} F; G)$ such that the given diagram commutes. Since $\lambda$ is continuous, the projective topology generated by $\Lambda$ satisfies (1) and (2). By (U2), $\mathcal{S}$ contains the projective topology generated by $\Lambda$. Therefore $\Lambda \in L(E \otimes_{\pi}; F)$.
(5): By (4) of the tensor product.
(6): Let $U \in \cn_{E}(0)$ and $V \in \cn_{F}(0)$ be balanced. For any $\sum_{j = 1}^{n} x_{j} \otimes y_{j} \in E \otimes_{\pi} F$, then there exists $\lambda > 0$ such that $\seqf{x_j}\subset \lambda U$ and $\seqf{y_j}\subset \lambda V$. In which case, $\sum_{j = 1}^{n} x_{j} \otimes y_{j} \subset \lambda \Gamma (U \otimes V)$, so $\fB$ is a collection of convex, circled, and radial sets. Since $\fB$ defines a locally convex topology that satisfies (1) and (2), $\mathcal{S}$ contains the topology defined by $\fB$.
On the other hand, for any convex and circled set $W \in \cn_{E \otimes_\pi F}(0)$, there exists $U \in \cn_{E}(0)$ and $V \in \cn_{F}(0)$ such that $U \otimes V \subset W$. In which case, $W \supset \Gamma(U \otimes V)$, so $\fB$ is a fundamental system of neighbourhoods at $0$ for $E \otimes_{\pi} F$.$\square$
Remark 9.9.1. In constructing the projective tensor product, it may be more natural to obtain its topology as a projective topology using its universal property. However, doing so requires taking a least upper bound across all continuous linear maps defined on $E \times F$, a collection too big to be a set. As such, constructing it as a projective topology is logically dubious, or at the very least beyond my abilities.
Definition 9.9.2 (Cross Seminorm). Let $E, F$ be locally convex spaces over $K \in \RC$. For any convex circled sets $U \in \cn_{E}(0)$ and $V \in \cn_{F}(0)$, let $p: E \to [0, \infty)$ and $q: F \to [0, \infty)$ be their gauges. For any $z \in E \otimes_{\pi} F$, let
then
$\rho$ is a continuous seminorm on $E \otimes_{\pi} F$.
$\rho$ is the gauge of $\Gamma(U \otimes V)$.
For any $x \in E$ and $y \in F$, $\rho(x \otimes y) = p(x)q(Y)$.
$\rho$ is a norm if and only if $[\cdot]_{U}$ and $[\cdot]_{V}$ are norms.
and the seminorm $\rho = p \otimes q$ is the cross seminorm of $p$ and $q$. Moreover,
If the seminorms $\seqi{p}$ define the topology on $E$, and the seminorms $\seqj{q}$ define the topology on $F$, then the seminorms $\bracsn{p_i \otimes q_j| (i, j) \in I \times J}$ define the topology on $E \otimes_{\pi} F$.
Proof [III.6.3, SW99]. (1): Let $\lambda \in K$, then for any $\seqf{(x_j,y_j)}\subset E \times F$,
and
so for any $z \in E \otimes_{\pi} F$, $|\lambda|\rho(z) = \rho(\lambda z)$.
Let $z, z' \in E \otimes F$, $\seqf{(x_j,y_j)}, \bracsn{(x_j',y_j')}_{1}^{m} \subset E \times F$ such that $z = \sum_{j = 1}^{n} x_{j} \otimes y_{j}$ and $z' = \sum_{j = 1}^{m} x_{j}' \otimes y_{j}'$, then
and
so $\rho$ satisfies the triangle inequality.
(2): Let $z \in \Gamma(U \otimes V)$, then there exists $\seqf{(x_j, y_j)}\subset U \times V$ and $\seqf{\lambda_j}\subset K$ such that $\sum_{j = 1}^{n} |\lambda_{j}| \le 1$ and $z = \sum_{j = 1}^{n} \lambda x_{j} \otimes y_{j}$. In which case,
so $\Gamma(U \otimes V) \subset \bracs{\rho < 1}$.
Let $z \in \bracs{\rho < 1}$, then there exists $\seqf{(x_j, y_j)}\subset E \times F$ such that $z = \sum_{j = 1}^{n}x_{j} \otimes y_{j}$ and $\sum_{j = 1}^{n} p(x_{j})q(x_{j}) < 1$. Let $\eps > 0$ such that $\sum_{j = 1}^{n}(p(x_{j}) + \eps)(q(x_{j}) + \eps) < 1$, then
and $\Gamma(U \otimes V) \supset \bracs{\rho < 1}$.
(3): Let $x \in U$ and $y \in V$. By the Hahn-Banach Theorem, there exists $\phi \in E^{*}$ and $\psi \in F^{*}$ such that $\dpn{x, \phi}{E}= p(x)$, $\dpn{y, \psi}{F}= q(x)$, $|\phi| \le p$, and $|\psi| \le q$. By (U1) of the projective tensor product, there exists $\Phi \in (E \otimes_{\pi} F)^{*}$ such that the following diagram commutes
For any $z \in E \otimes_{\pi} F$ and $\seqf{(x_j, y_j)}\subset E \times F$ such that $z = \sum_{j = 1}^{n} x_{j} \otimes y_{j}$,
As the above holds for all such $\seqf{(x_j, y_j)}\subset E \times F$, $|\Phi| \le \rho$. Since $\Phi(x \otimes y) = p(x)q(y)$, $\rho(x \otimes y) = p(x)q(y)$ as well.
(5): By (6) of Definition 9.9.1.$\square$
Theorem 9.9.3. Let $E, F$ be metrisable locally convex spaces over $K \in \RC$, then for any $z \in E \td{\otimes}_{\pi} F$, there exists $\seq{\lambda_n}\subset K$ and $\seq{(x_j, y_j)}\subset E \times F$ such that:
$\sum_{n \in \natp}|\lambda_{n}| < \infty$.
$\limv{n}x_{n} = 0$ and $\limv{n}y_{n} = 0$.
$z = \sum_{n = 1}^{\infty} \lambda_{n} x_{n} \otimes y_{n}$.
Proof. Let $\seq{p_n}$ and $\seq{q_n}$ be increasing sequences of continuous seminorms that induce the topology on $E$ and $F$, respectively. For each $n \in \natp$, let $r_{n} = p_{n} \otimes q_{n}$, and $\td r_{n}$ be the continuous extension of $r_{n}$ to $E \td{\otimes}_{\pi} F$.
Let $u \in E \td{\otimes}_{\pi} F$, then there exists $\seq{u_n}\subset E \otimes_{\pi} F$ such that $\td r_{n}(u - u_{n}) < 2^{-n}/n^{2}$ for all $n \in \natp$. For each $N \in \natp$, let $v_{N} = u_{N+1}- u_{N}$, then
Since $r_{N} = p_{N} \otimes q_{N}$, there exists $\bracsn{(x_{N, k}, y_{N, k})}_{1}^{n_N}\subset X \times Y$ such that $v_{N} = \sum_{k = 1}^{n_N}x_{N, k}\otimes y_{N, k}$ and
By rescaling, assume without loss of generality that there exists $\bracsn{\lambda_{N, k}}_{1}^{n_N}$ such that
$v_{N} = \sum_{k = 1}^{n_N}\lambda_{N, k}x_{N, k}\otimes y_{N, k}$.
For each $1 \le k \le n_{N}$, $p_{N}(x_{N, k}), q_{N}(x_{N, k}) \le 1/M$.
$\sum_{k = 1}^{n_N}|\lambda_{k}| \le 2^{-N+2}$.
From here, let $\seqf{(x_j, y_j)}\subset X \times Y$ such that $u_{1} = \sum_{j = 1}^{n} x_{j} \otimes y_{j}$, then
where $x_{N, k}\to 0$ and $y_{N, k}\to 0$ as $N \to \infty$, and $\sum_{N \in \natp}\sum_{k = 1}^{n_N}|\lambda_{N, k}| < \infty$.$\square$