\section{The Projective Tensor Product}
\label{section:projective-tensor-product}

\begin{definition}[Projective Tensor Product]
\label{definition: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:
    \begin{enumerate}
        \item $E \otimes_\pi F$ is a locally convex space over $K$. 
        \item $\iota \in L^2(E, F; E \otimes_\pi F)$ is a continuous bilinear map. 
        \item[(U1)] 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:
        \[
        \xymatrix{
        E \times F \ar@{->}[rd]_{\lambda} \ar@{->}[r]^{\iota} & E \otimes F \ar@{->}[d]^{\Lambda} \\
        & G
        }
        \]
        \item[(U2)] For any topology $\topo$ on $E \otimes_\pi F$ satisfying (1) and (2), $\topo$ is coarser than the topology on $E \otimes_\pi F$. 

        \item $E \otimes_\pi F$ is the linear span of $\iota(E \times F)$. 
        \item 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
        \[
        \fB = \bracsn{\aconv(U \otimes V)| U \in \cn_E(0), V \in \cn_F(0)}
        \]

        is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$.
    \end\{enumerate\}

    
    The space $E \otimes_\pi F$ is the \textbf{projective tensor product} of $E$ and $F$, and the mapping $\iota \in L^2(E, F; E \otimes_\pi F)$ is the \textbf{canonical embedding}.

    The space $E \widetilde{\otimes}_\pi F$ denotes the Hausdorff completion of $E \otimes_\pi F$.
\end{definition}
\begin{proof}
    Let $E \otimes_\pi F = E \otimes F$ be the \hyperref[tensor product]{definition: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 \autoref{proposition:lc-projective-topology}, $\mathcal{S}$ 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 \hyperref[tensor product]{definition: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 \hyperref[tensor product]{definition: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, $\sum_{j = 1}^n x_j \otimes y_j \subset \lambda \aconv (U \otimes V)$, 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, $W \supset \aconv(U \otimes V)$, so $\fB$ is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$. 
\end{proof}

\begin{remark}
\label{remark:projective-construction}
    In constructing the \hyperref[projective tensor product]{definition: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 \textit{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. 
\end{remark}



\begin{definition}[Cross Seminorm]
\label{definition: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 \hyperref[gauges]{definition:gauge}. For any $z \in E \otimes_\pi F$, let
    \[
    \rho(z) = \inf\bracs{\sum_{j = 1}^n p(x_j)q(y_j) \bigg | \seqf{(x_j,y_j)} \subset E \times F, z = \sum_{j = 1}^n x_j \otimes y_j}
    \]

    then
    \begin{enumerate}
        \item $\rho$ is a continuous seminorm on $E \otimes_\pi F$.
        \item $\rho$ is the gauge of $\aconv(U \otimes V)$. 
        \item For any $x \in E$ and $y \in F$, $\rho(x \otimes y) = p(x)q(Y)$. 
        \item $\rho$ is a norm if and only if $[\cdot]_U$ and $[\cdot]_V$ are norms. 
    \end{enumerate}

    and the seminorm $\rho = p \otimes q$ is the \textbf{cross seminorm} of $p$ and $q$. Moreover, 
    \begin{enumerate}
        \item[(5)] 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 $E \otimes_\pi F$. 
    \end\{enumerate\}

    
\end{definition}
\begin{proof}[Proof {{\cite[III.6.3]{SchaeferWolff}}}. ]
    (1): Let $\lambda \in K$, then for any $\seqf{(x_j,y_j)} \subset E \times F$, 
    \[
    |\lambda| \sum_{j = 1}^n p(x_j)q(y_j) = \sum_{j = 1}^n p(\lambda x_j)q(y_j)
    \]

    and
    \[
    \lambda\sum_{j = 1}^n x_j \otimes y_j =  \sum_{j = 1}^n \lambda x_j \otimes y_j
    \]

    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
    \[
    z + z' = \sum_{j = 1}^n x_j \otimes y_j + \sum_{j = 1}^m x_j' \otimes y_j'
    \]

    and
    \[
    \rho(z + z') \le \sum_{j = 1}^n p(x_j)q(y_j) + \sum_{j = 1}^m p(x_j')q(y_j')
    \]

    so $\rho$ satisfies the triangle inequality. 

    (2): Let $z \in \aconv(U \otimes V)$, 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, 
    \begin{align*}
        \rho(z) &\le \sum_{j = 1}^n p(\lambda x_j)q(y_j) = \sum_{j = 1}^n |\lambda_j|p(x_j)q(y_j) \\
        &< \sum_{j = 1}^n |\lambda_j| \le 1
    \end{align*}

    so $\aconv(U \otimes V) \subset \bracs{\rho < 1}$.

    Let $z \in \bracs{\rho < 1}$, then there exists $\seqf{(x_j, y_j)} \subset E \times F$ such that $z = \sum_{j = 1}^nx_j \otimes y_j$ 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
    \[
    z = \sum_{j = 1}^n (p(x_j) + \eps)(q(x_j) + \eps) \cdot \underbrace{\frac{x_j}{p(x_j) + \eps}}_{\in \bracs{p < 1} = U} \otimes \underbrace{\frac{y_j}{q(x_j) + \eps}}_{\in \bracs{q < 1} = V} \in \aconv(U \otimes V)
    \]

    and $\aconv(U \otimes V) \supset \bracs{\rho < 1}$.
    
    (3): Let $x \in U$ and $y \in V$. By the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, 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 \hyperref[projective tensor product]{definition:projective-tensor-product}, there exists $\Phi \in (E \otimes_\pi F)^*$ such that the following diagram commutes
    \[
    \xymatrix{
    E \times F \ar@{->}[rd]_{\phi \cdot \psi} \ar@{->}[r]^{\iota} & E \otimes_\pi F \ar@{->}[d]^{\Phi} \\
    & K
    }
    \]
    
    For any $z \in E \otimes_\pi F$ and $\seqf{(x_j, y_j)} \subset E \times F$ such that $z = \sum_{j = 1}^n x_j \otimes y_j$, 
    \[
    \Phi(z) = \sum_{j = 1}^n \Phi(x_j \otimes y_j) = \sum_{j = 1}^n \phi(x_j)\psi(y_j) \le \sum_{j = 1}^n p(x_j)q(y_j)
    \]

    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 \autoref{definition:projective-tensor-product}. 
\end{proof}

\begin{theorem}[{{\cite[III.6.4]{SchaeferWolff}}}]
\label{theorem:metrisable-tensor-product}
    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:
    \begin{enumerate}
        \item $\sum_{n \in \natp}|\lambda_n| < \infty$. 
        \item $\limv{n}x_n = 0$ and $\limv{n}y_n = 0$.
        \item $z = \sum_{n = 1}^\infty \lambda_n x_n \otimes y_n$. 
    \end\{enumerate\}

    
\end{theorem}
\begin{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
    \begin{align*}
        r_N(v_N) &= \td r_n(u_{N+1} - u_n) \le \td r_N(u - u_N) + \td r_{N}(u - u_{N+1}) \\
        &\le \td r_N(u - u_N) + \td r_{N+1}(u - u_{N+1}) < 2^{-N+1}/n^2
    \end{align*}

    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
    \[
        r_N(v_N) = \sum_{k = 1}^{n_N}p_N(x_{N, k})q_N(x_{N, k}) < 2^{-N+1}/n^2
    \]
    
    By rescaling, assume without loss of generality that there exists $\bracsn{\lambda_{N, k}}_1^{n_N}$ such that\
    \begin{enumerate}
        \item $v_N = \sum_{k = 1}^{n_N}\lambda_{N, k}x_{N, k} \otimes y_{N, k}$.
        \item For each $1 \le k \le n_N$, $p_N(x_{N, k}), q_N(x_{N, k}) \le 1/M$.
        \item $\sum_{k = 1}^{n_N}|\lambda_k| \le 2^{-N+2}$.
    \end\{enumerate\}

    
    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
    \[
    u = u_1 + \sum_{N = 1}^\infty v_N = \sum_{j = 1}^n x_j \otimes y_j + \sum_{N = 1}^\infty \sum_{k = 1}^{n_N}\lambda_{N, k}x_{N, k} \otimes y_{N, k}
    \]

    where $x_{N, k} \to 0$ and $y_{N, k} \to 0$ as $N \to \infty$, and $\sum_{N \in \natp}\sum_{k = 1}^{n_N}|\lambda_{N, k}| < \infty$.
    
\end{proof}