Definition 9.9.2: Cross Seminorm | Jerry's Digital Garden

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

\[\rho(z) = \inf\bracs{\sum_{j = 1}^n p(x_j)q(y_j) \bigg | \seqf{(x_j,y_j)} \subset E \times F, z = \sum_{j = 1}^n x_j \otimes y_j}\]

then

$\rho$ 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 $\seqf{(x_j,y_j)}\subset E \times F$,

\[|\lambda| \sum_{j = 1}^{n} p(x_{j})q(y_{j}) = \sum_{j = 1}^{n} p(\lambda x_{j})q(y_{j})\]

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 $z \in \Gamma(U \otimes V)$, 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,

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

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

\[z = \sum_{j = 1}^{n} (p(x_{j}) + \eps)(q(x_{j}) + \eps) \cdot \underbrace{\frac{x_{j}}{p(x_{j}) + \eps}}_{\in \bracs{p < 1} = U}\otimes \underbrace{\frac{y_{j}}{q(x_{j}) + \eps}}_{\in \bracs{q < 1} = V}\in \Gamma(U \otimes V)\]

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

\[\xymatrix{ E \times F \ar@{->}[rd]_{\phi \cdot \psi} \ar@{->}[r]^{\iota} & E \otimes_\pi F \ar@{->}[d]^{\Phi} \\ & K }\]

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 Definition 9.9.1.$\square$

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