Unable to understand an application of Minkowski's inequality

Question

Consider the following exerpt from the paper "Non-linear Quantum Processes" by Segal:

with the norm $\|F\|=\left(\int\|F(x)\|^p \, d x\right)^{1 / p}$, then the operator $T_1^{\prime}: F \rightarrow G$, where $G(x)=\int K_1(x, y) F(y) \, d y$, exists and is a contraction from $L_p\left(M_1, \mathbf{B}\right)$ to $L_q\left(M_1, \mathbf{B}\right)$. For the mapping $y \rightarrow K_1(x, y) F(y)$ is easily seen to be strongly measurable from $M_1$ to $\mathbf{B}$, for each $x$; and $$ \|G(x)\| \leqq \int K_1(x, y)\|F(y)\| \, d y, $$ i.e., $\|G(\cdot)\|=T_1(\|F(\cdot)\|)$, so that ||$G(\cdot)\left\|_{\mathbf{B}}\right\|_q \leqq\|\| F(\cdot)\left\|_{\mathbf{B}}\right\|_p$.

I'm unable to understand the equation $\|G(x)\| \leqq \int K_1(x, y)\|F(y)\|\, d y$. What does the norm in $\|F(y)\|$ mean? Are we just taking the absolute value $|F(y)|$? I remember being told that this inequality concerns Minkowski's inequality, but I'm unable to see this from the notation. Minkowski's inequality should have implied $$\|G(x)\|_p\leq \int \|K_1(x,y)F(y)\|_p\, dy$$ Any help with this would be great.

Answer 1

$\newcommand\B{\mathbf B}$The function $F$ takes values in a separable Banach space $\B$, and $\|F(y)\|$ is the $\B$-norm of $F(y)$. We have $$G(x)=\int K_1(x,y)F(y)dy,$$ where $K_1$ is positive. So, for any $b^*$ in the dual space $\B^*$ with $\|b^*\|\le1$, we have
$$b^*(G(x))=\int K_1(x,y)b^*(F(y))dy\le\int K_1(x,y)\|F(y)\|dy.$$ Thus, $$\|G(x)\|\le\int K_1(x,y)\|F(y)\|dy. \quad\Box$$