Proposition 27.2.4.label Let $G$ be a locally compact group and $f, h, g \in C_{c}^{+}(G)$, then:

1. (1)

   If $g \ne 0$, then $(f: g) < \infty$.

2. (2)

   $(f, h: g) \le (h: g) + (h: g)$.

3. (3)

   For each $\lambda \ge 0$, $(\lambda f: g) = \lambda(f: g)$.

4. (4)

   If $f \le h$, then $(f: g) \le (h: g)$.

5. (5)

   $(f: g) \le (f: h)(h: g)$.

6. (6)

   $(f: g) \ge \norm{f}_{u}/\norm{g}_{h}$.

7. (7)

   For each $x \in G$, $(L_{x}f: g) = (f: g)$.