习题 12, § 1

a) Show that the bilinear mapping $(g, \mu) \mapsto g \cdot \mu$ of $\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$ into $\mathscr{M}(X; \mathbf{C})$ is continuous when $\mathscr{C}(X; \mathbf{C})$ is equipped with the topology of compact convergence and $\mathscr{M}(X; \mathbf{C})$ with the quasi-strong topology (Exer. 8) or with the topology of strictly compact convergence.

b) 证明当$\mathscr{C}(X; \mathbf{C})$配备紧致收敛的拓扑和$\mathscr{M}(X; \mathbf{C})$配备模糊拓扑时，$\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$到$\mathscr{M}(X; \mathbf{C})$的双线性映射$(g, \mu) \mapsto g \cdot \mu$是$(\mathscr{S}, \mathscr{T})$次连续的，其中$\mathscr{S}$表示$\mathscr{C}(X; \mathbf{C})$有限子集集合和模糊边界$\mathscr{M}(X; \mathbf{C})$子集$\mathscr{T}$集合。

c) 证明当$\mathscr{C}(X; \mathbf{C})$配备紧致收敛的拓扑，$\mathscr{M}(X; \mathbf{C})$和$\mathscr{M}_+(X)$配备模糊拓扑时，$\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}_+(X)$到$\mathscr{M}(X; \mathbf{C})$的映射$(g, \mu) \mapsto g \cdot \mu$是连续的。

d) 给出紧致空间$X$的例，使得当$\mathscr{C}(X; \mathbf{C})$配备一致收敛的拓扑和模糊的$\mathscr{M}(X; \mathbf{C})$时，$\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$到$\mathscr{M}(X; \mathbf{C})$的双线性映射$(g, \mu) \mapsto g \cdot \mu$不连续拓扑。