习题 11, § 1

a) Show that when $\mathscr{M}(X; \mathbf{C})$ is equipped with the quasi-strong topology (resp. the topology of strictly compact convergence, resp. the vague topology), the bilinear form $(f, \mu) \mapsto \langle f, \mu \rangle$ is $(\mathscr{S}, \mathscr{T})$-hypocontinuous, where $\mathscr{T}$ is the set of vaguely bounded subsets of $\mathscr{M}(X; \mathbf{C})$, and $\mathscr{S}$ is the set of bounded subsets of $\mathscr{H}(X; \mathbf{C})$ that are contained in $\mathscr{H}(X, K; \mathbf{C})$ for a suitable compact set $K$ (resp. the set of strictly compact subsets of $\mathscr{H}(X; \mathbf{C})$, resp. the set of finite subsets of $\mathscr{H}(X; \mathbf{C})$) (cf. TVS, III, §5, Exer. 12).

b) 对于$X$的每个紧致子集$K$，证明当$\mathscr{M}(X; \mathbf{C})$配备准强拓扑时，双线性形式$(f, \mu) \mapsto \langle f, \mu \rangle$在$\mathscr{H}(X, K; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$上连续；当$\mathscr{M}_+(X)$配备模糊拓扑时，它在$\mathscr{H}(X, K; \mathbf{C}) \times \mathscr{M}_+(X)$上是连续的（利用Exer.10）。

c) 假设$X$是仿紧且非紧的。证明当$\mathscr{M}_+(X)$配备强拓扑（在这种情况下与准强拓扑相同）时，双线性形式$(f, \mu) \mapsto \langle f, \mu \rangle$在$\mathscr{H}(X; \mathbf{C}) \times \mathscr{M}_+(X)$上不连续。

d) 给出例，其中$X$是紧致的，而双线性形式$(f, \mu) \mapsto \langle f, \mu \rangle$在$\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$上不连续，当$\mathscr{M}(X; \mathbf{C})$配备紧致收敛的拓扑时（TVS，IV, §1，Exer.11）。