Suppose X n converges weakly to X with respect to the topology of C[0, 1]. Prove that X n converges weakly to X with respect to the topology of D[0, 1]. This is the converse to Let A be an index set,...

Suppose X_n
converges weakly to X with respect to the topology of C[0, 1]. Prove that X_n
converges weakly to X with respect to the topology of D[0, 1].

This is the converse to Let A be an index set, and suppose the collection of functions

is precompact in D[0, 1], i.e., its closure is compact.

(1) Prove

(2) Prove

Chapter 35