Measure, Integration & Real Analysis · Problem 6A.16

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · metric-space-completion

Problem

Suppose

(U, d)

is a metric space. Let

W

denote the set of all Cauchy sequences of elements of

U

. (a) For

(f_1, f_2, \ldots)

and

(g_1, g_2, \ldots)

W

, define

(f_1, f_2, \ldots) \equiv (g_1, g_2, \ldots)

to mean that

\lim_{k \to \infty} d(f_k, g_k) = 0.

Show that

\equiv

is an equivalence relation on

W

. (b) Let

V

denote the set of equivalence classes of elements of

W

under the equivalence relation above. For

(f_1, f_2, \ldots) \in W

, let

(f_1, f_2, \ldots)\,\hat{}\,

denote the equivalence class of

(f_1, f_2, \ldots)

. Define

d_V : V \times V \to [0, \infty)

d_V\bigl((f_1, f_2, \ldots)\,\hat{}\,,\, (g_1, g_2, \ldots)\,\hat{}\,\bigr) = \lim_{k \to \infty} d(f_k, g_k).

Show that this definition of

d_V

makes sense and that

d_V

is a metric on

V

. (c) Show that

(V, d_V)

is a complete metric space. (d) Show that the map from

U

V

that takes

f \in U

(f, f, f, \ldots)\,\hat{}\,

preserves distances, meaning that

d(f, g) = d_V\bigl((f, f, f, \ldots)\,\hat{}\,,\, (g, g, g, \ldots)\,\hat{}\,\bigr)

for all

f, g \in U

. (e) Explain why (d) shows that every metric space is a subset of some complete metric space.