ECE500/600
ENGINEERING ANALYTICAL
TECHNIQUES
Vector Spaces: Metric Spaces
HW 25 0129
I. METRIC SPACES: QUESTION
[Moon and Stirling, 2000] Let X be an arbitrary set. Show that the function defined by
is a metric.
II. METRIC SPACES: QUESTION
[Moon and Stirling, 2000] Let (X, d) be a metric space. Show that
is a metric on X. What significant feature does this metric possess?
III. METRIC SPACES: QUESTION
[Moon and Stirling, 2000] In defining the metric of the sequence space ℓ∞ (0, ∞) as
d∞ (x,y) = sup|x(n) - y(n)| , n
“sup” is used instead of “max” . To see the necessity of this definition, define the sequences {x(n)} and {y(n)} by
Show that d∞ (x,y) > |x(n) - y(n)| , yn ≥ 1.
IV. METRIC SPACES: QUESTION
[Moon and Stirling, 2000] Let
(a) Draw the set B.
(b) Determine the boundary of B.
(c) Determine the interior of B.
V. METRIC SPACES: QUESTION
[Moon and Stirling, 2000] The fact that a sequence is Cauchy depends upon the metric employed. Let fn (t) be the sequence of functions
in the metric space (C[a,b], d∞), where
Show that,
Hence, conclude that in this metric space, fn is not a Cauchy sequence.