Test 1: Take Home
1. Let X denote the set of all irrational numbers x with √
2 ≤ x ≤ 2√2, and
with the usual metric d(x, y) = |x − y|. Prove that X is not compact.
2. Let (X, d) denote any metric space. The metric space X is called “totally
bounded” when, for every > 0, there exists finitely many neighborhoods
N(xi) (i = 1, . . . n) such that X ⊆ ∪n
i=1N(xi). The metric space is
“bounded” when { d(x, y) | x, y ∈ R } is a bounded subset of R.
(a) Give an example of a bounded metric space that is not totally bounded.
(b) Prove that every totally bounded metric space is bounded
(c) Prove that a metric space is compact if and only if it is both complete
and totally bounded.
3. Let R
n denote the usual n-dimensional Euclidean space, with its Euclidean
norm
||x|| =vuutXni=1|xi|2
and corresponding metric d(x, y) = ||x − y||, with x, y ∈ R
n. Given an
n × n matrix T, define
||T|| ≡ sup { ||T x|| | ||x|| ≤ 1 } .
(a) Prove that, for all n × n matrices X and Y , that ||XY || ≤ ||X||||Y||.
(b) Prove that
||T|| = inf { M ∈ R | ||T x|| ≤ M||x|| for all x ∈ Rn}.
(c) With x ∈ R
n, find ||Cx|| when Cx is the n × n matrix with the
coordinates of x in the first column and zeros elsewhere.
(d) With x ∈ R
n, find ||Dx|| when Dx is the n × n diagonal matrix with
the coordinates of x on the main diagonal, and zeros elsewhere.
(e) With x ∈ R
n, find ||Rx|| when Rx is the n × n matrix with the
coordinates of x in the first row and zeros elsewhere.
4. Let T be an n × n matrix, with ||T|| defined as in the previous problem.
Prove that sup { |α| | α an eigenvalue of T }1