1. hw 10
(1) Compute the operator norm of the matrix A =1 1 0 1
.
(2) Let 1 ≤ p < ∞. In class, we defined the operator norm of a linear transformation T : Rn→ Rm by
kT kop= max
kxkRn=1kT xkRm
Here kxkRn is the Euclidean norm of x. In fact, we can define the operator norm of T using other norms on Rn not only the Euclidean norm of x. For each x = (x1, · · · , xn) ∈ Rn, we recall that the p-norm of x is defined to be
kxkp =
p
X
i=1
|xi|p
!1/p
. We define the operator p-norm of T to be
kT kop,p = max
kxkp=1
kT xkp.
Hence the usual operator norm of T is in fact the operator 2-norm: kT kop= kT kop,2. Similarly, the infinity norm of x is defined to be
kxk∞= max{|x1|, · · · , |xn|}.
The operator ∞-norm of T is defined to be kT kop,∞= max
kxk∞=1kT xk∞. Consider T = LA where A is an m × n matrix A = [aij].
(a) Let ai be the i-th column vector of A. Show that kT kop,1 = max{ka1k1, · · · , kank1}.
(b) Let bj be the j-th row vector of A. Show that kT kop,∞= max{kb1k1, · · · , kbmk1}.
(c) Define the p-norm of T by kT kp =
m
X
i=1 n
X
j=1
|aij|p
1/p
and the ∞-norm of T by
kT k∞= max{|aij| : 1 ≤ i ≤ m, 1 ≤ j ≤ n}.
Is kT kp = kT kop,p? or kT k∞ = kT kop,∞? Find kT kp and kT kop,p for p = 1, 2 and kT |∞and kT kop,∞ for T = LA where A is the 2 × 2 matrix in Exercise (1).
(d) Can you determine kT kop,p?
(3) Let k · k1 and k · k2 be norms on a real vector space V such that there exists C > 0, kvk1≤ Ckvk2
for any v ∈ V.
(a) Show that if {vn} is a Cauchy sequence in (V, k·k2), then it is a Cauchy sequence in (V, k · k1).
(b) Show that if {vn} is convergent to v in (V, k · k2), then it is convergent to v in (V, k · k1).
1
2
(c) Let > 0 and v ∈ V. Let Bi(v, ) be the subset of V consisting of w ∈ V so that kw − vki< for any i = 1, 2. Prove that B2(v, ) ⊆ B1(v, C) for any v ∈ V.
(d) Let U be a subset of V. Prove that if U is open in (V, k · k1), then it is open in (V, k · k2).
(4) Let w : [0, 1] → R be a positive continuous function. On C[0, 1], we define kf kw =
Z 1 0
|f (x)|w(x)dx.
(a) Prove that k · kw is a norm on C[0, 1]. (Hint: consider the hw 9 (11) in Calculus I).
(b) Prove that there exists C > 0 so that
kf kw ≤ Ckf k∞ for any f ∈ C[0, 1].
(c) Is k · kw equivalent to k · k∞?1 If not, find a sequence {fn} in C[0, 1] such that {fn} is convergent in k · kw but not in k · k∞.
(d) Prove or disprove that k · kw is equivalent to k · k1? Here kf k1=
Z 1
0
|f (x)|dx.
Hint: show that the maximum and minimum of w are both positive real num- bers.
(5) Let (K, d) be a sequentially compact metric space and (V, k · kV) be a real normed space. Let C(K, V ) be the space of all continuous maps from K into V. Then C(K, V ) forms a real vector space.
(a) Let f ∈ C(K, V ). Show that f is bounded, i.e. there exists M > 0 such that kf (x)kV ≤ M for any x ∈ K.
Hint: show that the function K → R defined by x 7→ kf (x)kV is continuous on K.
(b) Using (a), we define
kf kC(K,V )= sup
x∈K
kf (x)kV. Show that k · kC(K,V ) defines a norm on C(K, V ).
(c) Prove that (C(K, V ), k · kC(K,V )) is a real Banach space if (V, k · k) is a real Banach space. Hint: let {fn} be a Cauchy sequence in (C(K, V ), k · kC(K,V )).
Then for any x ∈ K,
kfn(x) − fm(x)kV ≤ kfn− fmkC(K,V ) for any n, m ≥ 1. Use completeness of (V, k · kV) to obtain
f (x) = lim
n→∞fn(x) in V .
1On a normed vector space V, two norms k · k1and k · k2are said to be equivalent if there exist C, c > 0 so that ckvk1≤ kvk2≤ Ckvk1
for any v ∈ V.
3
(d) Take V = Mmn(R) and k · kV to be the operator norm on Mmn(R). A function f : [0, 1] → Mmn(R) is a matrix valued function
f (x) =
f11(x) · · · f1n(x) ... . .. ... fm1(x) · · · fmn(x)
for x ∈ [0, 1].
Prove that f : [0, 1] → Mmn(R) is continuous if and only if fij : [0, 1] → R is continuous for any i, j. Hence C([0, 1], Mmn(R)) is the space of all matrix valued continuous functions on [0, 1].
(e) Let f ∈ C([0, 1], Mmn(R)). We define
kf kb = max{kfijk∞: 1 ≤ i ≤ m, 1 ≤ j ≤ n}.
Prove that kf kb is a norm on C([0, 1], Mmn(R)). Prove or disprove that kf kb is equivalent to kf kC([0,1],Mmn(R)).
(f) Let (V, h·, ·iV) be a real inner product space. For f, g ∈ C([0, 1], V ), we define hf, giC([0,1],V )=
Z 1 0
hf (x), g(x)iVdx.
Prove that h·, ·iC([0,1],V ) defines an inner product on C([0, 1], V ).
(g) Let f, g ∈ C([0, 1], Mmn(R)). We define hf, gi =
Z 1 0
Tr(f (x)tg(x))dx.
Prove or disprove that h·, ·i defines an inner product on C([0, 1], Mmn(R)). Here f (x)t means that the transpose of the matrix f (x).
