HW 4

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mth 266, homework 4
due february 20, 2012
problem 1:
Let V be Hilbert and W a subspace. Suppose : W C is a linear functional
which satisfies ||(v)|| C||v|| for any v W . Show that there is a unique extension of to V
which satisfies the same bound on V .
problem 2:
Suppose V is Banach, A, B B(V, V ). Show that ||AB|| ||A|| ||B||. Is it true
that ||AB|| = ||A|| ||B||?
problem 3:
Suppose V1, V2 are Banach, A B(V1, V2). Show that
||A|| = sup ||Au|| = sup ||Au||
||u||=1
||u||1
problem 4:
Suppose V is Hilbert, A B(V, V ). Show that
||A|| =
sup
< Au, v >
||u||=||v||=1
problem 5:
Suppose V is Hilbert, A B(V, V ). Show that ||A+|| = ||A||.
problem 6:
Suppose V is Hilbert, B : V x V C has the properties: i) B(u + v, w) =
B(u, w) + B(v, w); ii) B(w, u + v) = B(w, u) + B(w, v); |B(u, v)| C||u|| ||v||; for
any , C, u, v, w V . Show that there exists a bounded operator A on V such that
B(u, v) =< u, Av >.
problem 7:
Let A : l2 l2 be defined by A(a1, a2, . . .) = (0, a1, a2, . . .). Find ||A|| and A+. Is
A unitary?
problem 8:
Let A : l2 l2 be defined by A(a1, a2, a3, a4, . . .) = (a2, a1, a4, a3, . . .). Find ||A||
and A+. Is A unitary?
problem 9:
We have seen in class that L2([0, 1]) L1([0, 1]). Is it true that l2 l1?
problem 10:
If a l2 and
k2|a
k
k |2, show that a l1.
1