# 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

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