Dual Spaces

Question 1

T is a Linear Operator

(or transformation)

Answer 1

T(av + bw) = aT(v) + b T(w)

Question 2

Operator Norm ||T||

of a bounded linear operator T

Answer 2

sup{ || T(x) || : ||x|| =1}

Question 3

bounded linear operator T

is equivalent to

Answer 3

T continous

T continuous at 0

Ker is a closed set

Question 4

If dim V < ∞, then every

linear operator is…

Answer 4

bounded

Question 5

What is a Banach space?

Answer 5

a complete normed linear space

Question 6

What is a dual space on a vector space V?

(denoted V*)

Answer 6

V*

is the collection of continuous linear

transformations from V* into R

(V* forms a vector space)

Question 7

What is a Hilbert Space?

Answer 7

Hilbert Space is a Banach space whose norm

is an inner product

(Banach space is a complete n.l.s.)

Question 8

a measurable function (X –> R)

is essentially bounded​ if…

Answer 8

f(x) | ≤ K a.e.

Question 9

||f||_∞ ​is ….

Answer 9

the smallest essential bound for f

Question 10

L^∞​ is…

Answer 10

the set of measurable and essentially

bonded functions

Question 11

If X is sigma-finite

(X is the countable union of measurable sets with finite measure)

and

F is in L¹* then…

Answer 11

there exists g in L^∞ such that F(f) = ∫ fg

for all f in L ¹​

Question 12

|| ||sup = || ||∞

when?

Answer 12

for continous f on a connected set K

(in Rⁿ​)

Question 13

uniform limit of continuous functions is….

Answer 13

continuous

Question 14

What is C_b(K) ?

for K compact

Answer 14

set of continuous and bounded functions

K –> R

Question 15

C_b(K) is …

Answer 15

a complete n.l.s. with || || sup