Function Spaces

Question 1

a norm on a vector space

is a function || • || ….

Answer 1

1) positive definite
2) scalar: a || v || = || av ||
3) triangle

Question 2

normed linear space is a…

Answer 2

linear space (aka vector space)

with a norm

a n.l.s. implies a metric space

(metric does not imply n.l.s!)

Question 3

“sup norm”

Answer 3

is a norm on the set of bounded functions

f: X –> R

|| f ||_sup = sup over x of | f(x) |

Question 4

|| • ||₁ is equivalent to || • ||₂

if ….

Answer 4

|| v ||₁ ≤ K | v ||2

and

|| v ||₂ ≤ L | v ||1

(for all v)

Question 5

When is L^p a normed linear space?

Answer 5

when p ≥ 1

theorem called

“Minkowski”

Question 6

When are all norms on a vector space equivalent?

Answer 6

when the vector space is

finite dimensional

Question 7

a linear operator

(or transformation) is…

Answer 7

a function from one vector space to another

such that

“splits over +”

T(v + w) = T(v) + T(w)

and

scalar multiplication

T(av) = a T(v)

Question 8

When can we conclude

fg in L¹?

Answer 8

if |fg| ≤ some integrable function

Question 9

If 1 ≤ p ≤ ∞ and f, g in L^p

then

what can we say about

|| f + g ||_p

Answer 9

|| f + g ||_p ≤ || f ||_p + || g ||_p

Question 10

Holder’s Inequality says…

Answer 10

If 1 < p, q < ∞ with 1/p + 1/q = 1

and

f in L^p and g in L^q then

fg is in L¹ and

∫ |fg| ≤ || f ||_p || g ||_q

Question 11

“Risz-Fischer” says what about

L^p for p ≥ 1

Answer 11

that L^P is a complete space

(complete means every Cauchy seq converges)

Question 12

f: R –> R

has compact support if

Answer 12

the closure of {x | f(x) ≠ 0}

is compact

“f ≠ 0 on a compact set (including boundary)”

Question 13

To construct a function in

L^p but not in L^q …

Answer 13

consider 1/x^a

if ap < 1 then in L^p

Question 14

What can we say about L^p and L^q in a

finite measure space

Answer 14

If q ≥ p ≥ 1 then

L^q is contained in L^p

Question 15

What is the closure of a set A?

Answer 15

intersection of all closed sets containing A

“smallest closed set containing A”

or

in a metric space:

A and

set of all limits of convergent sequences

“A and limit points of A”