Q.5 Flashcards
1
Q
PARTIAL ORDER on X
A
⪯ if
1. x⪯x, ∀x∈X
2. x⪯y and y⪯x ⟹ x=y
3. x⪯y and y⪯z ⟹ x ⪯ z
X nonempty set
2
Q
TOTAL ORDER
A
partial order ⪯ where x ⪯ y or y ⪯ x, ∀x,y∈X
3
Q
UPPER BOUND for V⊆X
A
y∈X
if x⪯y, ∀x∈X
- ⪯ partial order on X
4
Q
MAXIMAL ELEMENT of X
A
y∈X
if y⪯x ⟹ y=x
5
Q
ZORN’S LEMMA
A
let X ≠ ∅ be partially ordered
if every totally ordered subset of X has an upper bound in X, then X has a maximal element
6
Q
HAHN-BANACH THEOREM
A
ASSUME
1. X is linear space
2. V ⊆ X is a proper linear subspace
3. p:X->[0, ∞) is a semi-norm
4. f∈L(V,K) satisfies |f(x)|<=p(x), ∀x∈V
THEN
there exists F∈L(X,K) such that
F|V = f and |F(x)|<=p(x), ∀x∈X
7
Q
HAHN-BANACH THEOREM (2)
A
if X NLS and V ⊆ X is a linear subspace, then for all f∈V’ there exists F∈X’ such that F|V=f and ||F||=||f||
