ECE 580 / Math 587
SPRING 2011
Correspondence
# 13
March 28, 2011
Notes On
HAHNBANACH THEOREM
Extension Form
We start with a definition – that of a
sublinear functional
.
Definition.
Let
X
be a real vector space (not necessarily normed). A map
p
:
X
→
R
is called a
sublinear
functional
if it satisfies the following two properties:
p
(
x
+
y
)
≤
p
(
x
) +
p
(
y
)
∀
x, y
∈
X
(subadditive)
p
(
αx
) =
αp
(
x
)
∀
α
≥
0
,
∀
x
∈
X
(positive homogeneity)
Note
that “norm” is a sublinear functional.
The next result is a useful property of sublinear functionals.
Lemma.
Let
M
⊂
X
be a subspace, and
f
a linear functional on
M
, such that for some sublinear functional
p
,
f
(
x
)
≤
p
(
x
)
∀
x
∈
M.
Let
x
o
be a fixed element of
X
. Then, for any real number
c
, the following are equivalent :
f
(
x
) +
λc
≤
p
(
x
+
λx
o
)
∀
x
∈
M, λ
∈
R
(1)
−
p
(
−
x
−
x
o
)
−
f
(
x
)
≤
c
≤
p
(
x
+
x
o
)
−
f
(
x
)
∀
x
∈
M
(2)
Furthermore, there is a real number
c
satisfying
(2)
and hence
(1)
.
Proof :
To show that (1)
⇒
(2), first set
λ
= 1, and then set
λ
=
−
1 and replace
x
by
−
x
; then (2) is
immediate. To show the converse, first take
λ >
0, and replace
x
by
x/λ
on the RHS inequality, to obtain
(1). Now take
λ <
0, and replace
x
by
x/λ
on the LHS inequality, to arrive again at (1). For
λ
= 0, (1) is
always satisfied, from the definitions of
f
and
p
. To produce the desired
c
, let
x, y
be arbitrary elements out
of
M
. Then,
f
(
x
)
−
f
(
y
) =
f
(
x
−
y
)
≤
p
(
x
−
y
)
≤
p
(
x
+
x
o
) +
p
(
−
y
−
x
o
)
by subadditivity
⇒
−
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 Spring '08
 Staff
 Linear Algebra, Vector Space, Topological vector space, XO, normed linear space

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