Page 209 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Lemma: Let M be a linear subspace of a normed linear space N let f be a functional defined on M.
If x N such that x M and if M = M + [x ] is the linear subspace of N spanned by M and x , then
o o o o o
f can be extended to a functional f defined on M s.t.
o o
f = f .
o
Proof: We first prove the following lemma which constitutes the most difficult part of this
theorem.
Lemma: Let M be a linear subspace of a normed linear space N let f be a functional defined on M.
If x N such that x M and if M = M + [x ] is the linear subspace of N spanned by M and x , then
o o o o o
f can be extended to a functional f defined on M s.t.
o o
f = f .
o
Proof: The lemma is obvious if f = o. Let then f 0.

Case I: Let N be a real normed linear space.
Since x M, each vector y in M is uniquely represented as
o o
y = x + x , x M and R.
o
This enables us to define

f : M R by
o o
f (y) = f (x + x ) = f (x) + r ,
o o o o
where r is any given real number … (1)
o
We show that for every choice of the real number r , f is not only linear on M but it also extends
o o
f from M to M and
o
f = f .
o
Let x , y M . Then these exists x and y M and real scalars and such that
1 1 o
x = x + x and y = y + x ,
1 o 1 o
Hence, f (x + y ) = f (x + x + y + x )
o 1 1 o o o
= f (x + y + ( + ) x )
o o
= f (x + y) + ( + ) r , r is a real scalar … (2)
o o
Since f is linear M, f (x + y) = f (x + y) … (3)
From (2) and (3) it follows after simplification that
f (x + y ) = f (x) + r + f (y) + r
o 1 1 o o
= f (x + x ) + f (y + x )
o o o o
= f (x ) + f (y )
o 1 o 1
f (x + y ) = f (x ) + f (y ) … (4)
o 1 1 o 1 o 1
Let k be any scalar. Then if y M , we have
o
f (ky) = f [k (x + x )]
o o o
= f (k + k x )
o x o
But f (k + k x ) = f (kx) + k r
o x o o
= k f (x) + k r
o

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