Page 219 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 18.1.2 Definition: Reflexive Mapping

If the map J : N N** defined by
J (x) = F x N,
x
is onto also, then N (or J) is said to be reflexive (or reflexive mapping). In this case we write
N = N**, i.e., if N = N**, then N is reflexive.

Note Equality in the above definition is in the sense of isometric isomorphism under the
natural imbedding. Since N** must always be a complete normed linear space, no
incomplete space can be reflexive.

18.1.3 Properties of Natural Imbedding of N into N**

I. Let N be a normed linear space. If x N, then
x = sup {|f (x)|: f N* and f = 1}.

Using natural imbedding of N into N**, we have for every x N,
F (f) = f (x) and F = x .
x x
Now, Fx = sup {|F (f)|} sup {|f(x)|, f N*}
x
f 1 f 1
therefore, x = sup {|f (x)|: f N*, f = 1}.
II. Every normed linear space is a dense linear subspace of a Banach space.
Let N be a normed linear space. Let

J : N N** be the natural imbedding of N into N**.
The image of the mapping is linear subspace J (N) N**. Let J(N) be the closure of N(N)
in N**.

Since N** is a Banach space, its closed subspace J(N) is also a Banach space. Hence if we
identity N with J(N), then J(N) is a dense subspace of a Banach space.

18.1.4 Theorems and Solved Examples

Theorem 1: Let N be an arbitrary normal linear space. Then each vector x in N induces a functional
Fx on N* defined by
F (f) = f(x) for every f N* such that F = x .
x x
Further, the mapping J : N N** : J (x) = F for every x N defines and isometric isomorphisms
x
of N into N**.
Proof: To show that F is actually a function on N*, we must prove that F is linear and bounded
x x
(i.e. continuous).
We first show F is linear.
x

212 LOVELY PROFESSIONAL UNIVERSITY