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Linear functionals
Definition 1: a linear functional
f
is a linear operator with its domain in a vector spaceX
and its range in a scalar fieldF
defined inX
.
The norm can be a linear functional \|\cdot\|: X \to F
under the condition that the norm is linear. Otherwise, it would solely be a functional.
Definition 2: a bounded linear functional
f
is a bounded linear operator with its domain in a vector spaceX
and its range in a scalar fieldF
defined inX
.
Dual space
Definition 3: the set of linear functionals on a vector space
X
is defined as the algebraic dual spaceX^*
ofX
.
From this definition we have the following.
Theorem 1: the algebraic dual space
X^*
of a vector spaceX
is a vector space.
??? note "Proof:"
Will be added later.
Furthermore, a secondary type of dual space may be defined as follows.
Definition 4: the set of bounded linear functionals on a normed space
X
is defined as dual spaceX'
.
In this case, a rather interesting property of a dual space emerges.
Theorem 2: the dual space
X'
of a normed space(X,\|\cdot\|_X)
is a Banach space with its norm\|\cdot\|_{X'}
given by
|f|{X'} = \sup{x \in X\backslash {0}} \frac{|f(x)|}{|x|X} = \sup{\substack{x \in X \ |x|_X = 1}} |f(x)|,
for all
f \in X'
.
??? note "Proof:"
Will be added later.