A linear functional on a vector space 
over a field 
is a linear transformation 
. It satisfies the following
properties for all 
and 
(Axler 2026, p. 105).
1. 
,
and
2. 
.
For a real vector space, 
; for a complex vector
space, 
.
All linear functionals on 
form its dual vector space.
On a real or complex normed space, a linear functional is continuous iff it is bounded, meaning that 
for some constant 
and every 
(Schenker 2008, p. 6-1). The continuous
linear functionals form the dual normed space.
Linearity alone does not imply continuity. For example,
on the vector space of real polynomials
with the supremum norm on ![[0,1]](/images/equations/LinearFunctional/Inline14.svg)
, the linear functional 
is unbounded, since 
and 
for every positive
integer 
.
Generalized functions are continuous linear functionals on suitable function spaces of test functions.
