Let
E
E
be an arbitrary set and
F
(
E
)
\mathcal {F}(E)
a linear space composed of all complex valued functions on
E
E
. Let
H
\mathcal {H}
be a (possibly finite-dimensional) Hilbert space with inner product
(
,
)
H
{(,)_\mathcal {H}}
. Let
h
:
E
→
H
{\mathbf {h}}:E \to \mathcal {H}
be a function and consider the linear mapping
L
L
from
H
\mathcal {H}
into
F
(
E
)
\mathcal {F}(E)
defined by
(
F
,
h
(
p
)
)
H
{({\mathbf {F}},{\mathbf {h}}(p))_\mathcal {H}}
. We let
H
~
\tilde {\mathcal {H}}
denote the range of
L
L
. Then we assert that
H
~
\tilde {\mathcal {H}}
becomes a Hilbert space with a reproducing kernel composed of functions on
E
E
, and, moreover, it is uniquely determined by the mapping
L
L
, in a sense. Furthermore, we investigate several fundamental properties for the mapping
L
L
and its inverse.