Since the proofs of the following theorems are not hard we omit them.
Theorem 3.1.
The sequence spaces
are linear spaces over the complex field
.
Theorem 3.2.
The space
is a linear topological space paranormed by
In what follows, we will show inclusion theorems between spaces
.
Theorem 3.3.
if and only if
Proof.
Let
and
. Then we get
hence
. Conversely, let us suppose that
and
. Then there exists a
such that
for every
. Suppose that
, then there exists a sequence of natural numbers
such that
. Hence we can write
Therefore,
, which is contradiction.
Corollary 3.4.
Let
be an Orlicz function. Then
if and only if
Theorem 3.5.
Let
,
,
be Orlicz functions which satisfy the
-condition and
, and
seminorms. Then
(1)
,
(2)
,
(3)
,
(4)If
is stronger than
, then
, and
(5)If
is equivalent to
, then
.
Proof.
Proof is similar to [14, Theorem 2.5].
Corollary 3.6.
Let
be an Orlicz function which satisfy the
-condition. Then
.
Theorem 3.7.
Let
be a sequence of Orlicz functions. Then the sequence space
is solid and monotone.
Proof.
Let
, then there exists
such that
for every
. Let
be a sequence of scalars with
for all
. Then from properties of Orlicz functions and seminorm, we get
which proves that
is solid space and monotone.