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Let us set and, then by using Definition 2.2, Lemmas 2.4, 2.5, and (3.2), we can have (3.4).
Using Definition 2.2 we have.
Using definition of paranorm we get.
Hence, using definition (1.4), we know.
Using definition of we may write (1.11).
Using Definition 2.2, we derive that.
Using Definition 3.1-3.4 3.1-3.4llowing lemma is immediathe
Using Definition 2.2, we have are bounded for all.
By using Definition 2.1, it is easy to have the following property of f.
By using Definition 2.3, it is easy to get that (2.4) is equivalent to (2.5).
Next, we will prove Theorem 2.4 by using Definition 1.1 and Theorem 2.1.
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