Before stating the main results, let us fix the rest of our notation. We assume, unless stated otherwise, throughout this section that is a unital Banach algebra. Also if is a semiinner product module and is a seminorm on , we put , and if is a positive linear functional on , we put . Let be a finite set of orthogonal elements in such that be idempotent, we set and .
Dragomir in [8, Lemma 4] shows that in a Hilbert space , the condition
is equivalent to the condition
where and , . But for semiinner product modules we have the following lemma, which is a generalization of [7, Lemma 1].
Lemma 3.1.
Let be a semiinner product module and , . Then
if and only if
Proof.
Follows from the equalities:
Remark 3.2.
By making use of the previous Lemma 3.1, we may conclude the following statements.
(i)Let be an inner product module and let be a finite set of orthogonal elements in such that are idempotent, then inequality (3.3) implies that
(ii)Let be an inner product module and be a finite set of orthogonal elements in such that are idempotent. If is a seminorm on then inequality (3.3) implies that
and if is a positive linear functional on from inequality (3.3) and [2, Section 37 Lemma 6(iii)], we get
(iii)Let be a proper algebra, let be an inner product module, and let be a finite set of orthogonal elements in such that are idempotent. Since for every , inequality (3.3) is valid only if
We are able now to state our first main result.
Theorem 3.3.
Let be an inner product module and let be a finite set of orthogonal elements in such that are idempotent. If , , , are real numbers and such that
hold, then one has the inequality
Proof.
By [11, Lemma 2] or, a straightforward calculation shows that for every
Therefore
Analogously, for every , we have
The equalities (3.10), (3.13), and (3.14) imply that
Since
therefore the Schwarz's inequality (2.1) holds, that is,
Finally, using the elementary inequality for real numbers
on
we get
Remark 3.4.

(i)
Let be an inner product module and let be a finite set of orthogonal elements in such that are idempotent. If and are such that
and if we put , and , then, by (3.15) and (3.16), we have
These and (3.11) imply that
Therefore, (3.11) is a refinement and a simple formulation of [9, Theorem 4.1.].

(ii)
If for , we set
then similarly (3.11) is a refinement and a simple form of [9, Corollary 4.3].
Corollary 3.5.
Let be a Banach algebra, let be an inner product module, and let be a finite set of orthogonal elements in such that are idempotent. If , , are real numbers and such that
hold, then one has the inequality
Proof.
Using the schwarz's inequality (2.6), we have
The assumptions (3.26) and the elementary inequality for real numbers (3.19) will provide the desired result (3.27).
Example 3.6.
Let be a Hermitian Banach algebra and let be the Pták function on . If is a semiinner product module and with the properties that
then we have
That is interesting in its own right.
Corollary 3.7.
Let be a proper algebra, let be an inner product module, and let be a finite set of orthogonal elements in such that are idempotent. If , are real numbers and such that
hold, then one has the inequality
Proof.
Using the strong Schwarz's inequality (2.3), we have
The assumptions (3.31) and the elementary inequality for real numbers (3.19) will provide (3.32).
The following companion of the Grüss inequality for positive linear functionals holds.
Theorem 3.8.
Let be an inner product module, let be a positive linear functional on , and let be a finite set of orthogonal elements in such that are idempotent. If , , , are real numbers and such that
hold, then one has the inequality
Proof.
By taking on both sides of (3.12), we have
Analogously
Now, using Aczl's inequality for real numbers, that is, we recall that
and the Schwarz's inequality for positive linear functionals, that is,
we deduce (3.35).