Download Improvement of Bessel's Inequality in Inner Product Spaces and more Study notes Literature in PDF only on Docsity!
A NOTE ON BESSEL’S INEQUALITY
S.S. DRAGOMIR
Abstract. A monotonicity property of Bessel’s inequality in inner product
spaces is given.
- Introduction
Let X be a linear space over the real or complex number field K. A mapping
(·, ·) : X × X → K is said to be a positive hermitian form if the following conditions
are satisfied:
(i) (αx + βy, z) = α (x, z) + β (y, z) for all x, y, z ∈ X and α, β ∈ K;
(ii) (y, x) = (x, y) for all x, y ∈ X;
(iii) (x, x) ≥ 0 for all x ∈ X.
If ‖x‖ := (x, x)
1 (^2) , x ∈ X denotes the semi-norm associated to this form and
(ei) i∈I is an orthornormal family of vectors in X, i.e., (ei, ej ) = δij (i, j ∈ I), then
one has the following inequality [15]:
(1.1) ‖x‖
2 ≥
i∈I
|(x, ei)|
2 for all x ∈ X,
which is well known in the literature as Bessel’s inequality.
Indeed, for every finite part H of I, one has:
x −
i∈H
(x, ei) ei
2
x −
i∈H
(x, ei) ei, x −
j∈H
(x, ej ) ej
= ‖x‖
2 −
i∈H
|(x, ei)|
2 −
j∈H
|(x, ej )|
2
i,j∈H
(x, ei) (ej , x) δij
= ‖x‖
2 −
i∈H
|(x, ei)|
2 ,
for all x ∈ X, which proves the assertion.
The main aim of this paper is to improve this result as follows.
- Results
The following theorem holds.
Theorem 1. Let X be a linear space and (·, ·) 2
1 two hermitian forms on
X such that ‖·‖ 2 is greater than or equal to ‖·‖ 1 , i.e., ‖x‖ 2 ≥ ‖x‖ 1 for all x ∈
X. Assume that (ei) i∈I is an orthornormal family in (X; (·, ·) 2 ) and (fi) i∈J is an
Date: January 05, 2000.
1991 Mathematics Subject Classification. Primary 26D15; Secondary 46C99.
Key words and phrases. Bessel’s inequality, Inner product spaces.
1
2 S.S. DRAGOMIR
orthornormal family in (X; (·, ·) 1 ) such that for any i ∈ I there exists a finite K ⊂ J
so that
(F) ei =
j∈K
αj fj , αj ∈ K (j ∈ K) ,
then one has the inequality:
(2.1) ‖x‖
2 2
i∈I
|(x, ei) 2
2 ≥ ‖x‖
2 1
j∈J
(x, fj ) 1
2 ≥ 0 ,
for all x ∈ X.
In order to prove this fact, we require the following lemma.
Lemma 1. Let X be a linear space endowed with a positive hermitian form (·, ·)
and (gk) k=1,n be an orthornormal family in (X; (·, ·)). Then
x −
n ∑
k=
λkgk
2
≥ ‖x‖
2 −
n ∑
k=
|(x, gk)|
2 ≥ 0 ,
for all λk ∈ K and x ∈ X (k = 1, ..., n).
Proof. We will prove this fact by induction over “n”.
Suppose n = 1. Then we must prove that
‖x − λ 1 g 1 ‖
2 ≥ ‖x‖
2 − |(x, g 1 )|
2 , x ∈ X, λ 1 ∈ K.
A simple computation shows that the above inequality is equivalent with
|λ 1 |
2 − 2 Re (x, λ 1 g 1 ) + |(x, g 1 )|
2 ≥ 0 , x ∈ X, λ 1 ∈ K.
Since Re (x, λ 1 g 1 ) ≤ |(x, λ 1 g 1 )|, one has
|λ 1 |
2 − 2 Re (x, λ 1 g 1 ) + |(x, g 1 )|
2 ≥ |λ 1 |
2 − 2 |λ 1 | |(x, g 1 )| + |(x, g 1 )|
2
≥ (|λ 1 | − |(x, g 1 )|)
2 ≥ 0
for all λ 1 ∈ K and x ∈ X, which proves the statement.
Now, assume that (2.2) is valid for “(n − 1)”. Then we have:
x −
n ∑
k=
λkgk
2
(x − λngn) −
n− 1 ∑
k=
λkgk
≥ ‖x − λngn‖
2 −
n− 1 ∑
k=
|(x − λngn, gk)|
2
= ‖x − λngn‖
2 −
n− 1 ∑
k=
|(x, gk)|
2 ≥ ‖x‖
2 − |(x, gn)|
2 −
n− 1 ∑
k=
|(x, gk)|
2
= ‖x‖
2 −
n ∑
k=
|(x, gk)|
2 ,
for all λk ∈ K, x ∈ X (k = 1, ..., n) , and the proof of the lemma is complete.
4 S.S. DRAGOMIR
and since ‖y‖ 2 ≥ ‖y‖ 1 , the inequality (2.3) is obtained.
Remark 1. For a different proof of (2.3), see also [5].
Now, we will give some natural applications of the above theorem.
- Applications
(1) Let (X; (·, ·)) be an inner product space and (ei) i∈I an orthornormal family
in X. Assume that A : X → X is a linear operator such that ‖Ax‖ ≤ ‖x‖
for all x ∈ X and (Aei, Aej ) = δij for all i, j ∈ I. Then one has the
inequality
‖x‖
2 −
i∈I
|(x, ei)|
2 ≥ ‖Ax‖
2 −
i∈I
|(Ax, Aei)|
2 ≥ 0
for all x ∈ X.
The proof follows by the hermitian forms (x, y) 2 = (x, y) and (x, y) 1
(Ax, Ay) for x, y ∈ X and for the family (fi) i∈I = (ei) i∈I
(2) If A : X → X is such that ‖Ax‖ ≥ ‖x‖ for all x ∈ X, then, with the
previous assumptions, we also have
0 ≤ ‖x‖
2 −
i∈I
|(x, ei)|
2 ≤ ‖Ax‖
2 −
i∈I
|(Ax, Aei)|
2 ,
for all x ∈ X.
(3) Suppose that A : X → X is a symmetric positive definite operator with
(Ax, x) ≥ ‖x‖
2 for all x ∈ X. If (ei) i∈I is an orthornormal family in X
such that (Aei, Aej ) = δij for all i, j ∈ I, then one has the inequality
0 ≤ ‖x‖
2 −
i∈I
|(x, ei)|
2 ≤ (Ax, x) −
i∈I
|(Ax, ei)|
2 ,
for all x ∈ X.
The proof follows from the above theorem for the choices (x, y) 1 = (Ax, y)
and (x, y) 2 = (x, y) , x, y ∈ X. We omit the details.
For other inequalities in inner product spaces, see the papers [1]-[14] and [7]-[6]
where further references are given.
References
[1] S.S. DRAGOMIR, A refinement of Cauchy-Schwartz inequality, G.M. Metod. (Bucharest),
8 (1987), 94-95.
[2] S.S. DRAGOMIR, Some refinements of Cauchy-Schwartz inequality, ibid, 10 (1989), 93-95.
[3] S.S. DRAGOMIR and B. MOND, On the Boas-Bellman generalisation of Bessel’s inequality
in inner product spaces, Italian J. of Pure and Appl. Math., 3 (1998), 29-38.
[4] S.S. DRAGOMIR and B. MOND, On the superadditivity and monotonicity of Gram’s in-
equality and related results, Acta Math. Hungarica, 71 (1-2) (1996), 75-90.
[5] S.S. DRAGOMIR and B. MOND, On the superadditivity and monotonicity of Schwartz’s
inequality in inner product spaces, Contributions Macedonian Acad. Sci. and Arts, 15 (2)
(1994), 5-22. [6] S.S. DRAGOMIR, B. MOND and Z. PALES, On a supermultiplicity property of Gram’s
determinant, Aequationes Mathematicae, 54 (1997), 199-204.
[7] S.S. DRAGOMIR, B. MOND and J.E. PE CARI ˇ C, Some remarks on Bessel’s inequality in´
inner product spaces, Studia Univ. “Babes-Bolyai”, Math., 37 (4) (1992), 77-86.
[8] S.S. DRAGOMIR and J. SANDOR, On Bessel’s and Gram’s inequalities in prehilbertian
spaces, Periodica Math. Hungarica, 29 (3) (1994), 197-205.
BESSEL’S INEQUALITY 5
[9] S.S. DRAGOMIR and J. S ANDOR, Some inequalities in prehilbertian spaces,´ Studia Univ.
“Babe¸s-Bolyai”, Mathematica, 1, 32 , (1987), 71-78.
[10] W.N. EVERITT, Inequalities for Gram determinants, Quart. J. Math., Oxford, Ser. (2),
8 (1957), 191-196.
[11] T. FURUTA, An elementary proof of Hadamard theorem, Math. Vesnik, 8( 23 )(1971), 267-
[12] C.F. METCALF, A Bessel-Schwartz inequality for Gramians and related bounds for deter-
minants, Ann. Math. Pura Appl., (4) 68 (1965), 201-232.
[13] D.S. MITRINOVI ´C, Analytic Inequalities, Springer-Verlag, Berlin-Heidelberg and New York,
[14] C.F. MOPPERT, On the Gram determinant, Quart. J. Math., Oxford, Ser (2), 10 (1959),
161-164.
[15] K. YOSHIDA, Functional Analysis, Springer-Verlag, Berlin, 1966.
School of Communications and Informatics, Victoria University of Technology, P.O.
Box 14428, Melbourne City MC, Victioria 8001, Australia
E-mail address: sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
