A Note on Bessel’s Inequality

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S.S. DRAGOMIR

Abstract. A monotonicity property of Bessel’s inequality in inner product spaces is given.

1. _Introduction

Let X be a linear space over the real or complex number field_K. A mapping (·,·) :X×X →_Kis said to be apositive hermitian formif the following conditions are satisfied:

(i) (αx+βy, z) =α(x, z) +β(y, z) for allx, y, z∈X andα, β∈K;

(ii) (y, x) = (x, y) for all x, y∈X; (iii) (x, x)≥0 for allx∈X.

If kxk := (x, x)12 _{, x}_∈ _X _{denotes the semi-norm associated to this form and} (ei)i∈I is an orthornormal family of vectors inX, i.e., (ei, ej) =δij (i, j∈I), then

one has the following inequality [15]:

(1.1) kxk2≥X

i∈I

|(x, ei)|2 for allx∈X,

which is well known in the literature as Bessel’s inequality. Indeed, for every finite partH ofI, one has:

0 ≤

x−X i∈H

(x, ei)ei

2

= 

x−X i∈H

(x, ei)ei, x−

X

j∈H

(x, ej)ej





= kxk2−X i∈H

|(x, ei)|2−

X

j∈H

|(x, ej)|2+

X

i,j∈H

(x, ei) (ej, x)δij

= kxk2−X i∈H

|(x, ei)|2,

for allx∈X,which proves the assertion.

The main aim of this paper is to improve this result as follows. 2. _Results

The following theorem holds.

Theorem 1. Let X be a linear space and (·,·)₂,(·,·)₁ two hermitian forms on

X such that k·k₂ is greater than or equal to k·k₁, i.e., kxk₂ ≥ kxk₁ 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.

1991Mathematics Subject Classification. Primary 26D15; Secondary 46C99. Key words and phrases. Bessel’s inequality, Inner product spaces.

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orthornormal family in(X; (·,·)₁)such that for anyi∈Ithere exists a finiteK⊂J

so that

(F) ei =

X

j∈K

αjfj, αj ∈K (j∈K),

then one has the inequality:

(2.1) kxk2₂−X i∈I

|(x, ei)2| 2

≥ kxk2₁−X j∈J

(x, fj)1

2 ≥0,

for allx∈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

(2.2)

x− n

X

k=1

λkgk

2

≥ kxk2− n

X

k=1

|(x, gk)| 2

≥0,

for allλk∈Kandx∈X (k= 1, ..., n).

Proof. We will prove this fact by induction over “n”. Supposen= 1. Then we must prove that

kx−λ1g1k 2

≥ kxk2− |(x, g1)| 2

, x∈X, λ1∈K.

A simple computation shows that the above inequality is equivalent with

|λ1|2−2 Re (x, λ1g1) +|(x, g1)| 2

≥0, x∈X, λ1∈K.

Since Re (x, λ1g1)≤ |(x, λ1g1)|, one has |λ1|

2

−2 Re (x, λ1g1) +|(x, g1)| 2

≥ |λ1| 2

−2|λ1| |(x, g1)|+|(x, g1)| 2

≥ (|λ1| − |(x, g1)|)2≥0

for allλ1∈Kandx∈X, which proves the statement.

Now, assume that (2.2) is valid for “(n−1)”. Then we have:

x− n

X

k=1

λkgk

2

=

(x−λngn)− n−1

X

k=1

λkgk

≥ kx−λngnk2− n−1

X

k=1

|(x−λngn, gk)|2

= kx−λngnk 2

− n−1

X

k=1

|(x, gk)| 2

≥ kxk2− |(x, gn)| 2

− n−1

X

k=1

|(x, gk)| 2

= kxk2− n

X

k=1

|(x, gk)| 2

,

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Proof. (Theorem) LetH be a finite part of I. Sincek·k₂ is greater than k·k₁, we have:

kxk2₂−X i∈H

|(x, ei)2| 2 =

x−X i∈H

(x, ei)2ei

2 2 ≥

x−X i∈H

(x, ei)2ei

2 1

, x∈X.

Since, by (F), we may state that for anyi∈H there exists a finiteK⊂J with ei=

X

j∈K

(ei, fj)₁fj,

we have

x−X i∈H

(x, ei)2ei

2 1 =

x−X i∈H

(x, ei)2

X

j∈K

(ei, fj)1fj

2 1 =

x−X j∈K

X

i∈H

(x, ei)2ei, fj

! 1 fj 2 1

for allx∈X.

Applying the above lemma for (·,·) = (·,·)₁, (gk)k=1,n = (fj)_j_∈_K, we can

conclude that

x−X j∈K

λjfj

2 1

≥ kxk2₁−X j∈K

(x, fj)₁

2

, x∈X,

where

λj =

X

i∈H

(x, ei)2ei, fj

!

1

∈K (j∈K).

Consequently, we have:

kxk2₂−X i∈H

|(x, ei)2| 2

≥ kxk2₁−X j∈K

(x, fj)1

2

≥ kxk2₁−X j∈J

(x, fj)1

2

for allx∈X andH a finite part ofI, from where results (2.1). The proof is thus completed.

Corollary 1. Let k·k₁,k·k₂:X →_R+be as above. Then for allx, y∈X, we have the inequality:

(2.3) kxk2₂kyk2₂− |(x, y)₂|2≥ kxk2₁kyk2₁− |(x, y)₁|2≥0,

which is an improvement of the well known Cauchy-Scwartz inequality.

Proof. Ifkyk₂= 0,then (2.3) holds with equality. If kyk_i 6= 0, (i= 1,2), then for {e1} = n_k_yy_k

2 o

, {f1} = n_k_yy_k 1

o

, the above theorem yields that

kxk2₂kyk2₂− |(x, y)₂|2

kyk2₂ ≥

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and sincekyk₂≥ kyk₁,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. 3. _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 thatkAxk ≤ kxk

for all x ∈ X and (Aei, Aej) = δij for all i, j ∈ I. Then one has the

inequality

kxk2−X i∈I

|(x, ei)|2≥ kAxk2−

X

i∈I

|(Ax, Aei)|2≥0

for allx∈X.

The proof follows by the hermitian forms (x, y)₂ = (x, y) and (x, y)₁ = (Ax, Ay) forx, y∈X and for the family (fi)i∈I = (ei)i∈I.

(2) If A : X → X is such that kAxk ≥ kxk for all x ∈ X, then, with the previous assumptions, we also have

0≤ kxk2−X i∈I

|(x, ei)| 2

≤ kAxk2−X i∈I

|(Ax, Aei)| 2

,

for allx∈X.

(3) Suppose that A : X → X is a symmetric positive definite operator with (Ax, x) ≥ kxk2 for all x∈ X. If (ei)i∈I is an orthornormal family in X

such that (Aei, Aej) =δij for alli, j∈I, then one has the inequality

0≤ kxk2−X i∈I

|(x, ei)|2≤(Ax, x)−

X

i∈I

|(Ax, ei)|2,

for allx∈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.

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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.

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[7] S.S. DRAGOMIR, B. MOND and J.E. PE ˇCARI ´C, Some remarks on Bessel’s inequality in inner product spaces,Studia Univ. “Babes-Bolyai”, Math.,37(4) (1992), 77-86.

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8(1957), 191-196.

[11] T. FURUTA, An elementary proof of Hadamard theorem,Math. Vesnik,8(23)(1971), 267-269.

[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, 1970.

[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.

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