INEQUALITIES INVOLVING THE INNER PRODUCT
DORIAN POPA AND IOAN RAŞA
Technical University of Cluj-Napoca
Department of Mathematics
Str. C. Daicoviciu 15
Cluj-Napoca, Romania
EMail: {Popa.Dorian, Ioan.Rasa}@math.utcluj.ro
Inequalities Involving
the Inner Product
Dorian Popa and Ioan Raşa
vol. 8, iss. 3, art. 86, 2007
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Received:
20 February, 2007
Accepted:
16 July, 2007
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Communicated by:
S.S. Dragomir
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2000 AMS Sub. Class.:
26D20.
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Key words:
Inner product, Schwarz inequality.
Abstract:
The paper contains inequalities related to generalizations of Schwarz’s inequality.
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Contents
1
Introduction
3
2
The Results
4
Inequalities Involving
the Inner Product
Dorian Popa and Ioan Raşa
vol. 8, iss. 3, art. 86, 2007
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1.
Introduction
Let (H, h·, ·i) be a Hilbert space over the field K = R or K = C. Then for all
x, a, b ∈ H the following inequality holds:
1
2
ha, xihx, bi − ha, bikxk ≤ 1 kak kbk kxk2 .
(1.1)
2
2
In particular, for a = b (1.1) reduces to Schwarz’s inequality.
For historical remarks, proofs, extensions, generalizations and applications of
(1.1), see [1] – [5] and the references given therein.
In this paper we consider a suitable quadratic form and derive inequalities related
to (1.1). More precisely, we obtain estimates involving the real and the imaginary
part of the expression whose absolute value is contained in the left hand of (1.1).
Inequalities Involving
the Inner Product
Dorian Popa and Ioan Raşa
vol. 8, iss. 3, art. 86, 2007
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2.
The Results
Let v1 , . . . , vn (n ≥ 2) be linearly independent vectors in H, and v := v1 + · · · + vn .
Consider the matrix


hv1 , v − v1 i hv2 , v − v1 i · · · hvn , v − v1 i
 hv , v − v i hv , v − v i · · · hv , v − v i 
 1
2
2
2
n
2 

A := 
..
..
..


.
.
···
.


hv1 , v − vn i hv2 , v − vn i · · · hvn , v − vn i
Theorem 2.1.
(i) The matrix A has real eigenvalues λ1 ≤ · · · ≤ λn ; moreover, λ1 ≤ 0 and
λn ≥ 0.
(ii) The following inequalities hold:
(2.1)
λ1 kxk2 ≤
n
X
hvi , xihx, vj i ≤ λn kxk2 ,
x ∈ H.
i,j=1
i6=j
Dorian Popa and Ioan Raşa
vol. 8, iss. 3, art. 86, 2007
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Proof. Let Hn denote the linear subspace of H generated by v1 , . . . , vn . Consider
the linear operator T : Hn → Hn defined by
T x = hx, viv −
Inequalities Involving
the Inner Product
n
X
hx, vi ivi ,
x ∈ Hn .
i=1
Then for all x, y ∈ Hn we have
hT x, yi = hx, vihv, yi −
n
X
i=1
hx, vi ihvi , yi
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= hy, vihx, vi −
n
X
hy, vi ihx, vi i
i=1
*
x, hy, viv −
=
n
X
+
hy, vi ivi
= hx, T yi.
i=1
We conclude that T is self-adjoint, hence it has real eigenvalues λ1 ≤ · · · ≤ λn and:
Inequalities Involving
the Inner Product
Dorian Popa and Ioan Raşa
2
2
λ1 kxk ≤ hT x, xi ≤ λn kxk ,
(2.2)
x ∈ Hn .
vol. 8, iss. 3, art. 86, 2007
On the other hand,
T vj = hvj , viv −
n
X
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hvj , vi ivi
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i=1
=
n
X
(hvj , vi − hvj , vi i)vi =
i=1
n
X
hvj , v − vi ivi
i=1
for all j = 1, 2, . . . , n.
This means that A is the matrix of T with respect to the basis {v1 , v2 , . . . , vn } of
Hn , and so λ1 ≤ · · · ≤ λn are the eigenvalues of the matrix A.
Now remark that
n
X
hT x, xi = hx, vihv, xi −
hx, vi ihvi , xi
i=1
=
n
X
i=1
hvi , xi
n
X
i=1
hx, vi i −
n
X
i=1
hvi , xihx, vi i
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=
n
X
hvi , xihx, vj i,
x ∈ Hn .
i,j=1
i6=j
Combined with (2.2), this gives
(2.3)
2
λ1 kxk ≤
n
X
hvi , xihx, vj i ≤ λn kxk2 ,
x ∈ Hn .
i,j=1
i6=j
Inequalities Involving
the Inner Product
Dorian Popa and Ioan Raşa
Let x ∈ Hn , x 6= 0, hx, vi i = 0, i = 1, 2, . . . , n − 1.
From (2.3) we infer that
vol. 8, iss. 3, art. 86, 2007
λ1 ≤ 0 ≤ λn .
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Let y ∈ H. Then y = x + z, x ∈ Hn , z ∈ Hn⊥ and kyk2 = kxk2 + kzk2 , so that
kyk2 ≥ kxk2 . Moreover,
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(2.4)
hvi , yi = hvi , x + zi = hvi , xi,
i = 1, . . . , n.
Using (2.3) and (2.4), we get
λ1 kyk2 ≤ λ1 kxk2 ≤
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n
X
hvi , yihy, vj i ≤ λn kxk2 ≤ λn kyk2
i,j=1
i6=j
and this concludes the proof.
Corollary 2.2. Let a, b, x ∈ H. Then
p
1
1
2
(2.5) Re ha, xihx, bi − kxk ha, bi ≤ kxk2 kak2 kbk2 − (Imha, bi)2 .
2
2
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Proof. If a and b are linearly dependent, (2.5) can be verified directly. Otherwise it
is a consequence of Theorem 2.1.
Indeed for n = 2, v1 = a, v2 = b, the eigenvalues of the matrix A are
p
λ1,2 = Reha, bi ± kak2 kbk2 − (Imha, bi)2
and
hT x, xi = 2 Rehx, aihb, xi,
x ∈ H.
Inequalities Involving
the Inner Product
Dorian Popa and Ioan Raşa
vol. 8, iss. 3, art. 86, 2007
Remark 1.
(i) When K = R, (2.5) coincides with (1.1).
(ii) Let K = C. Applying Corollary 2.2 to the vectors ia, b, x we get
1
2
Im ha, xihx, bi − kxk ha, bi (2.6)
2
p
1
≤ kxk2 kak2 kbk2 − (Reha, bi)2 .
2
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References
[1] M.L. BUZANO, Generalizzazione della disuguglianza di Cauchy-Schwarz (Italian), Rend. Sem. Mat. Univ. e Politech. Torino, 31(1971/73), 405–409(1974).
[2] S.S. DRAGOMIR, Refinements of Buzano’s and Kurepa’s inequalities in inner
product spaces, Facta Univ. (Niš), Ser. Math. Inform., 20 (2005), 65–73.
[3] S.S. DRAGOMIR, A potpourri of Schwarz related inequalities in inner product
spaces (I), J. Inequal. Pure Appl. Math., 6(3) (2005), Art. 59. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=532].
[4] J.E. PEČARIĆ, On some classical inequalities in unitary spaces, Mat. Bilten, 16
(1992), 63–72.
[5] T. PRECUPANU, On a generalization of Cauchy-Buniakowski-Schwarz inequality, Anal. St. Univ. "Al. I. Cuza" Iaşi, 22(2) (1976), 173–175.
Inequalities Involving
the Inner Product
Dorian Popa and Ioan Raşa
vol. 8, iss. 3, art. 86, 2007
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