THE HYPO-EUCLIDEAN NORM OF AN N−TUPLE OF VECTORS IN INNER PRODUCT SPACES AND APPLICATIONS

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Hypo-Euclidean Norm S.S. Dragomir vol. 8, iss. 2, art. 52, 2007

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THE HYPO-EUCLIDEAN NORM OF AN N −TUPLE OF VECTORS IN INNER PRODUCT SPACES AND

APPLICATIONS

S.S. DRAGOMIR

School of Computer Science and Mathematics Victoria University

PO Box 14428, Melbourne City 8001, VIC, Australia

EMail:sever.dragomir@vu.edu.au URL:http://rgmia.vu.edu.au/dragomir

Received: 19 March, 2007

Accepted: 28 April, 2007

Communicated by: J.M. Rassias

2000 AMS Sub. Class.: Primary 47C05, 47C10; Secondary 47A12.

Key words: Inner product spaces, Norms, Bessel’s inequality, Boas-Bellman and Bombieri inequalities, Bounded linear operators, Numerical radius.

Abstract: The concept of hypo-Euclidean norm for ann−tuple of vectors in inner product spaces is introduced. Its fundamental properties are established. Upper bounds via the Boas-Bellman [1]-[3] and Bombieri [2] type inequalities are provided.

Applications forn−tuples of bounded linear operators defined on Hilbert spaces are also given.

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1. Introduction

Let(E,k·k)be a normed linear space over the real or complex number fieldK. On Kⁿendowed with the canonical linear structure we consider a normk·k_nand the unit ball

B(k·k_n) :={λ = (λ1, . . . , λn)∈Kⁿ| kλk_n≤1}. As an example of such norms we should mention the usualp−norms

(1.1) kλk_n,p:=

( max{|λ1|, . . . ,|λn|} if p=∞;

(Pn

k=1|λ_k|^p)¹^p if p∈[1,∞).

The Euclidean norm is obtained forp= 2, i.e., kλk_n,2 =

n

X

k=1

|λ_k|²

!¹₂ .

It is well known that on Eⁿ := E × · · · ×E endowed with the canonical linear structure we can define the followingp−norms:

(1.2) kXk_n,p :=

( max{kx₁k, . . . ,kx_nk} if p=∞;

(Pn

k=1kx_kk^p)¹^p if p∈[1,∞);

whereX = (x₁, . . . , x_n)∈Eⁿ.

For a given normk·k_nonKⁿwe define the functionalk·k_h,n:Eⁿ →[0,∞)given by

(1.3) kXk_h,n:= sup

(λ1,...,λn)∈B(^k·kn)

n

X

j=1

λ_jx_j ,

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whereX = (x₁, . . . , x_n)∈Eⁿ. It is easy to see that:

(i) kXk_h,n≥0for anyX ∈Eⁿ;

(ii) kX+Yk_h,n≤ kXk_h,n+kYk_h,nfor anyX, Y ∈Eⁿ; (iii) kαXk_h,n =|α| kXk_h,nfor eachα∈KandX ∈Eⁿ;

and thereforek·k_h,nis a semi-norm onEⁿ.This will be called the hypo-semi-norm generated by the normk·k_nonXⁿ.

We observe thatkXk_h,n= 0if and only ifPn

j=1λ_jx_j = 0for any(λ₁, . . . , λ_n)∈ B(k·k_n).If there existsλ⁰₁, . . . , λ⁰_n 6= 0such that(λ⁰₁,0, . . . ,0),(0, λ⁰₂, . . . ,0), . . . , (0,0, . . . , λ⁰_n)∈B(k·k_n)then the semi-norm generated byk·k_nis a norm onEⁿ.

If byB^n,p withp ∈ [1,∞]we denote the balls generated by thep−normsk·k_n,p onKⁿ,then we can obtain the following hypo-p-norms onXⁿ:

(1.4) kXk_h,n,p := sup

(λ1,...,λn)∈Bn,p

n

X

j=1

λ_jx_j , withp∈[1,∞].

Forp= 2,we have the Euclidean ball inKⁿ,which we denote byBn, Bⁿ=

(

λ = (λ₁, . . . , λ_n)∈Kⁿ

n

X

i=1

|λ_i|² ≤1 )

that generates the hypo-Euclidean norm onEⁿ,i.e.,

(1.5) kXk_h,e:= sup

(λ1,...,λn)∈Bn

n

X

j=1

λ_jx_j .

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Moreover, ifE =H, H is a Hilbert space overK, then the hypo-Euclidean norm onHⁿwill be denoted simply by

(1.6) k(x₁, . . . , x_n)k_e := sup

(λ1,...,λn)∈Bn

n

X

j=1

λ_jx_j , and its properties will be extensively studied in the present paper.

Both the notation in (1.6) and the necessity of investigating its main properties are motivated by the recent work of G. Popescu [9] who introduced a similar norm on the Cartesian product of Banach algebraB(H)of all bounded linear operators onH and used it to investigate various properties ofn−tuple of operators in Multivariable Operator Theory. The study is also motivated by the fact that the hypo-Euclidean norm is closely related to the quadratic formPn

j=1|hx, x_ji|² (see the representation Theorem2.2) that plays a key role in many problems arising in the Theory of Fourier expansions in Hilbert spaces.

The paper is structured as follows: in Section 2we establish the equivalence of the hypo-Euclidean norm with the usual Euclidean norm on Hⁿ, provide a representation result and obtain some lower bounds for it. In Section3, on utilising the classical results of Boas-Bellman and Bombieri as well as some recent similar results obtained by the author, we give various upper bounds for the hypo-Euclidean norm.

These are complemented in Section4with other inequalities betweenp−norms and the hypo-Euclidean norm. Section5 is devoted to the presentation of some conditional reverse inequalities between the hypo-Euclidean norm and the norm of the sum of the vectors involved. In Section6, the natural connection between the hypo- Euclidean norm and the operator normk(·, . . . ,·)k_e introduced by Popescu in [9] is investigated. A representation result is obtained and some applications for operator inequalities are pointed out. Finally, in Section7, a new norm for operators is introduced and some natural inequalities are obtained.

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2. Fundamental Properties

Let (H;h·,·i) be a Hilbert space over K and n ∈ N, n ≥ 1. In the Cartesian product Hⁿ := H × · · · × H, for the n−tuples of vectors X = (x₁, . . . , x_n), Y = (y₁, . . . , y_n)∈Hⁿ,we can define the inner producth·,·iby

(2.1) hX, Yi:=

n

X

j=1

hx_j, y_ji, X, Y ∈Hⁿ, which generates the Euclidean normk·k₂ onHⁿ,i.e.,

(2.2) kXk₂ :=

n

X

j=1

kx_jk²

!¹₂

, X ∈Hⁿ.

The following result connects the usual Euclidean normk·kwith the hypo-Euclidean normk·k_e.

Theorem 2.1. For anyX ∈Hⁿwe have the inequalities

(2.3) kXk₂ ≥ kXk_e ≥ 1

√nkXk₂, i.e.,k·k₂ andk·k_eare equivalent norms onHⁿ.

Proof. By the Cauchy-Bunyakovsky-Schwarz inequality we have

(2.4)

n

X

j=1

λ_jx_j

≤

n

X

j=1

|λ_j|²

!¹₂ _n X

j=1

kx_jk²

!¹₂

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for any(λ₁, . . . , λ_n) ∈ Kⁿ.Taking the supremum over (λ₁, . . . , λ_n) ∈ Bⁿ in (2.4) we obtain the first inequality in (2.3).

If byσwe denote the rotation-invariant normalised positive Borel measure on the unit sphere ∂Bn ∂Bn= (λ₁, . . . , λ_n)∈Kⁿ

Pn

i=1|λ_i|² = 1

whose existence and properties have been pointed out in [10], then we can state that

Z

∂Bn

|λ_k|²dσ(λ) = 1 n and (2.5)

Z

∂Bn

λ_kλ_jdσ(λ) = 0 if k6=j, k, j= 1, . . . , n.

Utilising these properties, we have kXk²_e = sup

(λ1,...,λn)∈Bn

n

X

k=1

λ_kx_k

2

= sup

(λ1,...,λn)∈Bn

" _n X

k,j=1

λ_kλ_jhx_k, x_ji

#

≥ Z

∂Bn

" _n X

k,j=1

λ_kλ_jhx_k, x_ji

#

dσ(λ) =

n

X

k,j=1

Z

∂Bn

λ_kλ_jhx_k, x_ji dσ(λ)

= 1 n

n

X

k=1

kx_kk² = 1

nkXk²₂,

from where we deduce the second inequality in (2.3).

The following representation result for the hypo-Euclidean norm plays a key role in obtaining various bounds for this norm:

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Theorem 2.2. For anyX ∈HⁿwithX = (x₁, . . . , x_n),we have

(2.6) kXk_e = sup

kxk=1 n

X

j=1

|hx, x_ji|²

!¹₂ .

Proof. We use the following well known representation result for scalars:

(2.7)

n

X

j=1

|z_j|² = sup

(λ1,...,λn)∈Bn

n

X

j=1

λjzj

2

, where(z₁, . . . , z_n)∈Kⁿ.

Utilising this property, we thus have (2.8)

n

X

j=1

|hx, x_ji|²

!¹₂

= sup

(λ1,...,λn)∈Bn

* x,

n

X

j=1

λ_jx_j +

for anyx∈H.

Now, taking the supremum overkxk= 1in (2.8) we get sup

kxk=1 n

X

j=1

|hx, x_ji|²

!¹₂

= sup

kxk=1

"

sup

(λ1,...,λn)∈Bn

* x,

n

X

j=1

λ_jx_j +

#

= sup

(λ1,...,λn)∈Bn

"

sup

kxk=1

* x,

n

X

j=1

λ_jx_j +

#

= sup

(λ1,...,λn)∈Bn

n

X

j=1

λ_jx_j ,

since, in any Hilbert space we have thatsup_kuk=1|hu, vi|=kvkfor eachv ∈H.

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Corollary 2.3. IfX = (x₁, . . . , x_n)is ann−tuple of orthonormal vectors, i.e., we recall that kx_kk = 1 and hx_k, x_ji = 0 for k, j ∈ {1, . . . , n} with k 6= j, then kXk_e≤1.

The proof is obvious by Bessel’s inequality.

The next proposition contains two lower bounds for the hypo-Euclidean norm that are sometimes better than the one in (2.3), as will be shown by some examples later.

Proposition 2.4. For anyX = (x1, . . . , xn)∈Hⁿ\ {0}we have

(2.9) kXk_e≥











1 kXk₂

Pn

j=1kx_jkx_j ,

√1 n

Pn j=1x_j

.

Proof. By the definition of the hypo-Euclidean norm we have that, if(λ⁰₁, . . . , λ⁰_n)∈ Bn,then obviously

kXk_e≥

n

X

j=1

λ⁰_jxj

. The choice

λ⁰_j := kx_jk

kXk₂, j ∈ {1, . . . , n},

which satisfies the condition (λ⁰₁, . . . , λ⁰_n) ∈ Bⁿ will produce the first inequality while the selection

λ⁰_j = 1

√n, j ∈ {1, . . . , n}, will give the second inequality in (2.9).

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Remark 1. Forn = 2,the hypo-Euclidean norm onH² k(x, y)k_e = sup

(λ,µ)∈B2

kλx+µyk= sup

kzk=1

|hz, xi|²+|hz, yi|²¹₂

is bounded below by

B₁(x, y) := 1

√2 kxk²+kyk²¹₂ , B₂(x, y) := kkxkx+kykyk

kxk²+kyk²¹₂ and

B3(x, y) := 1

√2kx+yk.

IfH = Cendowed with the canonical inner producthx, yi :=xy¯wherex, y ∈ C, then

B₁(x, y) = 1

√2 |x|²+|y|²¹₂ , B2(x, y) = ||x|x+|y|y|

|x|²+|y|²¹₂ and

B₃(x, y) = 1

√2|x+y|, x, y ∈C.

The plots of the differences D₁(x, y) := B₁(x, y)− B₂(x, y) and D₂(x, y) :=

B₁(x, y)−B₃(x, y)which are depicted in Figure1and Figure2, respectively, show that the bound B1 is not always better than B2 or B3. However, since the plot of

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D₃(x, y) := B₂(x, y)−B₃(x, y)(see Figure 3) appears to indicate that, at least in the case ofC²,it may be possible that the boundB₂ is always better thanB₃,hence we can ask in general which bound from (2.6) is better for a givenn ≥2?This is an open problem that will be left to the interested reader for further investigation.

-10 -5 10

-4 -2

0 0

5 2

y 4

6

x 8

0 10

5 -5 -10

10

Figure 1: The behaviour ofD₁(x, y)

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-5 0 y -4 5

10 -2

5 0

0

-5 10

2

x -10

4 6 8 10

Figure 2: The behaviour ofD2(x, y)

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-5 0 y 5 10

0

x 5 0.5

0 -5

1

10 -10 1.5

2 2.5

3 3.5

Figure 3: The behaviour ofD3(x, y)

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3. Upper Bounds via the Boas-Bellman and Bombieri Type Inequalities

In 1941, R.P. Boas [3] and in 1944, independently, R. Bellman [1] proved the following generalisation of Bessel’s inequality that can be stated for any family of vectors {y1, . . . , yn}(see also [8, p. 392] or [5, p. 125]):

(3.1)

n

X

j=1

|hx, y_ji|² ≤ kxk²



max

1≤j≤nky_jk²+ X

1≤j6=k≤n

|hy_k, y_ji|²

!¹₂



for any x, y₁. . . , y_n vectors in the real or complex inner product space (H;h·,·i). This result is known in the literature as the Boas-Bellman inequality.

The following result provides various upper bounds for the hypo-Euclidean norm:

Theorem 3.1. For anyX = (x₁, . . . , x_n)∈Hⁿ,we have

(3.2) kXk²_e ≤











1≤j≤nmax kx_jk²+ P

1≤j6=k≤n

|hx_k, x_ji|²

!¹₂ ,

1≤j≤nmax kx_jk²+ (n−1) max

1≤j6=k≤n|hx_k, x_ji|;

(3.3) kXk²_e ≤

"

1≤j≤nmax kx_jk²

n

X

j=1

kx_jk²

+ max

1≤j6=k≤n{kx_jk kx_kk} X

1≤j6=k≤n

|hx_j, x_ki|

#¹₂

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and

(3.4) kXk⁴_e ≤











1≤j≤nmax kx_jk²

n

P

j=1

kx_jk²+ (n−1)kXk²_e max

1≤j6=k≤n|hx_j, x_ki|, kXk²_e max

1≤j≤nkx_jk²+ max

1≤j6=k≤n{kx_jk kx_kk} P

1≤j6=k≤n

|hx_j, x_ki|. Proof. Taking the supremum overkxk = 1in (3.1) and utilising the representation (2.6), we deduce the first inequality in (3.2).

In [4], we proved amongst others the following inequalities

(3.5)

n

X

j=1

c_jhx, y_ji

2

≤ kxk²×











1≤j≤nmax |c_j|²

n

P

j=1

ky_jk²,

n

P

j=1

|c_j|² max

1≤j≤nky_jk², +kxk²×







1≤j6=k≤nmax {|c_jc_k|} P

1≤j6=k≤n

|hy_j, y_ki|, (n−1)

n

P

j=1

|c_j|² max

1≤j6=k≤n|hy_j, y_ki|, for any y₁, . . . , y_n, x ∈ H and c₁, . . . , c_n ∈ K, where (3.5) should be seen as all possible configurations.

The choicec_j =hx, y_ji, j ∈ {1, . . . , n}will produce the following four inequalities:

(3.6)

" _n X

j=1

|hx, y_ji|²

#2

≤ kxk²×











1≤j≤nmax |hx, y_ji|²Pⁿ

j=1

ky_jk²,

n

P

j=1

|hx, yji|² max

1≤j≤nkyjk²,

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+kxk²×







1≤j6=k≤nmax {|hx, y_ji| |hx, y_ki|} P

1≤j6=k≤n

|hy_j, y_ki|, (n−1)

n

P

j=1

|hx, y_ji|² max

1≤j6=k≤n|hy_j, y_ki|.

Taking the supremum overkxk = 1and utilising the representation (2.6) we easily deduce the rest of the four inequalities.

A different generalisation of Bessel’s inequality for non-orthogonal vectors is the Bombieri inequality (see [2] or [8, p. 397] and [5, p. 134]):

(3.7)

n

X

j=1

|hx, y_ji|² ≤ kxk² max

1≤j≤n

( _n X

k=1

|hy_j, y_ki|

) ,

for any x ∈ H, where y₁, . . . , y_n are vectors in the real or complex inner product space(H;h·,·i).

Note that, the Bombieri inequality was not stated in the general case of inner product spaces in [2]. However, the inequality presented there easily leads to (3.7) which, apparently, was firstly mentioned as is in [8, p. 394].

On utilising the Bombieri inequality (3.7) and the representation Theorem 2.2, we can state the following simple upper bound for the hypo-Euclidean normk·k_e. Theorem 3.2. For anyX = (x₁, . . . , x_n)∈Hⁿ,we have

(3.8) kXk²_e ≤ max

1≤j≤n

( _n X

k=1

|hx_j, x_ki|

) .

In [6] (see also [5, p. 138]), we have established the following norm inequalities:

(3.9)

n

X

j=1

α_jz_j

2

≤n¹^p⁺¹^t⁻¹

n

X

k=1

|α_k|²





n

X

k=1 n

X

j=1

|hz_j, z_ki|^q

!^u_q



1 u

,

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where ¹_p + ¹_q = 1, ¹_t + ¹_u = 1 and1 < p ≤ 2,1 < t ≤ 2 and α_j ∈ C, z_j ∈ H, j ∈ {1, . . . , n}.

An interesting particular case of (3.9) obtained for p = q = 2, t = u = 2 is incorporated in

(3.10)

n

X

j=1

α_jz_j

2

≤

n

X

k=1

|α_k|²

n

X

j,k=1

|hz_j, z_ki|²

!¹₂ .

Other similar inequalities for norms are the following ones [6] (see also [5, pp.

139-140]):

(3.11)

n

X

j=1

α_jz_j

2

≤n^p¹

n

X

k=1

|α_k|² max

1≤j≤n







" _n X

k=1

|hz_j, z_ki|^q

#¹_q



 ,

provided that1 < p ≤ 2and ¹_p + ¹_q = 1, α_j ∈ C, z_j ∈ H, j ∈ {1, . . . , n}. In the particular casep=q= 2,we have

(3.12)

n

X

j=1

α_jz_j

2

≤√ n

n

X

k=1

|α_k|² max

1≤j≤n

" _n X

k=1

|hz_j, z_ki|²

#¹₂ . Also, if1< m≤2,then [6]:

(3.13)

n

X

j=1

α_jz_j

2

≤n^m¹

n

X

k=1

|α_k|² ( _n

X

j=1

1≤k≤nmax |hz_j, z_ki|^l )¹_l

, where _m¹ +¹_l = 1.Form =l = 2,we get

(3.14)

n

X

j=1

α_jz_j

2

≤√ n

n

X

k=1

|α_k|²

" _n X

j=1

1≤k≤nmax |hz_j, z_ki|² #¹₂

.

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Finally, we can also state the inequality [6]:

(3.15)

n

X

j=1

αjzj

2

≤n

n

X

k=1

|αk|² max

1≤j,k≤n|hzj, zki|.

Utilising the above norm-inequalities and the definition of the hypo-Euclidean norm, we can state the following result which provides other upper bounds than the ones outlined in Theorem3.1and3.2:

(3.16) kXk²_e ≤











n¹^p⁺¹^t⁻¹







n

P

k=1 n

P

j=1

|hx_j, x_ki|^q

!^u_q





1 u

where ¹_p +¹_q = 1,

1

t +_u¹ = 1 and 1< p≤2, 1< t≤2;

n¹^p max

1≤j≤n







"

n

P

j=1

|hx_j, x_ki|^q

#¹_q





where ¹_p +¹_q = 1 and 1< p≤2;

n^m¹ ( n

P

j=1

1≤k≤nmax |hx_j, x_ki|^l )¹_l

where _m¹ +¹_l = 1 and 1< m≤2;

n max

1≤k≤n|hx_k, z_ji|;

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and, in particular,

(3.17) kXk²_e ≤











"

n

P

j,k=1

|hx_j, x_ki|²

#¹₂

;

√n max

1≤j≤n

_n P

k=1

|hx_j, x_ki|² ¹₂

;

√n

"

n

P

j=1

1≤k≤nmax

|hx_j, x_ki|² #¹₂

.

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4. Various Inequalities for the Hypo-Euclidean Norm

For ann−tupleX = (x₁, . . . , x_n)of vectors inH,we consider the usualp−norms:

kXk_p :=

n

X

j=1

kx_jk^p

!¹_p , wherep∈[1,∞),and denote withSthe sumPn

j=1x_j.

With these notations we can state the following reverse of the inequalitykXk₂ ≥ kXk_e,that has been pointed out in Theorem2.1.

Theorem 4.1. For anyX = (x₁, . . . , x_n)∈Hⁿ,we have (4.1) (0≤)kXk²₂− kXk²_e ≤ kXk²₁ − kSk². If

kXk²₍₂₎ :=

n

X

j,k=1

x_j+x_k 2

2

, then also

(0≤)kXk²₂− kXk²_e ≤ kXk²₍₂₎− kSk² (4.2)

(≤nkXk²₂− kSk²).

Proof. We observe, for anyx∈H,that

n

X

j=1

hx, xji

2

=

n

X

j=1

hx, xji

n

X

k=1

hx, xki=

n

X

j=1

hx, xji

n

X

k=1

hx, xki (4.3)

=

n

X

k=1

|hx, x_ki|²+ X

1≤j6=k≤n

hx, x_ji hx_k, xi

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≤

n

X

k=1

|hx, x_ki|²+

X

1≤j6=k≤n

hx, x_ji hx_k, xi

≤

n

X

k=1

|hx, x_ki|²+ X

1≤j6=k≤n

|hx, x_ji| |hx_k, xi|. Taking the supremum overkxk= 1,we get

(4.4) sup

kxk=1

n

X

j=1

hx, xji

2

≤ sup

kxk=1 n

X

k=1

|hx, x_ki|²+ X

1≤j6=k≤n

sup

kxk=1

|hx, x_ji| · sup

kxk=1

|hx_k, xi|. However,

sup

kxk=1

n

X

j=1

hx, x_ji

2

= sup

kxk=1

* x,

n

X

j=1

x_j +

2

=kSk², sup

kxk=1

|hx, x_ji|=kx_jk and sup

kxk=1

|hx, x_ki|=kx_kk forj, k ∈ {1, . . . , n},and by (4.4) we get

kSk² ≤ kXk²_e+ X

1≤j6=k≤n

kx_jk kx_jk

=kXk²_e+

n

X

j,k=1

kx_jk kx_kk −

n

X

k=1

kx_kk²

=kXk²_e+kXk²₁− kXk²₂,

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which is clearly equivalent with (4.1).

Further on, we also observe that, for anyx∈H we have the identity:

n

X

j=1

hx, x_ji

2

= Re

" _n X

k,j=1

hx, x_ji hx_k, xi

# (4.5)

=

n

X

k=1

|hx, xki|²+ X

1≤j6=k≤n

Re [hx, xji hxk, xi]. Utilising the elementary inequality for complex numbers

(4.6) Re (u¯v)≤ 1

4|u+v|², u, v ∈C, we can state that

X

1≤k6=j≤n

Re [hx, x_ji hx_k, xi]≤ 1 4

X

1≤k6=j≤n

|hx, x_ji+hx, x_ki|²

= X

1≤k6=j≤n

x,x_j+x_k 2

2

, and by (4.5) we get

(4.7)

n

X

j=1

hx, x_ji

2

≤

n

X

k=1

|hx, x_ki|²+ X

1≤k6=j≤n

x,x_j +x_k 2

2

for anyx∈H.

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Taking the supremum overkxk= 1in (4.7) we deduce

n

X

j=1

x_j

2

≤ kXk²_e + X

1≤k6=j≤n

xj +xk

2

=kXk²_e+

n

X

k,j=1

x_j +x_k 2

2

−

n

X

k=1

kx_kk²

which provides the first inequality in (4.2).

By the convexity ofk·k² we have

n

X

j,k=1

x_j +x_k 2

2

≤ 1 2

n

X

j,k=1

kx_jk²+kx_kk²

=n

n

X

k=1

kx_kk²

and the last part of (4.2) is obvious.

Remark 2. Forn = 2, X = (x, y)∈H²we have the upper bounds B₁(x, y) :=kxk²+kyk²− kx+yk²

= 2 (kxk kyk −Rehx, yi) and

B₂(x, y) :=kxk²+kyk²

for the differencekXk²₂−kXk²_e, X ∈H²as provided by (4.1) and (4.2) respectively.

IfH = Rthen B₁(x, y) = 2 (|xy| −xy), B₂(x, y) = x² +y².If we consider the function∆ (x, y) = B₂(x, y)−B₁(x, y)then the plot of∆ (x, y)depicted in Figure 4shows that the bounds provided by (4.1) and (4.2) cannot be compared in general, meaning that sometimes the first is better than the second and vice versa.

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-5 0 y 5 -200

10 5 10

0 -5 -10

-100

x 0

100 200

Figure 4: The behaviour of∆ (x, y)

From a different view-point we can state the following result:

(4.8) kSk² ≤ kXk_e



kXk_e+

n

X

k=1

kS−x_kk²

!¹₂



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and

(4.9) kSk²

≤ kXk_e





kXk_e+







1≤k≤nmax kS−x_kk²+ X

1≤k6=l≤n

|hS−x_k, S−x_li|²

!¹₂





1 2



, respectively.

Proof. Utilising the identity (4.5) above we have

(4.10)

n

X

j=1

hx, x_ji

2

=

n

X

k=1

|hx, x_ki|²+ Re

*

x, X

1≤j6=k≤n

hx, x_kix_j +

for anyx∈H.

By the Schwarz inequality in the inner product space(H,h·,·i), we have that Re

*

x, X

1≤j6=k≤n

hx, x_kix_j +

≤ kxk

X

1≤j6=k≤n

hx, x_kix_j (4.11)

=kxk

n

X

j,k=1

hx, x_kix_j−

n

X

k=1

hx, x_kix_k

=kxk

* x,

n

X

k=1

x_k + _n

X

j=1

x_j −

n

X

k=1

hx, x_kix_k

=kxk

n

X

k=1

hx, x_ki(S−x_k) .

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Utilising the Cauchy-Bunyakovsky-Schwarz inequality we have (4.12)

n

X

k=1

hx, x_ki(S−x_k)

≤

n

X

k=1

|hx, x_ki|²

!¹₂ _n X

k=1

kS−x_kk²

!¹₂

and then by (4.10) – (4.12) we can state the inequality:

(4.13)

n

X

j=1

hx, x_ji

2

≤

n

X

k=1

|hx, x_ki|²

!¹₂ 



n

X

k=1

|hx, x_ki|²

!¹₂ +

n

X

k=1

kS−x_kk²

!¹₂



for anyx∈H,kxk= 1.Taking the supremum overkxk= 1we deduce the desired result (4.8).

Now, following the above argument, we can also state that (4.14)

* x,

n

X

j=1

x_j +

2

≤

n

X

k=1

|hx, x_ki|²+kxk

n

X

k=1

hx, x_ki(S−x_k) for anyx∈H.

Utilising the inequality (4.15)

n

X

j=1

α_jz_j

2

≤

n

X

j=1

|α_j|²







1≤j≤nmax kz_jk²+ X

1≤j6=k≤n

|hz_j, z_ki|²

!¹₂



 ,

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whereα_j ∈C,z_j ∈H, j ∈ {1, . . . , n},that has been obtained in [4], see also [5, p.

128], we can state that (4.16)

n

X

k=1

hx, x_ki(S−x_k)

≤

n

X

k=1

|hx, x_ki|²

!¹₂ 





1≤k≤nmax kS−x_kk²+ X

1≤k6=l≤n

|hS−x_k, S−x_li|²

!¹₂





1 2

for anyx∈H.

Now, by the use of (4.14) – (4.16) we deduce the desired result (4.9). The details are omitted.

Remark 3. On utilising the inequality:

(4.17)

n

X

j=1

αjzj

2

≤

n

X

j=1

|αj|²

1≤k≤nmax kzkk²+ (n−1) max

1≤k6=l≤n|hzk, zli|

, where α_j ∈ C, z_j ∈ H, j ∈ {1, . . . , n}, that has been obtained in [4], (see also [5, p. 130]) in place of (4.15) above, we can state the following inequality for the hypo-Euclidean norm as well:

(4.18) kSk²

≤ kXk_e

"

kXk_e+

1≤k≤nmax kS−x_kk² + (n−1) max

1≤k6=l≤n|hS−x_k, S−x_li|² ¹₂#

for anyX = (x1, . . . , xn)∈Hⁿ.

Other similar results may be stated by making use of the results from [6]. The details are left to the interested reader.

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5. Reverse Inequalities

Before we proceed with establishing some reverse inequalities for the hypo-Euclidean norm, we recall some reverse results of the Cauchy-Bunyakovsky-Schwarz inequality for real or complex numbers as follows:

Ifγ,Γ∈K(K=C,R)andαj ∈K,j ∈ {1, . . . , n}with the property that 0≤Re [(Γ−αj) (αj −γ¯)]

(5.1)

= (Re Γ−Reα_j) (Reα_j −Reγ) + (Im Γ−Imα_j) (Imα_j −Imγ) or, equivalently,

(5.2)

α_j − γ+ Γ 2

≤ 1

2|Γ−γ|

for eachj ∈ {1, . . . , n},then (see for instance [5, p. 9])

(5.3) n

n

X

j=1

|α_j|²−

n

X

j=1

α_j

2

≤ 1

4 ·n²|Γ−γ|². In addition, ifRe (Γ¯γ)>0,then (see for example [5, p. 26]):

n

X

j=1

|α_j|² ≤ 1 4 ·

n Re

h Γ + ¯¯ γ Pn j=1αj

io2

Re (Γ¯γ) (5.4)

≤ 1

4 ·|Γ +γ|² Re (Γ¯γ)

n

X

j=1

α_j

2

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and

(5.5) n

n

X

j=1

|α_j|²−

n

X

j=1

α_j

2

≤ 1

4· |Γ−γ|² Re (Γ¯γ)

n

X

j=1

α_j

2

. Also, ifΓ6=−γ,then (see for instance [5, p. 32]):

(5.6) n

n

X

j=1

|α_j|²

!¹₂

−

n

X

j=1

α_j

≤ 1

4n· |Γ−γ|²

|Γ +γ|. Finally, from [7] we can also state that

(5.7) n

n

X

j=1

|α_j|²−

n

X

j=1

α_j

2

≤nh

|Γ +γ| −2p

Re (Γ¯γ)i

n

X

j=1

α_j , providedRe (Γ¯γ)>0.

We notice that a simple sufficient condition for (5.1) to hold is that (5.8) Re Γ≥Reα_j ≥Reγ and Im Γ≥Imα_j ≥Imγ for eachj ∈ {1, . . . , n}.

We can state and prove the following conditional inequalities for the hypo-Euclidean normk·k_e :

Theorem 5.1. Letϕ, φ∈KandX = (x₁, . . . , x_n)∈Hⁿsuch that either:

(5.9)

hx, x_ji − ϕ+φ 2

≤ 1

2|φ−ϕ|