ON STRONG AND WEAK CONVERGENCE IN n-HILBERT SPACES | Soenjaya | Journal of the Indonesian Mathematical Society 164 449 1 PB

J. Indones. Math. Soc.
Vol. 19, No. 2 (2013), pp. 79–87.

ON STRONG AND WEAK CONVERGENCE IN
n-HILBERT SPACES
Agus L. Soenjaya
Department of Mathematics, National University of Singapore, Singapore
agus.leonardi@nus.edu.sg

Abstract. We discuss the concepts of strong and weak convergence in n-Hilbert
spaces and study their properties. Some examples are given to illustrate the concepts. In particular, we prove an analogue of Banach-Saks-Mazur theorem and
Radon-Riesz property in the case of n-Hilbert space.
Key words: Strong and weak convergence, n-Hilbert space.

Abstrak. Makalah ini menjelaskan konsep konvergensi kuat dan lemah dalam
ruang n-Hilbert dan mempelajari sifat-sifatnya. Beberapa contoh diberikan untuk
menjelaskan konsep-konsep tersebut. Khususnya, teorema analog dari Banach-SaksMazur dan sifat Radon-Riesz dibuktikan untuk kasus ruang n-Hilbert..
Kata kunci: Konvergensi kuat dan lemah, ruang n-Hilbert.

1. INTRODUCTION AND PRELIMINARIES
The notion of n-normed spaces was introduced by Gähler ([4]) as a generalization of normed spaces. It was initially suggested by the area function of a

triangle determined by a triple in Euclidean space. The corresponding theory of
n-inner product spaces was then established by Misiak ([10]). Since then, various
aspects of the theory have been studied, for instance the study of Mazur-Ulam
theorem and Aleksandrov problem in n-normed spaces are done in [1, 2], the study
of operators in n-Banach space is done in [5, 11], and many others.
In this paper, we will generalize the notion of weak convergence in Hilbert
space to the case of n-Hilbert space and study its properties. In particular, we will
expand on the results in [6]. We will also give an analogue of Radon-Riesz property
(on conditions relating strong and weak convergence) in the case of n-Hilbert space.
Furthermore, an analogue of the well-known Banach-Saks-Mazur theorem (on the
2000 Mathematics Subject Classification: Primary 40A05; Secondary 46C99.
Received: 17-07-2012, revised: 31-03-2013, accepted: 05-06-2013.
79

A.L. Soenjaya

80

strong convergence of a convex combination of a weakly convergent sequence) will
be given.

We begin with some preliminary results. Let X be a real vector space with
dim(X) ≥ n, where n is a positive integer. We allow dim(X) to be infinite. A
real-valued function k·, . . . , ·k : X n → R is called an n-norm on X n if the following
conditions hold:
(1)
(2)
(3)
(4)

kx1 , . . . , xn k = 0 if and only if x1 , . . . , xn are linearly dependent;
kx1 , . . . , xn k is invariant under permutations of x1 , . . . , xn ;
kαx1 , x2 , . . . , xn k = |α|kx1 , x2 , . . . , xn k for all α ∈ R and x1 , . . . , xn ∈ X;
kx0 + x1 , x2 , . . . , xn k ≤ kx0 , x2 , . . . , xn k + kx1 , x2 , . . . , xn k, for all
x0 , x1 , . . . , xn ∈ X.

The pair (X, k·, . . . , ·k) is then called an n-normed space. It also follows from the
definition that an n-norm is always non-negative.
Let X be a real vector space with dim(X) ≥ n, where n is a positive integer.
A real-valued function h·, ·|·, . . . , ·i : X n+1 → R is called an n-inner product on X
if the following conditions hold:

(1) hz1 , z1 |z2 , . . . , zn i ≥ 0, with equality if and only if z1 , z2 , . . . , zn are linearly
dependent;
(2) hz1 , z1 |z2 , . . . , zn i = hzi1 , zi1 |zi2 , . . . , zin i for every permutation (i1 , . . . , in )
of (1, . . . , n);
(3) hx, y|z2 , . . . , zn i = hy, x|z2 , . . . , zn i;
(4) hαx, y|z2 , . . . , zn i = αhx, y|z2 , . . . , zn i for every α ∈ R;
(5) hx + x′ , y|z2 , . . . , zn i = hx, y|z2 , . . . , zn i + hx′ , y|z2 , . . . , zn i.
The pair (X, h·, ·|·, . . . , ·i) is then called an n-inner product space.
Observe that any inner product space (X, h·, ·i) can be equipped with the
standard n-inner product:




hx, yi hx, z2 i · · · hx, zn i


 hz2 , yi hz2 , z2 i · · · hz2 , zn i 



(1)
hx, y|z2 , . . . , zn i := 

..
..
..
..


.
.
.
.


 hzn , yi hzn , z2 i · · · hzn , zn i 
where |A| denotes the determinant of A.
In that case, the induced standard n-norm on X is given by
q
kx1 , . . . , xn kS := det[hxi , xj i]

(2)

Note that the value of kx1 , . . . , xn kS is just the volume of the n-dimensional parallelepiped spanned by x1 , . . . , xn .
Further examples and results on n-normed space can be found in [8, 9]. In
particular, for the standard case, completeness in the norm is equivalent to that in
the induced standard n-norm.
Every n-inner product space is an n-normed space with the induced n-norm:
kx1 , . . . , xn k := hx1 , x1 |x2 , . . . , xn i1/2

(3)

On Strong and Weak Convergence

81

An analogue of Cauchy-Schwarz inequality also holds for n-inner product
space, i.e. for all x, y, z2 , . . . , zn ∈ X, we have
|hx, y|z2 , . . . , zn i| ≤ kx, z2 , . . . , zn kky, z2 , . . . , zn k

(4)

The following definitions are taken and inspired from [12].
Definition 1.1. A sequence {xk } in an n-normed space (X, k·, . . . , ·k) is said to
converge to x ∈ X if kxk − x, z2 , . . . , zn k → 0 as k → ∞ for all z2 , . . . , zn ∈ X.
Definition 1.2. A sequence {xk } in an n-normed space (X, k·, . . . , ·k) is a Cauchy
sequence if kxk − xl , z2 , . . . , zn k → 0 as k, l → ∞ for all z2 , . . . , zn ∈ X.
Definition 1.3. If every Cauchy sequence in an n-normed space (X, k·, . . . , ·k)
converges to an x ∈ X, then X is said to be complete. A complete n-inner product
space is called an n-Banach space. A complete n-inner product space is called an
n-Hilbert space.
2. STRONG AND WEAK CONVERGENCE
In this section, we will consider the notions of strong and weak convergence
in n-Hilbert space. The notion of (strong) convergence in 2-normed space has been
studied extensively in [12]. Here, we will focus more on the weak convergence and
the relationships between the two concepts. Let (X, h·, ·|·, . . . , ·i) be an n-Hilbert
space and k·, . . . , ·k be the induced n-norm.
Definition 2.1 (Strong convergence). A sequence (xk ) in X is said to converge
strongly to a point x ∈ X if kxk −x, z2 , . . . , zn k → 0 as k → ∞ for every z2 , . . . , zn ∈
X. In this case, we write xk → x.

Definition 2.2 (Weak convergence). A sequence (xk ) in X is said to converge
weakly to a point x ∈ X if hxk − x, y|z2 , . . . , zn i → 0 as k → ∞ for every
y, z2 , . . . , zn ∈ X. In this case, we write xk ⇀ x.
The following proposition is immediate from the definition.
Proposition 2.3. If (xk ) and (yk ) converges strongly (resp. weakly) to x and y
respectively, then (αxk + βyk ) converges strongly (resp. weakly) to αx + βy.
Here we mention some of the basic properties, the proofs of which can be
found in [6].
Proposition 2.4 (Continuity). The following results hold:
(1) The n-norm is continuous in each variable.
(2) The n-inner product is continuous in the first two variables.
Proposition 2.5. If (xk ) converges strongly (resp. weakly) to x and x′ , then
x = x′ .
Note that strong convergence implies weak convergence.
Proposition 2.6. If (xk ) converges strongly to x, then it converges weakly to x.

A.L. Soenjaya

82

Proof. Refer to [6].



However, the converse is not true in general. The following highlights some
of the way a sequence can fail to converge strongly.
Example 2.7. Let X = L2 [0, 1] which is a Hilbert space with the usual inner
product. Equip X with the standard 2-inner product. Define a sequence (fn ) by
fn (x) = sin nπx. Then for all g, h ∈ X,
hfn , g|hi = hfn , gihh, hi − hfn , hihh, gi
 Z
Z 1

 

2

g(x) sin nπx dx khk2  + 
≤

0

0

1




h(x) sin nπx dx khk2 kgk2 

so that fn ⇀ 0, where we used the Riemann-Lebesgue lemma. However,

1/2
kfn , hk = kfn k22 khk22 − hfn , hi2

As n → ∞, kfn , hk → √12 khk2 , which is not zero as long as h 6= 0 a.e., showing
that fn does not converge strongly to the zero function.
Example 2.8. Let (X, h·, ·i) be a separable infinite-dimensional Hilbert space with
(ek )∞
k=1 as an orthonormal basis. Equip X with the standard n-inner product. In

[6], it is proven that (ek ) converges weakly, but not strongly to 0.
Remark 2.9. More generally, if X is a separable Hilbert space and {φk } is an
orthonormal sequence in X. Then φk ⇀ 0 in the induced standard n-inner product.
Example 2.10. Let X = L2 (R), equipped with the standard 2-inner product. Define a sequence (fn ) by fn (x) = χ(n,n+1) (x), where χ is the characteristic function.
Then one can check that (fn ) converges weakly, but not strongly, to zero in X.
Remark 2.11. In [6], it is observed that in standard, finite-dimensional n-Hilbert
spaces, the notions of strong and weak convergence are equivalent.
We will now give an extension of Radon-Riesz property for n-Hilbert space.
Theorem 2.12. If xk ⇀ x, then
kx, z2 , . . . , zn k ≤ lim inf kxk , z2 , . . . , zn k
k→∞

(5)

If, in addition,
lim kxk , z2 , . . . , zn k = kx, z2 , . . . , zn k

k→∞

for all z2 , . . . , zn ∈ X, then xk → x.

Proof. Using weak convergence of (xk ) and Cauchy-Schwarz inequality,
kx, z2 , . . . , zn k2 = hx, x|z2 , . . . , zn i = lim hx, xk |z2 , . . . , zn i
k→∞

≤ kx, z2 , . . . , zn k lim inf kxk , z2 , . . . , zn k
k→∞

proving (5). Next, by expanding the n-norm,
kxk − x, z2 , . . . , zn k2 = kxk , z2 , . . . , zn k2 − 2hxk , x|z2 , . . . , zn i + kx, z2 , . . . , zn k2 → 0

On Strong and Weak Convergence

using the assumptions given. Hence xk → x.

83



Next, we give an analogue of Banach-Saks-Mazur theorem for the case of
n-Hilbert spaces.
Theorem 2.13. If xk ⇀ x in X and
m
1 X
kxi − x, z2 , . . . , zn k2 = 0
lim
m→∞ m2
i=1

(6)

for all z2 , . . . , zn ∈ X, then there exists a sequence (yk ) of finite convex combinations
of (xk ) such that yk → x (strongly).
Proof. Replacing xk by xk − x, we may assume xk ⇀ 0. Pick k1 = 1 and choose
k2 > k1 such that hxk1 , xk2 |z2 , . . . , zn i ≤ 1 for all z2 , . . . , zn . Inductively, given
k1 , . . . , km , pick km+1 > km such that
1
1
|hxk1 , xkm+1 |z2 , . . . , zn i ≤ , . . . , |hxkm , xkm+1 |z2 , . . . , zn i ≤
k
k
which is possible since by the weak convergence of (xk ), hxki , xk |z2 , . . . , zn i → 0 as
k → ∞ for 1 ≤ i ≤ m. Let
1
ym := (xk1 + . . . + xkm )
m
Then we have
m
m j−1
2 XX
1 X
2
2
hxni , xnj |z2 , . . . , zn i
kxni , z2 , . . . , zn k + 2
kym , z2 , . . . , zn k = 2
m i=1
m j=1 i=1
≤

m
m j−1
1 X
2 XX 1
2
,
z
,
.
.
.
,
z
k
+
kx
2
n
n
i
m2 i=1
m2 j=1 i=1 j − 1

=

m
2
1 X
kxni , z2 , . . . , zn k2 +
m2 i=1
m

so that ym → 0 strongly as m → ∞ as required.



Corollary 2.14. Let X be an Hilbert space equipped with the standard n-inner
product. Suppose xk ⇀ x in X and kxi k < M for all i, where M is a constant and
k · k is the norm induced by the inner product on X. Then there exists a sequence
(yk ) of finite convex combinations of (xk ) such that yk → x (strongly).
Proof. It suffices to check that (6) holds in this case. Clearly, kxi − xk2 is also
bounded in norm, say kxi − xk2 < M ′ . By Hadamard’s inequality,
m
M ′ kz2 k2 . . . kzn k2
1 X
2
kx
−
x,
z
,
.
.
.
,
z
k
≤
→0
i
2
n
m2 i=1
m
as m → ∞ for all z2 , . . . , zn ∈ X, hence the statement is proven.



A.L. Soenjaya

84

3. APPLICATIONS
In this section, we apply the theorems deduced earlier to L2 -space, L2 (X, µ),
where (X, µ) is a measure space, equipped with the usual inner product
Z
f (x)g(x) dµ(x)
hf, gi =
X

2

We then equip L (X) with the standard n-inner product. Note that when n = 1,
the following reduce to the familiar cases. Subsequently, |A| = det(A).
Proposition 3.1. Let fk ∈ L2 (X, µ), k = 1, 2, . . ., be such that
 R
R
R
2

X
 R X fk dµ
R fk h2 2 dµ · · · RX fk hn dµ

h dµ
···
h h dµ
X 2
X 2 n
 X h2 fk dµ
lim 
..
..
..
.
..
k→∞ 
.
.
 R
R
R .2

h f dµ X hn h2 dµ · · ·
h dµ
X n
X n k
for all h2 , . . . , hn ∈ L2 (X, µ). Then
 R
R

X
 RX fk g dµ
R fk h2 2 dµ

h
g
dµ
h dµ
2
X 2
 X
lim 
..
..
k→∞ 
.
.
 R
R

h g dµ X hn h2 dµ
X n
2

for all g, h2 , . . . , hn ∈ L (X, µ).

···
···
..
.
···

R
RX fk hn dµ
h h dµ
X 2 n
..
R .2
h dµ
X n






 = 0








 = 0




Proof. This follows from Proposition 2.6.



Proposition 3.2. Let fk ∈ L2 (X, µ), k = 1, 2, . . ., be such that
 R
R
R

(f − f )g dµ X (fRk − f )h2 dµ · · ·
(f − f )hn dµ
XR k
XR k

2

h
g
dµ
h
dµ
·
·
·
h h dµ
2
X
X 2
X 2 n

lim 
..
..
..
.
..
k→∞ 
.
.

R
R
R .2

h g dµ
h h dµ
···
h dµ
X n
X n 2
X n

for all g, h2 , . . . , hn ∈ L2 (X, µ). Then
 R
R
R

2


 R X f dµ

RX f h22 dµ · · · R X f hn dµ



h
f
dµ
dµ
·
·
·
h
h
h
dµ
X 2
X 2 n
 X 2




..
..
.
.
..
.


.
.
 R

R
R .2


h f dµ X hn h2 dµ · · ·
h dµ
X n
X n
 R
R
2

X
 R X fk dµ
R fk h2 2 dµ

h
f
dµ
h dµ
2
k
X 2
 X
≤ lim inf 
..
..
k→∞ 
.
.
 R
R

h f dµ X hn h2 dµ
X n k
2

for all h2 , . . . , hn ∈ L (X, µ).

···
···
..
.
···






 = 0




R
RX fk hn dµ
h h dµ
X 2 n
..
R .2
h dµ
X n











On Strong and Weak Convergence

85

If, in addition,
 R
R
R

2


X
 R X fk dµ
R fk h2 2 dµ · · · RX fk hn dµ 



h
f
dµ
dµ
·
·
·
h
h
h
dµ
2
n
2
k
X 2
X
 X

lim 


..
..
.
.
..
.
k→∞ 

.
.
 R

R
R .2


h f dµ X hn h2 dµ · · ·
h dµ
X n k
X n
 R
R
R
2

 R X f dµ
RX f h22 dµ · · · R X f hn dµ

h
f
dµ
h dµ
···
h h dµ
2
X 2
X 2 n
 X
= 
..
..
..
.
.

.
.
.
 R
R .2
R

h f dµ X hn h2 dµ · · ·
h dµ
X n
X n

for all h2 , . . . , hn ∈ L2 (X, µ), then
 R
R
2

(f − f )h2 dµ
X Rk
 R X (fk − f ) dµ

h
(f
−
f
)
dµ
h2 dµ
2
k
X
X 2

lim 
..
..
k→∞ 
.
.
 R
R

h
(f
−
f
)
dµ
h
h dµ
X n k
X n 2

···
···
..
.

R

(f
XR k

− f )hn dµ
h h dµ
X 2 n
..
R .2
h dµ
X n

···

for all h2 , . . . , hn ∈ L2 (X, µ).















 = 0




Proof. This follows from Theorem 2.12.



Proposition 3.3. Let fk ∈ L2 (X, µ), k = 1, 2, . . .,
 R
R

X
 RX fk g dµ
R fk h2 2 dµ · · ·

h
g
dµ
h dµ
···
2
X 2
 X
lim 
..
..
..
k→∞ 
.
.
.
 R
R

h
g
dµ
h
h
dµ
·
·
·
X n
X n 2
2

for all g, h2 , . . . , hn ∈ L (X, µ), and there exists a
 R
R
2

X
 R X fk dµ
R fk h2 2 dµ
m
h
f
dµ
h dµ
2
k
1 X 
X
X 2

lim

..
..
m→∞ m2

.
.
k=1 
R
R

h
f
dµ
h
h dµ
X n k
X n 2

Proof. This follows from Theorem 2.13.

X

n

constant M such that
R

· · · RX fk hn dµ 

···
h h dµ 
X 2 n

 = 0
..
..

.

R .2

···
h dµ

for all h2 , . . . , hn ∈ L2 (X, µ). Then there exists a
combinations of (fk ), such that
 R 2
R

X
 R X gk dµ
R gk h2 2 dµ · · ·

h
g
dµ
h dµ
···
X 2
 X 2 k
lim 
..
..
..
k→∞ 
.
.
.
 R
R

h
g
dµ
h
h
dµ
·
·
·
n
k
n
2
X
X
for all h2 , . . . , hn ∈ L2 (X, µ).

be such that
R


RX fk hn dµ 

h h dµ 
X 2 n
 = 0
..


R .2

h dµ

X

n

sequence (gk ) of finite convex

R
RX gk hn dµ
h h dµ
X 2 n
..
R .2
h dµ
X n






 = 0





A.L. Soenjaya

86

Remark 3.4. Note that we also have another equivalent formula for the standard
n-inner product and n-norm in L2 (X) as follows (see [3, 9])
Z
Z Z
1
det(F ) det(G) dµ(x1 ) . . . dµ(xn )
...
hf, g|h2 , . . . , hn i =
n! X X
X
|
{z
}
n times

where

and



f (x1 )
f (x2 ) · · ·

 h2 (x1 ) h2 (x2 ) · · ·

det(F ) = 
..
..
..

.
.
.

 hn (x1 ) hn (x2 ) · · ·



g(x1 )
g(x2 ) · · ·

 h2 (x1 ) h2 (x2 ) · · ·

det(G) = 
..
..
..

.
.
.

 hn (x1 ) hn (x2 ) · · ·









hn (xn ) 
f (xn )
h2 (xn )
..
.









hn (xn ) 
g(xn )
h2 (xn )
..
.

The standard n-norm is therefore

1/2
Z
Z Z
1

kf, h2 , . . . , hn k = 
. . . [det(F )]2 dµ(x1 ) . . . dµ(xn )
 n!

X
}
| X X{z
n times

The above propositions hold accordingly using this form of n-inner product and
n-norm.

Similarly, we can get other results concerning weak and strong convergence
in other n-Hilbert spaces. For instance, we mention the Sobolev space W s,2 (Ω) =
H s (Ω), which is a Hilbert space with inner product
Z
s Z
X
f (x)g(x) dx +
Di f (x) · Di g(x) dx
hf, gi =
Ω

i=1

Ω

s

We can equip H (Ω) with the standard n-inner product (1), and all the above
convergence results will hold accordingly.
REFERENCES
1. Chu, H.Y., Park, C., Park, W., ”The Aleksandrov problem in linear 2-normed spaces”, J.
Math. Anal. Appl. 289 (2004), 666–672.
2. Chu, H.Y., Ku, S.H., Kang, D.S., ”Characterizations on 2-isometries”, J. Math. Anal. Appl.
340 (2008), 621–628.
3. Chugh, R., Lather, S., ”On reverses of some inequalities in n-inner product spaces”, Int. J.
Math. Math. Sci. (2010), Article ID 385824.
4. Gähler, S., ”Lineare 2-normietre räume”, Math. Nachr. 28 (1965), 1–43.
5. Gozali, S.M., Gunawan, H., Neswan, O., ”On n-norms and n-bounded linear functionals in a
Hilbert space”, Ann. Funct. Anal. 1 (2010), 72–79.

On Strong and Weak Convergence

87

6. Gunawan, H., ”On convergence in n-inner product spaces”, Bull. Malays. Math. Sci. Soc. (2)
25 (2002), 11–16.
7. Gunawan, H., ”A generalization of Bessel’s inequality and Parseval’s identity”, Per. Math.
Hungar. (2002), 177–181.
8. Gunawan, H., Mashadi, ”On n-normed spaces”, Int. J. Math. Math. Sci. 27 (2001), 631–639.
9. Gunawan, H., ”The space of p-summable sequences and its natural n-norms”, Bull. Australian
Math. Soc. 64 (2001), 137–147.
10. Misiak, A., ”n-inner product spaces”, Math. Nachr. 140 (1989), 299–319.
11. Soenjaya, A.L., ”On n-bounded and n-continuous operator in n-normed space”, J. Indonesian
Math. Soc. Vol 18 No. 1 (2012), 45–56.
12. White, A., ”2-Banach spaces”, Math. Nachr. 42 (1969), 43–60.