Rev.Acad.Canar.Cienc., XI (Núms. 1-2), 29-40 (1999)
TENSOR PRODUCT AND 1-TWO PARANORMED STRUCTURES
D. K. Bhattacharya and T. Roy
ABSTRACT : The papar generalizas the idea of tensor pr9duct of two normed linear
spaces to that of two 1-paranormed spaces relativa to a given 1-paranormed space and
discusses different types of 1-paranorms which generate different types of 1-two paranorm
structures on the tensor product space. lt also considers the tensor product of a Banach
algebra and a complete 1-paranormed algebra relativa to a given complete 1-paranormed
algebra and shows that, in this case, the tensor product is a 1-two paranormed algebra.
Keywords : Tensor Product, 1-two Paranormed space, 1-two Parenormed algebra.
1. INTRODUCTION : The distinction between a ho·mogeneous norm and a
nonhomogenous norm was made by W. Orlicz (1950); he callad them a B-norm anda
F-norm respectively. Later on, B. Chaudhury and S. Nanda (1991) called such a
nonhomogeneous norm a paranorm. A Taylor ( 1958) defined a metric linear space and L.
Collatz ( 1966) defined a supermetric space. Moreover, they showed that every metric linear
space ora supermetric space (X, p) was such that p could induce such paranorm p on the
linear space x given by p(x) = p( x, 9) , 9 being the additive identity of X. D. K. Bhattacharya
and T. Roy (1997) defined a 1-paranormed space anda 1-paranormed algebra and studied
sorne properties of such an algebra. They also showed that every 1-paranormed space
was a paranormed space but the converse was not true. Also, it followed from the
definitions that a normed linear space was not a 1-paranorm space. As normed linear
spaces were obviously paranormed spaces, so it was remarked that normed linear spaces
and 1-paranormed spaces formad two distinct subsets of the set all paranormed spaces.
This necessitated further study of such 1-paranormed spaces and algebras.
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The idea of a two normed space was introduced by W. Orlicz (1950) and the
importance of such a space was subsequently highlighted by A. Alexiewicz and Z.
Semadeni [1954, 1958, 1960) and several others.
The idea of a two normed algebra was given by R. K. Srivastava (1990) while
studying the space of entire Dirichlet series. D. Bhattacharya and S. Manna (1997) defined
a two paranormed space and different types of two paranormed algebras and cited
examples from the set of entire Dirichlet series with different exponents.
Extensive work was done by R. Shattern ( 1950), A. Grothendieck ( 1955),
B. Gelbaum (1959), F. Bonsall and J. Duncan (1973) and many others on the tensor
product of two Banach spaces and on that of two Banach algebras. In the latter case, out
of the three possible types of norms on such a space, only one norm viz. projective norm
was an algebra norm. As in the definition of a two normed algebra, both types of norms
were to be algebra norms, so it remained open to investigate whether the tensor product
of two Banach algebras was a two normed algebra.
In the present paper, we answer the question in the affirmative by considering
the tensor product of a Banach algebra and a complete 1-paranormed algebra relativa to
another complete 1-paranormed algebra.
Throughout the paper, we use the definitions and examples as given in [4].
2.SOME NEW DEFINITIONS ANO EXAMPLES:
DEFINITION 2.1. Let X be a linear space equipped with two 1-paranorms p and p" where
p" is weaker than p. A sequence { x,,} e X is said to be 'Y -convergent to x0 e X , if there
exists k >O such that sup P(x,,) ~ k < oo and p"(x,, - Xo) ~O as n ~ 00 •
n
DEFINITION 2.2. A linear space x equipped with two 1-paranorms p and p" ( p" being
weaker than p) is callad a 1-two paranormed space if convergence of a sequence of x
means its v -convergence. lt is denotad by ( x, p, ·p·) and is abbreviated as 1-TPS.
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DEFINITION 2.3. A sequence {x,,} of a 1-TPS (x, p, p·) is called a Cauchy sequence if
there exists k > O such that sup p( x,,) s k < "° and if corresponding to e > o , arbitrary
n
small, there exists a positive integer m such that p'(><i. -xq) <e, for V p , q;::: m.
A 1-TPS is called t} -complete if every Cauchy sequence is t} -convergent in
the space.
EXAMPLE 2.1. The class of all real sequences x = {x,,} with componentwise addition
and scalar multiplication is a 1-TPS if p and p' are defined as p(x) = ~ 1 ~~.I and
p"(x) = i, _!_ _[3l_ . Other examples of 1-TPS are considered in the next two articles .
... 1 2" (1 + lx..D
3. BOUNDED LINEAR MAPS AND 1-TWO PARANORM SPACES
DEFINITION 3.1 . Let (X, P.). (Y, P,) be two 1-paranormed spaces over the field F. A
linear map T : X -+•Y is said to be bounded if P, ( T (p.~x) )) s k, V xe X, x* 0
(null element of X)
PROPOSITION 3.1 . Let B(X, Y) denote the set of all bounded linear maps
T : (X, P.)-+(Y, P,), Let
(a) p(T) = infk such that P, ( T (p.~x)))sk, V XEX, X*ª·
(b) p(T)=SU~ {p,(r ( p,~xJJJ. xeX , ue}
Then in both the cases pis a 1-paranorm on B(X,Y). Moreover these two p's are same.
PROOF. In order to show that (a) and (b) give 1-paranorms of T, we simply verify that in
both the cases.
p (a T) s ¡aj p(T) , laJ;::: 1
jaj p(T)< p(a T)< p(T), jaj < 1 ,aeF
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We have,
Pv( a T (p.(x))J<I~ Pv( T (P.Cx))J. x*e. it l~>l
<l~kwherepv(T(P.Cx))J s; k, 'v'x*e.
So Pv( a T (p.(x))J < 1a:1 k for sorne suitable 1a:1 <lo!·
and hence for I~ > l , P( a T) = inf 1 a' 1 k = 1 a' 1 inf k
=I a' 1 P(T) < I~ P(T)
Again for I~ = l , P( a T) = P(T) .
So, P( a T) s; I~ P(T) , 1~ ~ l.
Againl~Py(T(P.Cx))J < Pv(aT(p.(x))J < ~y(T(p.Cx))J. I~< l , 'v'x*ª ·
So, Pv(aT(P.Cx))J s; k' < k, ifpv(T(P.Cx))J s; k, 'v'x*9 .
Hence, p( a. T) = inf k' < inf k = P(T) ( when I~ < I) ·
Thus when I~ < I , P(a T) < P(T).
Againl~Pv(T(P.Cx))J < Pv(aT(P.Cx))J
=> I~ P,[ T (p.~x)lj : ;n1 k' '""' that P,[a T [p.(x)]) < k'
=> l~Pv(T(p.(x)) - P(aT)
=> Pv(T(P.Cx))J s; /~ P(aT), vx*e(a*o)
=> P(T) = inf (1~ P(a T)) < /~ p(a T)
i.e, I~ P(T) < P( a T) < P(T) when I~ < l .
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So, pis a 1-paranorm defined by (a). Further, for the case (b) we have,
~cxT) = s~p Py[cxT((Px(x)))] $1~ ~p Py(T(Px(x))) ~I~ ~ 1
Alsol~ s~p Py[r(Px(x))] < s~p Py[cxr(Px(x))] < s~p Py(r(Px(x))Jitl~ < 1
So p(cx T) $ I~ p(T) if I~ ~ l.
I~ p(T) < P( ex T) < p(T) if I~ < l.
Thus pis a 1-paranorm for the case (b) also.
To show that the above two 1-paranorms are same,
let us denote the right hand sides of (a) and (b) by M1 and M2.
Obviously, M2 ~ M1. Again, from (b) it follows that
Py( T (Px(x)) J ~ Mi, so that from (a) we have M2 ~ M1.
Hence M1 = M2 = p(T).
PROPOSITION 3.2. p·(r) = sup {Py (T (x)) ; Px(x) ~ 1} is a 1-paranorm on B(X, Y) and
p•(T) ~ P(T) , 'v' Te B (X, Y)
PROOF : p·(r) is obviously a 1-paranorm on B (X, Y). Nowwe show that p·(r) ~ P(T) .
First we see that S = {x e X/ Px(x) ~ 1}
e S1 = ({o} u {Px(x)}, x*O, x e x)
In fact, for those x which satisfy Px(x) ~ 1,
p.(Px(x)) $ Px~x) . Px(x) = 1 ' X* 9
X .
Butall x e X, x = -(), x e X , x*e maynotsatisfy Px(x) ~ 1;
Px X
In fact, for these x = ~() for which Px(x) > 1, Px X
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1 < p,( x') = p,( p,(x)) < p,(x) .
This implies that p,(x') can nevar be less than or equal to 1.
So pº(T) ~ p(T) where p(T) is a 1-paranorm given by (b).
PROPOSITION 3.3. Let B be the set of all bounded lir'tear transformations
T : (X , p,) -+ (Y , Py) where (X , p,) and (Y , py) are both 1-paranormed spaces.
Let pº(T) = s~p {Py(T (x)) , x E X. P,(x) ~ 1} and
p(T) = ·~+( T [p,(x) ll · X E X , x; ·)
Then (B. p, pº) is a 1-two paranormed space.
4. BOUNDED LINEAR MAPS ANO 1-TWO PARANORMED SPACES
DEFINITION 4.1. Let (X, p.), (Y, Py), (Z, p.)be three 1-paranomied spaces. Let 4> : X x Y-+ 2
be a bilinear map, 4> is said to be bounded if there exists
M >O such that P, 4> (p,(x) • Py~Y)) ~ M
V X E X' V y E y' X*0' Y*ª ·
The set of all such bounded bilinear maps is denotad by B (X. Y; z)
PROPOSITION 4.1.
Let X, Y, Z, 4> be given as in definition 4.1.
Then (8, p, pº) is a 1-two paranormed space, where
pº(4>) = sup {P,(4> (x,y)), P,(x) ~ 1, Ph) ~ 1}
'· y
P(•) = ',"d+ (p,(x) ' P,(Y)ll · x.O ' y;O)
are two 1-paranorms on (8, p, pº).
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5. TENSOR PRODUCT ANO 1-TWO PARANORMED ALGEBRA
5.1. TENSOR PRODUCT OF TWO 1-PARANORMED SPACES W.R.T. ANOTHER
1-PARANORMED SPACE
DEFINITION 5.1.
Let (X, P,), (Y, P). (Z, P.) be 1-paranormed spaces over F.
Let x. denote the set of all bounded linear maps from X to Z and let Y' denote the set of
all bounded linear functionals on Y Let 8 (x •. Y' ; Z) denote the set of all bounded bilinear
maps from x. x Y' --+ z. Given x e X, y e Y, let x®y denote an element of
B (X., Y' ; Z) defined by ( x ®y) (f, g) = g(x) f(y) for all 1-1 maps ~ e X~ and g e Y' .
Then (X® Y)., the algebraic tensor product relativa to Z, is defined by the linear span of
{x® y , x e x , y e Y} in 8 (X •. Y' ; Z) . When X and Y are both normed linear spaces
over F, the usual algebraic tensor product of X and Y follows by taking Z = F.
The following propositions now easily follow :
PROPOSITION 5.1.
Given u e X ® Y , there exists linearly independent sets {x;} e X , {Y,} e Y,
i = 1,2,3, ..... n, such that u = I x, ®y,.
i..:I
PROPOSITION 5.2.
lf u = I x, ®y, =O, where { x,} is a linearly independent set, then y,= O; i = 1,2, .. .. n.
i:I •
PROPOSITION 5.3.
lf { ><¡} ; i = 1,2, .... m and {Y¡}, j = 1,2, .... n be two linearly independent subsets of X and Y,
then {X;® y;} is a linearly independent subset of x ®y .
The above propositions lead to the following
THEOREM 5.1. Let X, Y, Z be three 1-paranormed spaces ovar F and let
'!' : (X® Y), --+X x Y be a bilinear map. Then corresponding to each bilinear map
4> : X x Y--+ z, there exists a unique linear map a : (X® Y), --+ Z such that 4> = aº'!' .
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PROPOSITION 5.4.
Let X, Z be two complete 1-paranormed algebras ovar F and let Y be a Banach algebra
ovar F. Let X, denote the set of all bounded linear maps from X to Z. Let
u= f x;®Y, E (X®Y), andw·, p·: (X®Y)--tR bedefinedas.
i=l
w"(u) = ~f (.~ 1g(Y;)1 p, (f (~)); 11g11 5 1, p,,(f) 5 1)
p"(u) = inf ( ;t, 11 Y; 11 s~p {p,(f (~)); P.,(f) s 1 , f E x,})
where the infimum is taken over all finite representations of u.
Then w· and p· are both 1-paranorms on (X® Y), and w"(u) s p" (u) , 7 u E (X 'Z Y j,.
PROOF: Obviously w· and p· are 1-paranorms on (X® Y),.
We simply verify that w"(u) s p"(u) .
Now f 1 <i,_y;) 1 p,(f(~)) s 11 911 f 11 Y. 11 P,(f(~))
1 = 1 t = 1
So, (t, 1 ~y;) 1 P,(f(~)) ; P., (f) s 1 , 11 g 11 s 1)
S 11911(t,11Y; 11 P,(f(~)); P,,(f) S 1, 11911 S 1)
S (;t, 11Y;11 P,(f(~)) ; P.,(f) S 1)
So, w"(u) = ~uf { ;t, 1 g(y;) 1 P,(f(x;)) ; 11 g 11 5 1 , P,.(f) 5 1 }
S { ¡~ 11 Y; 11 s~p (p,(f(~)), P.,(f) 5 1)}
Hence w"(u) s inf { ;~, 11 Y; 11 s~p (p,f(~)) ; P., (f) s 1 } = p" (u)
where the infimum is taken over all finita representations of u. Thu~ w"(u) s p" (u).
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PROPOSITION 5.5.
Let X, Y, Z, x. be defined as in proposition 5.4.
Let w(u) = sup [ f 1 g_y,) 1 Pz( f((~) )) ; 11 911 :::; 1 , Px, (f) :::; 1 , f :t:'0x,]
1. g , = 1 Pz f(~)
and P(u) = inf ¡.t , 11 y, lf ·~+lP.&~)) ll · P,(f) < 1 • f •• .. )
where infimum is taken over all finite representations of u. Then w and p are both
1-paranorms on (X® Y)z such that
(i) w(u) :::; P(u)' '<;/ UE X®Y.
Further (ii) w' (u) :::; w(u) and p'(u) :::; P(u) .
PROOF : (i) is proved as in prop. 5.4. and (ii) and (iii) followfrom definitions.
PROPOSITION 5.6.
Let X, Y, Z be defined as in prop. 5.4. Let x. denote the set of all bounded linear algebra
homomorphisms from X to Z.
Let p' be defined as in prop. 5.4. then ((X® Y)z, p') is a 1-paranormed algebra.
PROOF : We simply verify that p'(uv) :::; p'(u) p'(v) , '<fu , v e (X® Y)z.
Let u = f, x, ®y, ; v = f xi ®Y¡ .
i = 1 j = 1
m n
Then uv = L. L. ~ xi ®y, Y¡
p'(uv) = inf { ,t1 ¡ti 11 Y; Y¡ /1 s~p (Pzf( ~ xi) ; f E Xz, Px, (f) :::; 1)}
:::; inf { ;~ ¡ti 11 y, 1111 Y¡ 11 s~p (Pzf(~) f(x;)) ; Px,(f) :::; 1}
(as f is an algebra homomorphism)
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= pº(u) pº(v).
PROPOSITION 5.7.
Let X, Y, Z, f be defined as in prop. 5.6. and p be defined as in proposition 5.5.
Then ((X ® Y),. p) is a l•paranormed algebra.
PROOF : We verify that P(uv) ~ P(u) P(v) , V u, ve (X® Y),.
Now it follows from the definition of p(uv) that we are to consider only those
fe X, for which p,t( ~ xi) ~ l.
As P,(t(~ x¡)) 5 P,(t(~)) P,(t(xJ), so without any loss of generality, we may assume that
P,(f(x;)) = l , P,(t(xi)) = l .
Now p{uv) = ;nf [ fy 1 y, Y, 11 s~p •{{P,(~:'x,)) J; •Av,) < 1 · 1 ••., ]]
< ;n1 [ fy 1 y, y, 1 s~p [ [p,(1(~ x,)) · •A~ ~iJ]]
= inf ( ft 11 y, Yill)
5 inf (f1 11Y,1111Yi11)
= inf (f1 11y,11 /I Y¡ 11 s~p P,(f(~)) s~p P,(t(xi)); P.,(f) ~ 1)
[as s~p p,f()<j) = l = s~p P,(t(><¡))]
=in{~ 11 y, 11 s~p (p,f(x,)) ; P., (f) 51 , f E X,) x in{~ 11 yi 11 s~p p,f(xi) ; Px, (f) 51, f E X,)
= p"(u) pº(v) ~ P(u) P(v) .
This completes the proof.
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5.2. L-TWO PARANORMED ALGEBRA
DEFINITION 5.2.
A 1-two paranormed space (X, p, p") over F where X is also an algebra over F is called
a 1-two paranormed algebra (abbreviated as 1-TPA)
if V' x , y e X, p"(x y) s p"(x) p"(y) and P(x y) s p(x) p(y) .
From propositions 5.4. - 5.7. the following result readily follows :
THEOREM 5.2.
Let X, Z be two 1-paranormed algebras over F. Let Y be a normed algebra over F. Then
((X® Y) •. p, p") is a 1-two paranormed algebra over F where p·, p : (X® Y).--+ R are
defined respectively as p"(u) = inf (;*, 11 Y1 11 s~p [P.(f(x;)) , P., (f) s 1 , fe x. l)
~") = ;"' [t.11 ,, ~ ·~+(~;~))] · P •. <'l < , . , •• , J.
infimum being taken over all finite representations of u e (X® Y). , u f x¡ ®Y; .
i • l
CONCLUDING REMARKS :
1. A linear map ber...aen two 1-paranormed spaces is bounded if it is continuous. But the
converse is not true.
2. Unlike the space of bounded linear maps between two normed linear spaces, the
space of bounded linear maps between two 1-paranormed spaces is a two normed space
only because there exist 'two distinct nonequivalent norms in the latter case, which,
however, coincide in the former case.
3. The presence of two nonequivalent norms ultimately resultad in the introduction of two
distinct algebra 1-paranorms on the corresponding tensor product of 1-paranormed
algebras.
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