1.1 Some basic definitions
Linear space: Let X be a non-empty set whose elements are called vectors. Let F be any set
whose elements are called scalars where (F, +, ·) is a field. The set X is said to be a linear
space(ls) if
1) there is defined an internal composition in X called addition of vectors denoted by +, for
which (X, +) is an abelian group.
2) there is defined an external composition in X over F, called the scalar multiplication in
which aα X for all α X and a F. 3 the above two compositions satisfy the following postulates i a α +β) = aα + aβ
ii) (a + b) α = aα + bα
iii) (ab) α = a(bα)
iv)1 α = α
for every α, β X and a, b F and 1 is the unit element of F.
Almost linear space (als): An almost linear space(als) is a non empty set X together with
two mappings s: X x X → X and m:
For x , y
X and α
Let x, y, z
x X → X satisfying (i) – (viii) below.
we denote s(x ,y) by x + y and m(α , x ) by α x
X and α , β
i)
(x +y) + z = x + (y +z)
ii)
x +y = y +x
iii)
There exists an element 0
iv)
α ( x + y) = αx + α y
v)
(α +β ) x = αx + βx for α ≥0, β ≥0
vi)
α(β x ) = αβ ( x)
vii)
1x = x
viii)
0x = 0
X such that x+ 0 = x for each x
1 X.
Almost linear subspace: A non-empty set Y of an almost linear space X is called an almost
linear subspace of X, if for each a1, a2
For an almost linear space
Y and α
, s(a1, a2)
Y and m(α , a1)
Y.
we introduce the following two sets
:
0
:
Basis of an almost linear space: A subset B of an almost linear space X is called a basis of X
if for each x
X \ {0} there exist unique sets {b1,…,bn} ⊂ B and {α1,... ,α n} ⊂
(n depending on x) such that x =
i
bi (i=1,…,n) where α i > 0 for bi ∉ VX.
Almost linear functional: Let X be an almost linear space. A function f : X →
almost linear functional if f satisfies the following conditions. For x, y
i)
f(x+y) = f(x) +f(y)
ii)
f(α x) = α f(x)
iii) -f(-x) ≤ f(x) (or) f(w)
\ {0}
0 for every w
is called an
X and α (≥0)
.
The set of all almost linear functionals defined on an almost linear space X is denoted by X#.
Definition: Let X be an almost linear space. An almost linear sub-space Y # of X # is said to
be total over X if the relations x, y
X and f(x) = f(y) , for each f
Y# implies x = y .
Norm on a linear space: A norm || · || on a linear space X is a function satisfying the
following properties.
i)
|| x || > 0 if x
0
ii)
|| αx || = | α | ||x||
iii)
|| x – y ||
| ||x|| - ||y|| |
for x, y X and α is any scalar.
Normed linear space (nls): A linear space X together with || · || : X→
above properties i) – iii) is called normed linear space.
2 satisfying the
Norm on an almost linear space: A norm ||| · ||| on an almost linear space X is a function
satisfying the following conditions N1 – N3.
Let x, y, z
X and α
.
N1. |||x ||| = 0 if and only if x=0.
N2. |||α x ||| = |α | |||x |||
N3. |||x - z |||≤ ||| x – y ||| + |||y - z |||
Normed almost linear space (nals): An almost linear space X together with |||. ||| : X →
satisfying N1 – N3 is called a normed almost linear space .
Semi-metric: ρ is a semi-metric on a normed almost linear space X if it satisfies the
following properties.
For x ,y, z
X, v
VX and ,
0
,
1) ρ(x, v) = ||| x- v |||
2) ρ( x+ z, y + z) = ρ(x, y)
3) ρ( x,
y) = | | ρ(x, y)
4)| ||| x ||| - ||| y ||| | ≤ ρ(x, y) ≤ ||| x – y |||
5) lim{ρ( x, x)} = ρ(
0
x, x) when α→ 0
Metric: A metric on normed almost linear space d: X x X →
is defined as
d(x, y) = ||| vx - vy + ∑ (αi – βi) bi ||| , i=1 to n , x, y
VX, bi
X , vx , vy
B\ VX , αi , βi ≥ 0.
Strong normed almost linear spaces (snals): A strong normed almost linear space is a
normed almost linear space X together with a semi-metric ρ on X which satisfies the
following conditions.
For x, y, z
X and α
i)
||| x||| - ||| y ||| ≤ ρ( x, y) ≤ ||| x- y |||
ii)
ρ (x+z , y+z ) ≤ ρ (x , y)
iii)
The function α → ρ (α x , x) is continuous at α = 1
3 The dual space: Let X*= { f
X #: ||| f ||| < ∞}, then the space X * together with ||| . |||
defined by ||| f ||| = sup { |f (x)| : ||| x ||| ≤ 1} is called the dual space of the normed almost
linear space X.
Bi-dual space: The dual space of the dual space X * is called bi-dual space or second dual
space of X and is denoted by X **.
Convex: A subset A of an almost linear space X is said to be convex if for every a, b
A,
a ≠ b, the line segment [a, b] is a subset of A.
Strictly convex: The normed linear space E is said to be strictly convex with respect to its
linear subspace G if the conditions x, y
E, || x ||=|| y ||= ||(x + y)/2||=1, x-y
G
implies that x = y.
Cone: A subset K of an almost linear space X is called a cone with vertex xo if K contains
every point xo + r (x – xo), r > 0, where ever it contains x.
Property (P). The convex cone C has property (P) in X if the relations x,y
c
X, x+y
C and
C imply that max{|||x|||,|||y|||} ≤ max{|||x+c|||,|||y+c|||}
If C1 , C2 are convex cones of X, C1
C2 and C2 has property (P) in X then C1 also has
property (P) in X . Clearly the cone C = WX has property (P) in X.
Almost linear operator: A mapping T: X → Y is called an almost linear operator
with respect to C if the following conditions hold.
For x, x1 , x2
X and λ(≥0)
i)
T(x1 + x2 ) = T(x1)+T(x2) ,
ii)
T(λx) = λ T(x) and
iii)
T(WX) ⊂ C.
We denote the set of all such ‘T’ by L(X,(Y,C)).
For every normed almost linear space X, there is a natural map F: X → X ** such that
F(x) ( f ) = f(x), for every x
X and for every f
F(x) : X* → .
4 X * where f : X → , F(x)
X** where
Reflexive: The normed almost linear space X is called reflexive when the natural map
F: X → X ** is an isomorphism.
Definition: Let X be a normed almost linear space, over the real field
and G a nonempty
subset of X. For a bounded set A ⊂ X let us define
radG (A) = inf g
centG (A) = { g0
G
{ supa
||| a- g|||}
A
G: supa
A
||| a- g0 ||| = radG (A)}
Here the number radG (A) is called the chebyshev radius of A with respect to G, centG (A) is
called the chebyshev center of A with respect to G and an element g0 is called a best
simultaneous approximation of A with respect to G.
Definition: When A is singleton, say A = {x}, x
X , then radG (A) is the distance
of x to G, denoted by dist (x, G) and defined by dist (x, G) = inf g
G
||| x- g||| and
centG (A) is set of all best approximations of x out of G, denoted by PG (x) and is
defined by PG (x) = { g0
G:||| x- g0 |||= dist (x, G)}.
Definition: Let X be a normed almost linear space. The set G is said to be proximinal if PG(x)
is nonempty for each x
: Let
X\VX.
be a normed almost linear space and let
If for each g
the following way.
there exists
. we define
such that the following
conditions are hold.
i)
|||
ii)
|||
g||| = |||
|||
||| for each
, then
.
We denote the set
When
by
.
is a normed linear space, then
: Let
subset .
.
We have
If
g|||
|||
be a nals and
.
. We shall assign to each
in the following way. For g
(i) and (ii) of definition of
}. Since
let
, the set
a non-empty
:
Satisfying
is non-empty.
Complete metric space: A metric space X is called complete if every Cauchy sequence of
points in X has a limit point that is also in X.
5 in
Compact set: A set S of real numbers is called compact if every sequence in S has a
subsequence that converges to an element again contained in S.
Upper semi-continuous: We say that f is upper semi-continuous at x0 if for every ε > 0 there
exists a neighbourhood U of x0 such that f(x) ≤ f(x0) + ε for all x in U when f(x0) > -∞, and f(x)
tends to -∞ as x tends towards x0 when f(x0) = -∞.
Lower semi-continuous: We say that f is lower semi-continuous at x0 if for every ε > 0 there
exists a neighbourhood U of x0 such that f(x) ≥ f(x0) – ε for all x in U when f(x0) < +∞, and
f(x) tends to +∞ as x tends towards x0 when f(x0) = +∞.
Convex hull: A set of points is defined to be convex if it contains the line segments
connecting each pair of its points. The intersection of all convex sets containing X is called
the convex hull of a given set X.
Banach space: A Banach space is a vector space X over the field R of real numbers, or
over the field C of complex numbers, which is equipped with a norm and which
is complete with respect to that norm, that is to say, for every Cauchy sequence {xn} in X,
∞.
there exists an element x in X such that lim xn = x as n
Hausdorff space: We say that X is Hausdorff space if for every pair x, y
x
y there exists nbds U and V of x and y respectively such that U
Uniformly Kadec-Klee
0 such that the relations
there exists a δ
||
: The norm of a Banach space
||
Definition: Let
, imply that,||
: x g ,g
g ,
Definition: Let
property
g ||| have the property
of
such that for every | |
and
be a nals,
,
g ,
δ
δ
if it has property
0 there is a δ
is said
0 and a
g ,g ,
0 there is a δ
g,
such that δ
.
,
. The pair
. The pair
δ we have that
,
we have
and
and every
0,
if for every
1,2 …
g ,
6 is
δ.
1
||
0 and every
if for every
if for every
with |||
|| 1, V= .
be a normed almost linear space and
to have the property
function
, ||
X such that
,
.
is said to have the
0 such that for every g
. The pair
,
is said to
0 can be chosen independently
1.2 Brief survey of the work:
In recent days the research work on abstract algebra is very interesting and also very
useful to develop the subject algebra. We know the concepts linear space (Vector space) and
also normed linear space very well. By extending these concepts, some authors like S.
Gahler, Y.J. Cho, C. Diminnie, R. Freese, G. Godini, Sang Han Lee, Geetha S.Rao,
T.L. Bhaskaramurthi, Sang Mo Im, Kil-Woung Jun and many others developed new concepts
like linear 2-normed space, almost linear space and normed almost linear space.
R. Freese and S.Gahler [1] initially investigated the concept of linear 2-normed space
and has been extensively by Y.J. Cho, C. Diminnie, R. Freese and many others. G. Godini [24] introduced the concepts almost linear space, almost linear sub space, the special subsets
WX and VX of an almost linear space X, basis of an almost linear space, normed almost linear
space and strong normed almost linear space. G. Godini [5] established the property (P) in
normed almost linear space and introduced the concept almost linear operator and proved
some results relating to cones with property (P) and almost linear operators. G. Godini [3]
introduced the concepts chebyshev radius, chebyshev center and also an almost linear
functional on an almost linear space and established many results related to the best
approximation in normed almost linear space.
Sang Han Lee [7] proved that if a normed almost linear space X is reflexive then
X= WX +VX . He also proved that if X has a finite basis then X= WX + VX if and only if X is
reflexive. Sung Mo Im and Sang Han Lee [8] proved that the cardinality of bases of an almost
linear space is unique and also he established some results of bases of an almost linear space.
Sung Mo Im and Sang Han Lee [9] introduced a metric induced by norm on normed almost
linear space and established some results on metric induced by norm. Sung Mo Im and Sang
Han Lee [10], established a characterization of reflexivity of normed almost linear space in
which they proved that for a split normed almost linear space X= WX +VX , X is reflexive if
and only if WX and VX are reflexive. S. Elumalai, Y.J. Cho and S.S. Kim 11] introduced the
concept of linear 2- normed space and proved some of the results of best approximation sets
in linear 2-normed space. Geetha S. Rao and T.L. Bhaskeramurthi [12] extended some results
from the theory of best approximation in a normed linear space into a normed almost linear
space. Sang Han Lee and Kil-Woung Jun [13] describes some important results relating to a
metric on normed almost linear space. Sung Mo Im and Sang Han Lee [14] established the
completeness of a normed almost linear space B(X,(Y,C)). S.S. Dragomir [15] had given
7 some new characterization of best approximation in normed linear space. G. Apreutesei [16]
introduced the hyperspacial topologies on almost linear space.
S. Elumalai and R. Vijayaragavan [17] established some of the results of the best
simultaneous approximation in the context of linear 2-normed space. G. Apreutesei,
Nikoas E. Mastorakis, Anca Croitoru and Alina Gavrilut [18] deal with almost linear
topological space. S. Elumalai, R. Vijayaragavan [19] established some characterization of
best approximation in terms of 2-semi inner products and normalized duality mapping
associated with a linear 2-normed space. A. Khorasani and M. Abrishami Moghaddam [20]
studied the best approximation in probabilistic 2-normed spaces. Y. Dominic and M. Marudai
[21] proved some results on b-best approximation in uniformly convex 2-normed spaces.
8