ROMANIAN JOURNAL OF INFORMATION SCIENCE AND TECHNOLOGY Volume 18, Number 1, 2015, 69–78 Approximation of K othe-Bochner Spaces L(X)in case Xhas a… [626315]

ROMANIAN JOURNAL OF INFORMATION
SCIENCE AND TECHNOLOGY
Volume 18, Number 1, 2015, 69–78
Approximation of K othe-Bochner Spaces
L(X)in case Xhas a Schauder Basis
Ion CHIT  ESCU, R azvan-Cornel SFETCU, Oana COJOCARU
University of Bucharest, Faculty of Mathematics and Computer Science
Str. Academiei 14, 010014, Bucharest, Romania
E-mail: {[anonimizat], [anonimizat],
[anonimizat] }
Abstract. We consider K othe-Bochner spaces L(X) (XBanach space,
function norm) in case Xhas a Schauder basis. The canonical nite di-
mensional subspaces YofXgenerate the K othe-Bochner spaces L(Y);which
approximate L(X):Numerical computations in case Xis a Lorentz sequence
space are performed. We present a general computation and a special compu-
tation for a particular case, comparing the procedures and noticing that the
special computation is more economic.
Key-words: K othe spaces, K othe-Bochner spaces, Schauder basis, Lorentz
sequence spaces, measurable vector functions.
2010 Mathematics Subject Classication. 46A45, 46B45, 46E30, 65B10.
1. Introduction
The history of K¨ othe spaces begins with the famous paper [ 3], in which G. K¨ othe
and O. Toeplitz introduced the ancestors of K¨ othe spaces, the so called ”gestufte
R¨ aume” (which were special sequence spaces). Subsequently, G. K¨ othe proved new
properties concerning these sequence spaces. Whitin the more general framework of
measure spaces, the Nederland school (especially A. C. Zaanen and W. A. J. Lux-
emburg) developped the theory and proposed the name ”K¨ othe spaces” for the more
general spaces of measurable functions studied by them. The K¨ othe spaces generalise
the Orlicz, Lorentz and Marcinkiewicz spaces.
Increasing generality, the K¨ othe-Bochner spaces appeared: they are K¨ othe spaces
of measurable functions taking values in Banach spaces X.These K¨ othe-Bochner

70 I. Chit escu et al.
spaces constitute the subject of this paper. More precisely, we consider the case when
the Banach space Xpossesses a Schauder basis. Within this framework, we can study
the approximation of the elements of a given K¨ othe-Bochner space with elements in
the canonical subspaces.
Concrete computations and comparisons concerning the speeds of the approxima-
tion processes are performed at the end of this paper, in the special case of functions
having their values in Lorentz sequence spaces.
The monographs used as theoretical basis are [ 1], [4] and [ 6]. For Functional
Analysis, see [ 2]. For supplementary results, see also [ 5].
2. Preliminary Facts
Throughout the paper Kwill be the scalar field (either K=RorK=C),N=
{1,2,3, …}will be the set of natural numbers, R+= [0,∞) andR+= [0,∞)∪ {∞} .
For any nonempty set T, any vector space X, any x∈Xand any f:T→K,we
can define fx:T→Xvia (fx)(t) =f(t)x.For any A⊂T, the function φA:T→R+,
acting via φA(t) = 0 ,ift /∈AandφA(t) = 1 if t∈Ais the characteristic (indicator)
function of A.Write P(T) ={A|A⊂T}.
For any Banach space X, we shall denote by X′the dual of X.
From now on, we assume that Xis a Banach space having a Schauder basis
(en)n≥1.The corresponding biorthogonal functionals are ( x′
i)i≥1⊂X′.For any n,
Pn:X→Xis the canonical projection defined via Pn(∞∑
i=1αiei) =n∑
i=1αiei.One
knows that ( Pn)n≥1is a bounded sequence, i.e. sup
n∥Pn∥o=M <∞,where ∥Pn∥o
is the usual operator norm. We call Mthe constant of the Schauder basis. In case
M= 1, we say that the Schauder base is monotone. For any function f:T→X,we
dene the functions fn:T→X, f n(t) = (Pn◦f)(t), which will be very important in
the sequel.
IfXis a Banach space consisting of numerical sequences (i.e. an element of X
has the form x= (xn)n≥1with xn∈K), in many cases one has:
a)en∈Xfor any n, where en= (0,0, …,0,1,0, …), with 1 on the n−thplace;
b) (en)n≥1is a Schauder basis for X.
In this case, we shall say that ( en)nisthe canonical Schauder basis for X.
This happens if X=lp,1≤p <∞. In this case, one can see that ( en)nis a
monotone Schauder basis.
The remaining part of these preliminaries is dedicated to K¨ othe and K¨ othe-
Bochner spaces.
Let ( T,T, µ) be a measure space ( ϕ̸=T,T ⊂ P (T),Tσ-algebra, µ:T −→ R+
measure, µ σ-finite and complete). Let ( X,∥ ∥) be a Banach space.
Write MX(µ) ={f:T−→X|fisµ-measurable }.
Hence, for any f∈MX(µ), there exists a sequence ( fn)n⊂ E X(µ) such that
fn− →
nfpointwise µ−a.e., where EX(µ) is the space of µ-simple functions with values
inX.

Approximation of K othe-Bochner spaces L(X)in case Xhas a Schauder basis 71
Write M+(µ) ={u:T→R+|uisµ-measurable }.
A function norm is a function ρ:M+(µ)→R+with the properties:
i)ρ(u) = 0⇔u= 0µ−a.e.;
ii)ρ(u)≤ρ(v), whenever u≤v;
iii)ρ(u+v)≤ρ(u) +ρ(v);
iv)ρ(αu) =αρ(u),with the convention 0 · ∞= 0,for any u, v∈M+(µ) and any
α∈R+.
Example 2.1. For 1≤p <∞,
ρ(u) =∥u∥p= (∫
updµ)1
p.
We say that ρhas the Riesz-Fischer property (and write ρR-F) if
ρ(∞∑
n=1un)≤∞∑
n=1ρ(un),
for any sequence ( un)ninM+(µ).
We say that ρis of absolutely continuous type (and write ρ a.c. ) if
ρ(un)↓0,
for any decreasing sequence ( un)ninM+(µ) such that ρ(u1)<∞andun↓0 (point-
wise).
For any 1 ≤p <∞,∥ ∥pR-F and ∥ ∥pa.c.
For any f:T→X,we shall write |f|to designate the function |f|:T→R+,
acting via
|f|(t) =∥f(t)∥.
Iff∈MX(µ),then|f| ∈M+(µ) and one can compute
ρ|f|=ρ(|f|)≤ ∞.
Define L(X) ={f∈MX(µ)|ρ|f|<∞}.
Denition 2.2. The spaces L(X) are called (seminormed) K othe-Bochner spaces.
Then L(X) is seminormed with the seminorm
f→ρ|f|.
The null space of this seminorm is
N(X) ={f∈ L(X)|ρ|f|= 0}.
One can see that

72 I. Chit escu et al.
N(X) ={f∈MX(µ)|f= 0µ−a.e.}.
The associated normed space is
L(X) =L(X)/N(X),
normed with the norm
˜f→ρ|f|=

˜f

,
for any f∈˜f∈L(X).
Denition 2.3. The spaces L(X) are called K othe-Bochner spaces .
Theorem 2.4. IfρR-F, L(X)is a Banach space.
Example 2.5. Ifρ=∥ ∥p,L(X) =Lp(X), L(X) =Lp(X) (the Lebesgue
spaces).
In case X=K,we write M(µ),L, Linstead of MX(µ),L(X) and L(X)
respectively. If X=R, Lis a normed lattice.
3. Results
Theorem 3.1. Any f∈ L (X)can be uniquely written as follows (pointwise
convergence):
f=∞∑
i=1aiei,
where ai∈ L,for any i.
Proof. Fixing t∈Tandx′
n(where ( x′
i)iare the biorthogonal functionals of the
basis ( en)n) we get
an(t) =x′
n(f(t)),
hence
f(t) =∞∑
n=1an(t)en.
We got the functions ( an)n, an:T→K,which are µ−measurable because an=
x′
n◦f.We infer that all anare in L.To this end, let Fn= (Pn−Pn−1)◦fforn≥2
and notice that, for any iand any t, one has
∥Fi(t)∥ ≤2M∥f(t)∥,

Approximation of K othe-Bochner spaces L(X)in case Xhas a Schauder basis 73
where Mis the constant of the Schauder basis ( en)n.Because Fi(t) =ai(t)eiit follows
that
|ai| ≤2M
∥ei∥|f|,
completing the proof of the fact that ai∈ L. 
The main result is the following:
Theorem 3.2. (Approximation theorem) Assume ρ a.c. Then, for any f∈
L(X),one has fn− →
nfinL(X).
Proof. As previously, f=∞∑
i=1aiei,with ai∈ L.For any t∈T,
∥fn(t)∥=∥Pn(f(t))∥ ≤M∥f(t)∥, (1)
hence |fn| ≤M|f|,for any n∈N.
For any ,let us define un:T→R+,via
un(t) = sup
m∥fn+m(t)−f(t)∥.
Hence, unareµ-measurable functions and the sequence ( un)nis decreasing. Due to
the inequality ∥fn+m(t)−f(t)∥ ≤ ∥ fn+m(t)∥+∥f(t)∥,and ( 1) we get un≤(M+1)|f|,
hence ρ(un)<∞,for any n.
The next step is to notice that lim
n→∞un(t) = 0 ,for any t∈T.Indeed, fixing t∈T
andε >0, we get p(t, ε)∈Nsuch that ∥fp(t)−f(t)∥< ε, for any p≥p(t, ε),because
fn− →
nf(pointwise). Consequently, up(t)≤ε,for such p.
Finally, because ρ a.c. , it follows that ρ(un)− →
n0,and this implies ρ|f−fn| − →
n0,
because |f−fn| ≤un. 
Remark 3.3. It is not possible to drop the condition ρ a.c. , as the following
counterexample shows.
Counterexample 3.4. Take ( T,T, µ) = (N,P(N), µ),where µis the counting
measure. Take ρgiven via ρ(u) = sup
nu(n). One can see that ρis not a.c., because
ρ(un) = 1 ,for any n∈N,where un(m) = 0 ,ifm≤nandun(m) = 1 ,ifm > n
(clearly un↓0). Here classes and functions coincide ( L(X) =L(X)), because the
only negligible set is ϕ.
Take X=l2with the canonical Schauder basis. Take f∈ L (X),given via
f(m) =em.
Then f=∞∑
i=1aiei,ai=φ{i},fn=n∑
i=1aieiandρ|f−fn|= 1,for any n.
Hence ” fn− →
nfinL(X)” is false.
We would like to close with the following:

74 I. Chit escu et al.
Concrete example.
Again ( T,T, µ) = (N,P(N), µ),where µis the counting measure (all the functions
areµ-measurable).
We shall consider the function norm ρ=∥ ∥1,i.e.
ρ(u) =∞∑
n=1u(n),
for any u:N→R+.
Of course, ρisa.c.
The Banach space Xwill be a Lorentz sequence space defined as follows:
Take a numerical sequence w= (wn)nsuch that wn↓0 and∞∑
n=1wn=∞(e.g.
wn=1
n).
Take 1 ≤p <∞.The function norm ρ1:M+(µ)→R+(having the Riesz-Fischer
property) given via
ρ1(u) = sup (∞∑
n=1u(π(n))pwn)1
p
(the supremum being taken over all bijections π:N→N) generates
L1≡L1not=d(w, p).
Namely d(w, p) ={u:N→K|ρ1|u|<∞}which is a Banach space, called
Lorentz sequence space.
This space admits the canonical Schauder basis as monotone Schauder basis and
∥en∥=w1
p
1,for any n.
Because d(w, p)⊂c0= the sequences which have zero limit, one can compute the
norm in d(w, p) via
∥u∥=ρ1(u) = (∞∑
n=1(|u(n)|∗)pwn)1
p,
where ( |u(n)|∗)nis the decreasing rearrangement of ( |u(n)|)n.
As anounced, we shall take X=d(w, p),with 1 ≤p <∞arbitrary and w= (wn)n,
wn=1
n,hence we are concerned with L(d(w, p)).
We shall study a particular f∈ L (d(w, p)) in order to perform a numerical
example.
Lett∈Kwith|t|<1.
We shall define the function f:N→d(w, p) (see later), via
f(m) =am= (am;n)n≥1.

Approximation of K othe-Bochner spaces L(X)in case Xhas a Schauder basis 75
Here, for any m∈N,we have: am;n= 0,if 1≤n≤mandam;n=tn,ifn > m.
Via computation, one can see that (pointwise) f=∞∑
n=1Fnen(where Fn:N→K,
Fn(m) =am;n) and fn=n∑
i=1Fiei,for any n.
Indeed, one has f(m) =am∈d(w, p),namely am=∞∑
n=1am;nen,for any m.This
is seen as follows. Because
(|am;n|∗)n∈N= (|t|m+1,|t|m+2, …),
it follows that
∥am∥p=∞∑
n=1(|am;n|∗)pwn≤|t|mp|t|p
1− |t|p.
On the other hand, the series∞∑
n=1am;nen=∞∑
n=m+1am;nenconverges absolutely,
because ∥am;nen∥=|tn|w1
p
1,ifn > m and its sum is precisely am=f(m),due to the
fact that (for n > m )

am−n∑
i=1am;iei

=∞∑
i=n+1|t|ip1
i−n≤∞∑
i=n+1|t|ip− →
n0.
It is also seen that f∈ L(d(w, p)),because
ρ|f|=∞∑
m=1∥am∥ ≤|t|
(1− |t|p)1
p∞∑
m=1|t|m=|t|2
(1− |t|p)1
p(1− |t|).
After some (long) computations we obtain the following evaluation, valid for any
n∈N:
ρ|f−fn| ≤n|t|n+1 1
(1− |t|p)1
p(1− |t|). (2)
Let us prove ( 2). According to the previous computation, one has (pointwise) for
anymandn,
f−fn=∞∑
i=n+1Fiei⇒(f−fn)(m) =∞∑
i=n+1am;iei⇒ |f−fn|(m) =
=∥(0,0, …,0, am;n+1, am;n+2, …)∥,
where, for any h≥1,one has am;n+h= 0 (if n+h≤m,i.e.m > n ) and am;n+h=tn+h
(ifn+h > m, i.e. 1≤m≤n).
In order to compute ρ|f−fn|=∞∑
m=1|f−fn|(m),we must compute ( |f−fn|(m))p
for any mandn.Let us fix n.

76 I. Chit escu et al.
For 1≤m≤n,i.e.n+h > m, one has am;n+h=tn+h,for any h≥1 and the de-
creasing rearrangement of (0 ,0, …,0,|am;n+1|,|am;n+2|, …) is precisely ( |t|n+1,|t|n+2, …),
hence ( |f−fn|(m))p=∞∑
h=1(|t|n+h)p1
h≤ |t|np|t|p
1− |t|pand this implies
n∑
m=1|f−fn|(m)≤n|t|n|t|
(1− |t|p)1
p. (3)
Form > n, i.e.n+h≤m,one has am;n+h= 0,hence, writing i=n+h,we have
(f−fn)(m) =∞∑
i=n+1am;iei= (0,0, …,0, am;n+1, am;n+2, …) =
= (0,0, …,0, …,0, am;m +1, am;m +2, …).
The decreasing rearrangement of the last sequence is ( |t|m+1,|t|m+2, …),hence
(|f−fn|(m))p=∞∑
h=1(|t|m+h)p1
h≤ |t|mp|t|p
1− |t|p.
We get
∞∑
m=n+1|f−fn|(m)≤|t|
(1− |t|p)1
p∞∑
m=n+1|t|m=|t|
(1− |t|p)1
p·|t|n+1
1− |t|. (4)
From ( 3) and ( 4) we obtain
ρ|f−fn|=∞∑
m=1|f−fn|(m)≤|t|n+1
(1− |t|p)1
p(n+|t|
1− |t|)≤
n|t|n+1 1
(1− |t|p)1
p(1 +|t|
1− |t|) =n|t|n+1 1
(1− |t|p)1
p(1− |t|).
Of course, ρ|f−fn| − →
n0.
Take ε >0 arbitrarily. We want to evaluate nin order to have
ρ|f−fn|< ε. (5)
It will be sufficient to have
n|t|n+1 1
(1− |t|p)1
p(1− |t|)< ε. (6)
With an extra effort, one can prove that, in order to have ( 6), it will be sufficient
to have
n >2|t|2
(1− |t|p)1
p(1− |t|)3·1
ε−1 (7)
(inverse dependence upon ε).
Concrete numerical example.
Take t=1
2, p= 2 and ε= 0,01 =1
100.
We have

Approximation of K othe-Bochner spaces L(X)in case Xhas a Schauder basis 77
2|t|2
(1− |t|p)1
p(1− |t|)3·1
ε−1 =800√
3−1≈460,88021.
So, for n≥461, one has ρ|f−fn|<1
100.
Conclusion. Using the general procedure exhibited above, one needs to perform at
least 461 computational steps in order to obtain a two decimals precision.
Remark. The condition 7, which is very confortable to be applied, was obtained
from 6, writing |t|=1
1 +a, with a >0 and majorizing
n|t|n+1=n
(1 +a)n+1<n
(n+1)n
2a2=2
(n+ 1)a2=2
n+ 1·|t|2
(1− |t|)2.
This procedure induces (unfortunately) an augmentation of the critical level for
n, because the computation is not very sharp.
This can be seen using 6and working directly. One must have
n|t|n+1·1
(1− |t|p)1
p(1− |t|)< ε=1
100.
Let us work for t=1
2,p= 2 and take ε=1
100. We obtain
n
2n+1·1
√
3
2·1
2=n
2n−1·1√
3<1
100.
It will be sufficient to have
n
2n−1·1
3
2<1
100⇔n
2n−2·1
3<1
100.
Because
2n−2= (1 + 1)n−2= 1 + ( n−2) +(n−3)(n−2)
2+… >
>1 + (n−2) +(n−3)(n−2)
2=n−1 +(n−3)(n−2)
2,
we have
n
2n−2·1
3<n
n−1 +(n−3)(n−2)
2·1
3=2n
3n2−9n+ 12
and it will be sufficient to have
2n
3n2−9n+ 12<1
100⇔3n2−209n+ 12 >0.
Because ∆ = 2092−122= 43537 it will be sufficient to have

78 I. Chit escu et al.
n >209 +√
43537
6≈69,609203 .
Forn≥70, one has
ρ|f−fn|<1
100.
Conclusion. Working for this particular case and using the special procedure exhib-
ited above, we need to perform only 70 computational steps in order to obtain again
two decimals precision.
Final conclusion. Our paper is dedicated to the general idea of Numerical Analysis:
approximating mathematical objects via finite objects of the same kind. More pre-
cisely, here we use the canonical finite dimensional subspaces generated by a Schauder
basis to approximate all the vectors in the space. We obtained a general approxima-
tion formula.
It is easy to apply this general formula. Unfortunately, the results thus obtained
are not very sharp (for our example, the obtained critical level was 461). Using a par-
ticular (specifical) method for this example, we obtained the critical level 70, which
is by far better than the “general” level 461, furnishing a considerable more economic
computation. This fact is characteristic for Numerical Analysis.
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