On the existence of distinguished bases in a <span class="mathjax-formula">$V$</span>-space

C ^OMPOSITIO M ^ATHEMATICA

K ^UNG -W ^EI Y ^ANG

On the existence of distinguished bases in a V -space

Compositio Mathematica, tome 23, n

^o

3 (1971), p. 307-308

<http://www.numdam.org/item?id=CM_1971__23_3_307_0>

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307

ON THE EXISTENCE OF DISTINGUISHED BASES IN A V-SPACE

by

Kung-Wei Yang

COMPOSITIO MATHEMATICA, ^Vol. 23, ^Fasc. 3, 1971, pag. 307-308 Wolters-Noordhoff Publishing

Printed in the Netherlands

In a sequence of papers

[1 ],

^{Pierre Robert}

introduced,

among other

things,

^the

concepts

^{of a}

V-space

^{and a}

distinguished

^{basis and} proves the existence of

distinguished

^{bases in a}

V-space.

^In this note, ^we

present

a

completely

^different

proof

which make no use of the ’Modified Riesz’s Lemma’

[1, 1,

p.

12] 1),

^{of the} existence of a

distinguished

^{basis in a}

V-space.

^{For notation} and definitions we refer the reader to

[1 ].

THEOREM. A

V-space

^{admits a}

distinguished

^basis.

PROOF. Let X be a

V-space

^over the field F. Let

03A9(X) ^[1,

^p.

^16]

^{be order-}

ed

according

^to its natural order. For i E

Q(X)

^let

^{Xi =} {x

^{E X :}

^{|x| ~} i}.

Xi is clearly a F-linear space. Let X0 = {0},

^and

^{if i ~} 0, letXi

⁼

Xi/Xp(i),

where

p(i)

^{is the}

predecessor

^{of i}

(p(i)

^{could be}

equal

^to

0). Each Xi

^is

again

^{a F-linear} space. Choose a F-basis

Fi

^{of the} F-linear space

Xi (Ho = 0).

^{For each x E}

Hi

^{choose a}

unique x

^E

Xi

^{which is}

mapped

^{to x}

under the natural

projection Xi -+ Xi. (In

^the

following,

^{we shall con-}

sistently

^{use -} notation in this

fashion.)

^Let

^Hi

^{be the set of all such}

x’s in

Xi (Ho

⁼

0).

^We claim that H ⁼

UiEU(X) Hi

^{is a}

distinguished

^bases

of X. To prove this

assertion,

^{we have to}

(1) ^verify

^condition

(i)

^of

Definition 5.3 in

[1, 1],

^and

(2)

^{show that}

[H ] =

^X.

(1 )

^{First of all we} remark that if

{x1,

x2 , ’ ’ ’,

xj}

^{is a finite subset of H}

such

that |x1| = |x2|

^{=... =}

lxjl

^{and if} al , oc2 , ’ ’ ’,

03B1j

^{are all non-zero} elements in

F, then lot, xi

^{+ ...}

+ 03B1jxj

⁼

[x1|.

^{Now let}

(xi ,

x2 , ’ ’ ’,

x.1

be ^a finite subset of H. Rename the elements so

that IX11 = |x2|

^{= ··· =}

lxjl ^I

&#x3E;

|xj+1| ~ ···~ lxl.

^Let 03B11, 03B12, ···, an be non-zero elements in F.

Then |03B11 x1 + ···03B1jxj + ··· + an xn | = |03B11x1 + ··· + 03B1jxj| IX11

⁼

max1 ~ m ~ n |xm|.

(2)

Let x E X.

Suppose lxi i(1).

^Then

x ~ Xi(1).

^If

i(1) = 0,

^then

x E

[H ].

^If

i(1) ~ ^0,

^{let x be the}

image

^{of x under the map}

Xi(1)

^-

Xi(i).

Then x ⁼

03B111x11 + ··· +alnlxlnl

^{for some}

otlj e F (1 ^{~ j ~} nl)

^and

1. 2 1 am indebted to Professor Pierre Robert for these two comments.

308

x1j ~ Hi(1) ^{(1 ~ j ~ nl). Clearly} |x - (03B111x11

^{+ ...}

+ 03B11n1 x1n1) = i(2) i(1).

^If

i(2)

⁼

^0,

^{then x E}

^[H].

^If

i(2) ~ ^0,

^we

^repeat.

^{This process}

either terminates after a finite number of

steps

^{or continues}

indefinitely.

If it terminates after ^a finite number of

steps,

^{then x E}

[H]. If it

^continues

indefinitely,

^{then we have an infinite series}

with

03B1st ~ F

^and

xst ~ Hi(s).

^Since

^{lx -} 03A3sk=1 03A3t = 03B1ktxkt|

^{is a}

strictly

^de-

creasing

^function

of s,

the series

generated by

^{the above} process converges to x

2). Therefore, x E [H].

^This

completes

^the

proof.

REFERENCE

P. ROBERT

[1] On ^some non-Archimedean normed linear _spaces, I, II, ..., VI, Compositio Mathematica, ^Vol 19, pp. 1-77, 1968.

(Oblatum 21-VI-68) Department ^of Mathematics Western Michigan University KALAMAZOO, Michigan ⁴⁹⁰⁰¹ U.S.A.