ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXV (1985)
K a z i m i e r z W l o d a r c z y k (Lôdz)
Criteria of injectivity for mappings of Banach and locally convex spaces
Abstract. In the present paper there are given sufficient conditions for (Fréchet-) holomorphic mappings of complex Banach spaces and В Г -differentiable or В Г -analytic mappings in Hausdorff locally convex and sequentially complete topological vector spaces to be injections.
1. Introduction. The role and importance of sufficient conditions of injectivity in the extremal problems of the geometrical theory of functions are well known. The results of investigations made in this field so far, which derive their origin in the 30’s, together with a historical outline were included by F. G. Avchadiev and L. A. Aksent’ev in their extensive paper [1].
In the geometrical theory of functions, a separate direction of investigations concerning the search for general criteria of the injectivity of mappings defined in various spaces was taken in the 50’s (see [1] for details).
In the present paper, we shall extend this type of classical results of A. Ostrowski [4], [5] to the cases of complex Banach spaces as well as arbitrary Hausdorff locally convex and sequentially complete topological vector spaces.
2. Statement of results. In this paper, X and У denote any Banach spaces over the field С, E denotes an arbitrary Hausdorff locally convex and sequentially complete topological vector space over the field К (equal to R or С), Г stands for any set of continuous semi-norms p on E, inducing the topology of E, and U denotes a non-void open subset in X or E with the property that if x', x" e U , x' Ф x", then there exists a finite sequence of distinct points
(1) x ' = x l9 x 2, . .. , x„, x„+1 = x", n ^ l , belonging to the set U and such that, for each к = 1, . . . , n,
{x: x = xk + t{x k+1 — xfc), 0 ^ t < 1} c U.
If U с X, let H (U , У) stand for the set of all holomorphic mappings
4 0 2 K . W l o d a r c z y k
F : U -> Y, and if U c E, let DBr(U , E) denote the set of all mappings F : U -> E such that F are fir-differentiable in U if K = R, or fir-analytic in U if К= C. Standard results on holomorphic mappings can be found in [2]
and [3] and on fir-differentiable or fir-analytic mappings — in [6].
Since Г has the property that p(x) = 0 for all р е Г if and only if x = 0, one can define the quantity
П
Ar {U) = sup [inf ]T p (x k + 1- x k)/p {x ” - x ' ) : x ' ,x " e U ,
k = 1
x! Ф x", p {x " - x ' ) Ф 0, р е Г ] , where in the case of the infimum, all sequences of type (1) are considered.
Obviously, Ar {U) ^ 1 for any Г.
Let I E denote the identity mapping on E, and let ||*||r stand for the norm in the Banach space L Br(E, E ) of all fir-continuous linear mappings of E into E.
We prove the following theorems which are generalizations of the results of the papers by A. Ostrowski [4], [5].
Theorem 1. I f F e H { U , У) and if, fo r any x', x ” e U , x' Ф x ”, there exists a continuous linear form f e L ( Y , C) such that
1 n
J Re/ { X F '[x k + t(x k + ! - xk)] (xk+1- xk) dt} > 0
0 k = 1
fo r som e sequence o f type (1), then F is an injective mapping in U.
Theorem 2. I f F e D Br(U , E) and i f there exists а В Г -isomorphism T: E -> E such that
(2) sup{||/£-T F (x )||r: xe Щ < l/Xr(U)
or, in the case o f convexity o f U,
\\IE— TF'(x)\\r < 1 fo r x e U , then F is an injection in U.
3. Proof of Theorem 1. Let x', x " e U , x' Ф x". Then F {x") — F(x') — I [ F ( x t + 1) ~ F ( x t)]
k = 1 1 n
= 1 S F T x k + t(xk+1- x j ] ( x k+1- x k)dt.
0 k = l
In consequence,
l/ -[F (x")-F (x')]| » J R e / { £ F ' lx t + t(x k+ 1- x km x k^ - x k)}d t > 0,
0 k = l
and thus, Г (х ') Ф F (xn).
Injectivity fo r mappings o f Banach spaces 403
4. Proof of Theorem 2. Let x', x" be any fixed distinct points of the set U. Then p (x ” — x') Ф 0 for some р е Г and, in virtue of the definition of Яг ((7)
(3) inf £ p {x k+1- x k) ^ l r {U )p ( x " -x '),
k = l
where in the case of the infimum, all sequences of type (1) are considered. Let v' — y*^') y"0) vO) vO) — Y,/ И ^ 1 j — 1 9
Л X\ , X2 5 9 Xn. , Xn j X j 1, Z, . . ., be a sequence of sequences of type (1), such that
(4) lim £ p(x^+ l - x ^ ) = inf £ p (x k+1- x k).
j~*OD fc = l fe = l
Let h be a mapping defined by the formula h(x) = x —TF (x)
and let us suppose that F (x ') = F(x"). Then, for each j = 1, 2, ..., 0 = p { T [ F ( x " ) - F ( x ’)-]}
= p! I
k = 1
>\p [ Z (*$■ 1 - xk *)] - p { Z (*$ - i) - h ( 4 j))]}|
k = 1 k = 1
k = 1
Hence it appears, after applying the mean value theorem, that, for each j = 1 , 2 , . . . ,
"j
° ^ 1 - [ £ P ^ + i- ^ k V p ^ - ^ O js u p d l/ i'W llr : x e l / j .
Jc= 1
Consequently, taking account of (4), we have
0 ^ 1 — [inf ]T p(xk + 1 - x Jk)/p(x,,-x')]sup{||/z, (^)llr: x e U } , k= 1
where in the case of the infimum, all sequence of type (1) are considered, and next, taking (2) and (3) into account, we find that
0 > 1 — inf ]T p (x k+1 - x k)/[2r (l/ )p (x " -x ')] ^ 0,
k = 1
which is impossible. So, F (x ') Ф F(x").
40 4 K. W l o d a r c z y k
If U is a convex set, then
p { r [ F ( x " ) - F ( x ' ) ] } > p ( x " - x ') [ l - \ \ h 'l x ’ + e ( x " - x ' m r ] for some 0 ^ в ^ 1, whence we infer that F{x') Ф F(x"). 1
So, the theorem is proved.
Ex a m p l e. If G e D Br(U, E) and if F is a mapping defined by the formula F (x ) = x + pG (x), v e K ,
then, for v e V where
v = { v e K : |t)| < l/[Ar ([/)sup{||G'(x)||r : xsC/}]}, the mapping F is injective in U.
References
[1] F. G. A v ch ad iev , L. A. A k s e n t’ev, Ostiovnye rezuTtaty v dostatoZnych uslovijach odnolistnosti analitiëeskich funkcij, Usp. Mat. Nauk 30 (1975), 3-60.
[2] E. Hi lie, R. S. P h illip s , Functional analysis and semi-groups, Amer. Math. Soc. Colloq.
Publ. 31 (1957).
[3] L. N ach b in , Topology on spaces o f holomorphic mappings, Springer-Verlag, Berlin 1969.
[4] A. O stro w sk i, Un critère d’univalence des transformations dans un R", C. R. Acad. Sci.
Paris 247 (1958), 172-175.
[5] —; Un nouveau critère dunivalence des transformations dans un R”, ibidem 248 (1959), 348—
350.
[6] S. Y am am u ro , A theory o f differentiation in locally convex spaces, Memoirs Amer. Math.
Soc. 17 (212) (1979), 1-82.
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