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Differentiable Maps with Isolated Critical Points Are Not Necessarily Differentiable maps with isolated critical points are not necessarily open in infinite dimensional spaces Chunrong Feng* and Liangpan Li† *Department of Mathematical Sciences, Durham University, DH1 3LE, UK †School of Mathematics, Shandong University, Jinan, Shandong, 250100, China [email protected], [email protected] Abstract Jean Saint Raymond asked whether continuously differentiable maps with isolated critical points are necessarily open in infinite dimensional (Hilbert) spaces. We answer this question negatively by constructing counterexamples in various settings. Keywords: Differentiable map, critical point, open map, Banach space Mathematics Subject Classifications (2020): Primary 46G05; Secondary 46T20 1 Introduction It is well known [1, 5, 19] that C1 (continuously differentiable) maps without critical points between Banach spaces are open. Saint Raymond [23] asked whether such phenomenon still occurs if the given maps are relaxed to having isolated critical points in infinite dimensional (Hilbert) spaces. The purpose of the paper is to answer this question negatively by construct- ing counterexamples in various real Banach spaces including all separable ones. Back to finite dimensional spaces, Saint Raymond proved [23] that C1 vector fields with countably many critical points are open provided that the dimension of the ambient space is higher than 1. This result was rediscovered by the second-listed author [16] and is implicitly implied by Theorem 1 or 2 in [25] by Titus and Young. For the interest of readers, we refer to [2, 3, 4, 6, 9, 10, 12, 13, 14, 20, 21, 22, 24] for some relevant works in Euclidean spaces. Throughout the paper Banach spaces are assumed to be over the field R of real numbers. Our general idea is as follows. Let (X, ) be a real Banach space, and consider maps of k · k the form 1 arXiv:2104.01827v1 [math.FA] 5 Apr 2021 F : x X exp( ) x X, (1.1) ∈ 7→ − x s · ∈ ||| ||| where is another norm on X that is strictly weaker than , i.e. there exists a constant |||·||| k·k C > 0 such that C but both norms are not equivalent. Here s > 0 is some real ||| · ||| ≤ || · || number, and we use the convention that 1 = and exp( ) = 0. It is geometrically evident 0 ∞ −∞ that the unique fixed point x =0 of F is not an interior point of the image of the unit open ball under F , thus F is not an open map. After specifying the quadruple (X, , ,s) in k · k ||| · ||| several ways later on, we will always show that F is a C1 map with a unique critical point at the origin. Hence Saint Raymond’s question is answered negatively. 1 2 C. Feng and L. Li 2 Differentiable open maps Definition 2.1. A map F : X Y between two Banach spaces is said to be (Fr´echet) → differentiable at a point x X if there exists a (unique) bounded linear operator JF (x) from ∈ X into Y , called the derivative of F at x, such that F (x + h) F (x)= JF (x)h + o( h ) (h 0). − || || → Furthermore, x is said to be a regular (critical) point of F if JF (x) has (has not) an inverse in B(Y, X), the space of bounded linear operators from Y to X; and F is said to be C1 if it is differentiable everywhere and x JF (x) is a continuous map from X to B(X,Y ). A map 7→ between two topological spaces is said to be open if it maps open sets to open sets. We now make an initial study of maps of the form (1.1). Take a positive constant C such that x C x for all x X. For any h X 0 , ||| ||| ≤ k k ∈ ∈ \{ } F (h) F (0) F (h) 1 1 k − k = k k = exp( ) exp( ), h h − h s ≤ −Cs h s k k k k ||| ||| k k where the last term goes to zero as h 0 in (X, )). According to Definition 2.1, F is → k · k differentiable at the origin with derivative JF (0) = 0, the zero element of B(X) := B(X, X). Thus x = 0 is a critical point of F . Theorem 2.2. The map F defined by (1.1) is not open. Proof. If this was not true, then by considering F (0) = 0, there exists a δ (0, 1) such that ∈ for any given y X with y = 1, one can find an element xy in the unit open ball in (X, ) ∈ k k k·k such that δ 1 y = F (xy) = exp( s ) xy. 2 − xy · ||| ||| Obviously, xy must be of the form xy = ryy for some ry (0, 1). Consequently, ∈ δ 1 1 = exp( ) ry exp( ), 2 −rs y s · ≤ − y s y||| ||| ||| ||| which implies that 1 1 s y . ||| ||| ≥ ln 2 δ Therefore, is stronger than on X. It is assumed that is strictly weaker than ||| · ||| k · k ||| · ||| , so a contradiction is derived. This finishes the proof of Theorem 2.2. k · k Obviously, the map F defined by (1.1) is a bijective map on X. The proof of Theorem 2.2 implies that the inverse F −1 of F is not continuous at F (0) = 0 of X. We now can make a slightly stronger estimate as follows. Since is strictly weaker than , one can find ||| · ||| 1k · k yn elements yn X for large enough n N, such that yn = 1 and yn = . Define zn = , ∈ ∈ k k ||| ||| n n and let xn X be such that F (xn) = zn. Obviously, xn must be of the form xn = γnyn for ∈ some γn > 0. Hence ns 1 exp s γn = F (xn) = zn = , − γn · k k k k n ns or equivalently nγn = exp s , from which one can easily deduce that γn √n for sufficiently γn ≥ −1 large n. To conclude, we see that zn 0 while F (zn) as n goes to infinity. k k→ k k→∞ Differentiable maps with isolated critical points 3 3 First example Consider ∞ N l2 = x = (x ,x , ) R : x2 < 1 2 · · · ∈ k ∞ n Xk=1 o 1 1 ∞ 2 ∞ 2 2 2 xk 2 with standard norm x = xk . Take x = k , another norm on l , which || || k=1 ||| ||| k=1 is strictly weaker than . P P k · k 1 1 2 Theorem 3.1. F (x) = exp( 2 ) x is a C map on (l , ) with a unique critical point − |||x||| · k · k at the origin. Proof. In Section 2 we have shown that F is differentiable at the origin with derivative JF (0) = 0. In the following we will first show that F is also differentiable on l2 0 . Let x l2 0 \{ } ∈ \{ } and h l2 be fixed such that h < |||x|||. For any N N, denote ∈ || || 2 ∈ N N x = (x1,x2,...,xN , 0, 0,...), h = (h1, h2,...,hN , 0, 0,...). N We always let N be large enough so that hN < |||x ||| and |||x||| < xN . For such N, || || 2 2 ||| ||| 1 1 fN : t [0, 1] − = − R xN thN 2 N 2 ∈ 7→ + (xk+thk) ∈ ||| ||| k k=1 P is a smooth function, and 1 16 fN L∞ := sup fN (t) = sup || || | | xN + thN 2 ≤ x 2 t∈[0,1] t∈[0,1] ||| ||| ||| ||| since for any t [0, 1], ∈ xN x xN + thN xN hN xN hN ||| ||| || ||. ||| ||| ≥ ||| ||| − ||| ||| ≥ ||| ||| − || || ≥ 2 ≥ 4 Moreover, direct computation shows that N 2(x + th ) f ′ (t)= f 2 (t) k k h , N N k k Xk=1 N N 2 ′′ 2(x + th ) 2 2h f (t) = 2f 3 (t) k k h + f 2 (t) k . N N k k N k Xk=1 Xk=1 Hence 2 ′ 16 f L∞ 3 x h , k N k ≤ x 4 · k k · k k ||| ||| 3 2 ′′ 16 2 16 2 f L∞ (2 9 x + 2) h . k N k ≤ · x 6 · k k x 4 · · k k ||| ||| ||| ||| 4 C. Feng and L. Li More briefly, there exists a positive constant M depending only on x such that i d i fN M h (3.1) dti L∞ ≤ k k for i = 0, 1, 2. As a consequence, di efN M h i (3.2) dti L∞ ≤ k k for i = 0, 1, 2, where M is another positive constant c depending only M. Note that 1 1 F (x + h) F (x) = expc (x + h) exp x − − x + h 2 · − − x 2 · ||| ||| ||| ||| = lim efN (1) efN (0) x + efN (1) h N→∞ − · · d 1 d2 d efN (t) efN (t) x efN (0) efN (t) h, = lim + 2 + + N→∞ dt t=0 2 dt t=αN · dt t=βN · where the existence of αN (0, 1) and βN (0, 1) is ensured by Taylor’s theorem in calculus. ∈ ∈ Considering (3.2), one gets d M x F (x + h) F (x) lim efN (t) x lim efN (0) h k k + M h 2, − − N→∞ dt t=0 · − N→∞ · ≤ 2 k k c where c 1 lim efN (0) = exp( ), N→∞ − x 2 ||| ||| ∞ d 1 1 2x h lim efN (t) = exp( ) k k . N→∞ dt t=0 − x 2 · x 4 · k ||| ||| ||| ||| k=1 X According to the Cauchy-Schwarz inequality, the linear map ∞ 1 1 2x h h l2 exp( ) k k x l2 ∈ 7→ − x 2 · x 4 · k · ∈ ||| ||| ||| ||| Xk=1 is a bounded operator on (l2, ), where now h can represent an arbitrary element of l2. By k · k Definition 2.1, F is differentiable at x with derivative given by ∞ 1 1 2xkhk 1 JF (x)h = exp( ) x + exp( ) h. (3.3) − x 2 · x 4 · k · − x 2 · ||| ||| ||| ||| Xk=1 ||| ||| 2 2 It is easy to verify by composition rule that x JF (x) is a continuous map from l to B(l ), 7→ so F is C1.
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