Topological Properties of Operations on Spaces of Continuous Functions and Integrable Functions

Topological Properties of Operations on Spaces of Continuous Functions and Integrable Functions Holly Renaud University of Memphis hrenaud@memphis.edu May 3, 2018 Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 1 / 24

Outline 1 Definitions Metric spaces 2 Motivation: Rudin s Example 3 Scalar multiplication 4 Openness of basic operators Addition Multiplication on C([0, 1]) 5 Multiplication on C (n) [0, 1] <D> 6 Openness of Other Binary Maps Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 2 / 24

Definitions Definition Let E and F be topological spaces and f : E F a surjective map. Let x E. 1 f is open at x if and only if f maps every neighborhood of x onto a neighborhood of f (x). Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 3 / 24

Definitions Definition Let E and F be topological spaces and f : E F a surjective map. Let x E. 1 f is open at x if and only if f maps every neighborhood of x onto a neighborhood of f (x). 2 f is densely open (d-open) at x if and only if f maps every neighborhood of x onto a dense subset of a neighborhood of f (x). 3 f is weakly open (w-open) at x if and only if f maps every neighborhood of x onto a set with nonempty interior. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 3 / 24

Definitions Definition Let E and F be topological spaces and f : E F a surjective map. Let x E. 1 f is open at x if and only if f maps every neighborhood of x onto a neighborhood of f (x). 2 f is densely open (d-open) at x if and only if f maps every neighborhood of x onto a dense subset of a neighborhood of f (x). 3 f is weakly open (w-open) at x if and only if f maps every neighborhood of x onto a set with nonempty interior. 4 f is open (d-open or w-open) if f is open (respectively d-open or w-open) at x, for every x E. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 3 / 24

Metric Spaces Definition Let (E, d) and (F, D) be two metric spaces and f : E F a surjective map. Then 1 f is open (d-open or w-open) if and only if, for every x E and ɛ > 0, there exists δ > 0 such that f (B(x, ɛ)) B(f (x), δ), (resp. f (B(x, ɛ)) B(f (x), δ) or int(f (B(x, ɛ))) ). Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 4 / 24

Metric Spaces Definition Let (E, d) and (F, D) be two metric spaces and f : E F a surjective map. Then 1 f is open (d-open or w-open) if and only if, for every x E and ɛ > 0, there exists δ > 0 such that f (B(x, ɛ)) B(f (x), δ), (resp. f (B(x, ɛ)) B(f (x), δ) or int(f (B(x, ɛ))) ). 2 f is uniformly open if, for every ɛ > 0, there exists a δ > 0 such that f (B(x, ɛ)) B(f (x), δ), for all x E. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 4 / 24

Motivation Theorem (Open Mapping Theorem (Banach-Schauder Theorem)) Every continuous and linear surjective map between two Banach spaces is open. This property does not extend to bilinear maps. (Functional Analysis by Rudin. Problem 11 on p. 54) P : R R 2 R 2 given by P (t, (x, y)) = t (x, y) = (tx, ty). Is P open? Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 5 / 24

Motivation Theorem (Open Mapping Theorem (Banach-Schauder Theorem)) Every continuous and linear surjective map between two Banach spaces is open. This property does not extend to bilinear maps. (Functional Analysis by Rudin. Problem 11 on p. 54) P : R R 2 R 2 given by P (t, (x, y)) = t (x, y) = (tx, ty). Is P open? No. To show: P is not open at (0, (1, 0)). Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 5 / 24

Motivation Theorem (Open Mapping Theorem (Banach-Schauder Theorem)) Every continuous and linear surjective map between two Banach spaces is open. This property does not extend to bilinear maps. (Functional Analysis by Rudin. Problem 11 on p. 54) P : R R 2 R 2 given by P (t, (x, y)) = t (x, y) = (tx, ty). Is P open? No. To show: P is not open at (0, (1, 0)). Let ɛ = 1/2. For every δ > 0, we have that (0, δ/2) B((0, 0), δ), yet (0, δ/2) / B(0, ɛ) B((1, 0), ɛ). Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 5 / 24

Scalar Multiplication Let {A k } be a sequence of normed spaces over the same field F, with resp. norms k ( k N). Let A be c 0 ({A k } k ), l p ({A k } k ) (p 1) or l ({A k } k ). Lemma Let T : F A A be given by T (t, a) = t a = (ta i ) i=1,...n. The following statements are equivalent: 1 t 0 or (t, a) = (0, 0). 2 T is open at (t, a). 3 T is d-open at (t, a). Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 6 / 24

T is open at (t, a), for t 0, and T is open at (0, 0) If t 0, then ɛ > 0, let δ = t ɛ. Then B(ta, δ) {λy : λ B(t, ɛ) and y B(a, ɛ)}. For x B(ta, δ), set y = 1 x, λ = t. t Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 7 / 24

T is open at (t, a), for t 0, and T is open at (0, 0) If t 0, then ɛ > 0, let δ = t ɛ. Then B(ta, δ) {λy : λ B(t, ɛ) and y B(a, ɛ)}. For x B(ta, δ), set y = 1 x, λ = t. t If t = 0, a i = 0 for all i, then ɛ > 0, set δ = ɛ, with k such that k 1 k < ɛ. B(0, δ) B(0, ɛ) B(0, ɛ) = {(λy) : λ B(0, ɛ) and y B(0, ɛ)}. Given x B(0, δ), let λ = 1 k B(0, ɛ) and y = kx B(0, ɛ). Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 7 / 24

T is weakly open Recall: T (t, a) = t a = (ta i ) i=1,...n Lemma T is weakly open. O(T ) = {(t, a) : t 0} {(0, 0)} O(T ) is dense in F A Thus B(0, ɛ) B(a, ɛ) B(ɛ/2, ɛ/4) B(a, ɛ/4) B(ɛ/2 a, ɛ 2 /8) x ɛ 2 a < ɛ2 8 implies 2 ɛ x a < ɛ 4. ( ) ɛ 2 ( ɛ 2 ɛ x B 2 4), ɛ ( B a, ɛ ). 4 Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 8 / 24

Problem Let X be an algebra: when are the basic binary operations (addition, scalar product, and multiplication) open? Lemma Let X be a topological vector space. Then + : X X X is open. Proof. To show: If O 1, O 2 are open, then is open. O 1 + O 2 = {x + y : x O 1, y O 2 } = y O2 O 1 + {y} T : X X with T (u) = u + y is a homeomorphism, so T is open. O 1 + O 2 is a union of open sets Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 9 / 24

Addition is open Corollary Let X be a normed space. Then + : X X X is open (uniformly open). Proof. We have the following result (c.f. M.Balcerzak, A.Wachowicz, and W.Wilczyński, Studia Mathematica 170 (2005), no. 2, 203 209.): B(x 1, r 1 ) + B(x 2, r 2 ) = B(x 1 + x 2, r 1 + r 2 ). The result now follows from the definition of uniform openness. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 10 / 24

Multiplication The multiplication on C([0, 1]) is not open. Example (Fremlin) P : C([0, 1]) C([0, 1]) C([0, 1]), P(f, g) = f g 1 Let f (x) = x 1 2 and ɛ = 1 4 2 B(f 2, δ) B(f, 1 4 ) B(f, 1 4 ) f (0) = 1 2 ; f (1) = 1 2 All functions in B(f, 1 4 ) must vanish at some point f 2 + δ 2 > 0 does not vanish 3 P is not open at (f, f ) Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 11 / 24

Old Results Theorem (M.Balcerzak, A.Wachowicz, and W.Wilczyński, Studia Mathematica 170:2 (2005)) Let f C([0, 1]). Then f 2 Int(B 2 (f, r)) for all r > 0 if and only if either f 0 on [0, 1] or f 0 on [0, 1]. Theorem (A. Wachowicz, Real Analysis Exchange 34:2 (2009)) The multiplication on (C (n) [0, 1], m ) is weakly open ( f m = max 0 i n max x [0,1] f (i) (x) ). Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 12 / 24

Old Results Theorem (M.Balcerzak, A.Wachowicz, and W.Wilczyński, Studia Mathematica 170:2 (2005)) Let f C([0, 1]). Then f 2 Int(B 2 (f, r)) for all r > 0 if and only if either f 0 on [0, 1] or f 0 on [0, 1]. Theorem (A. Wachowicz, Real Analysis Exchange 34:2 (2009)) The multiplication on (C (n) [0, 1], m ) is weakly open ( f m = max 0 i n max x [0,1] f (i) (x) ). Consider f 1 = Σ n i=0 f (i) = Σ n i=0 max x [0,1] f (i) (x) Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 12 / 24

Old Results Theorem (M.Balcerzak, A.Wachowicz, and W.Wilczyński, Studia Mathematica 170:2 (2005)) Let f C([0, 1]). Then f 2 Int(B 2 (f, r)) for all r > 0 if and only if either f 0 on [0, 1] or f 0 on [0, 1]. Theorem (A. Wachowicz, Real Analysis Exchange 34:2 (2009)) The multiplication on (C (n) [0, 1], m ) is weakly open ( f m = max 0 i n max x [0,1] f (i) (x) ). Consider f 1 = Σ n i=0 f (i) = Σ n i=0 max x [0,1] f (i) (x) f m f 1 (n + 1) f m Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 12 / 24

Generalization of KKM Spaces Let C (n) ([0, 1]) be the space of all n differentiable functions on [0, 1]. Let D be a compact, connected subset of [0, 1] n+1. We set f <D> = sup { f (r 0 ) + Σ n i=1 f (i) (r i ) }, r <D> for r = (r 0, r 1, r 2,..., r n ), and f C (n) ([0, 1]). Lemma <D> is a norm on C (n) ([0, 1]) if and only if n i=0 π i(d) = [0, 1]. 1 is a particular case of D For treatment of the case where n = 1, c.f. K. Kawamura, H. Koshimizu, and T. Miura, Norms on C 1 ([0, 1]) and their isometries, preprint. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 13 / 24

New results: A Quasi-algebra norm Lemma Let D be a connected and compact subset of [0, 1] n+1 such that [0, 1] = π 0 (D) π 1 (D) π n (D). Then fg D (n + 1)2 n f D g D, for every f, g C (n) ([0, 1]), Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 14 / 24

Sketch of the proof: f, g C (n) ([0, 1]). Then for n = 0, we have fg f g, and for n = 1 we have fg D = fg + fg + g f 1 f D g D. For D [0, 1] n+1, we set f (j) k = max{ f (j) (x), with x π k (D)}. f g D = fg 0 + (f g) 1 + + (f g) (n) n n ( ) n f 0 g 0 + + f (n k) n g k) n k k=0 n n ( ) k = f (k j) k g (j) k j j=0 j=0 k=j n n k=j ( ) k f (k j) k j g (j) j (since k j, we have g (j) k g (j) j ) Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 15 / 24

fg D (n + 1)2 n f D g D f g D n n j=0 n j=0 (n + 1) since ( ) ( i+j j n ) j and n j i=0 k=j n j g (j) j ( ) k f (k j) k j i=0 ( i + j n g (j) j j=0 j n j=0 = (n + 1)2 n f D g D, ( i+j ) ( j (n + 1) n ) j. g (j) j ) f (i) i ( ) n n f (i) i j i=0 Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 16 / 24

New Results: A Theorem Let D be a connected and compact subset of [0, 1] n+1 such that [0, 1] = π 0 (D) π 1 (D) π n (D). Let C (n) [0, 1] D denote the space of all n-continuously differentiable functions endowed with the D. Theorem The multiplication on C (n) [0, 1] D is weakly open. The proof is adapted from a proof by Wachowicz for (C (n) [0, 1], m ). Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 17 / 24

Outline of the proof 1. Let π n (D) = [a, b]. Approximate f and g by polynomials with simple zeros and disjoint non-empty zero sets. For simplicity also denoted by f and g. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 18 / 24

Outline of the proof 1. Let π n (D) = [a, b]. Approximate f and g by polynomials with simple zeros and disjoint non-empty zero sets. For simplicity also denoted by f and g. 2. Define a partition of [a, b], a = x 0 < x 1 < x 2 < < x m = b such that f k even [x k, x k+1 ] 0, g k odd [x k, x k+1 ] 0 3. Extension Lemma. Let ϕ and h functions in C (n) [a, b]. Let η > 0 and x 0 [0, 1] such that ϕ (j) (x 0 ) h (j) (x 0 ) < η, with j = 0,, n. For every x [a, b], we set k(x) = h(x) + n j=0 ( ) (x ϕ (j) (x 0 ) h (j) x0 ) j (x 0 ) j! Then for every j {0, 1,..., n} we have k (j) (x 0 ) = ϕ (j) (x 0 ) and k B(h, eη). k is an extension of ϕ.. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 18 / 24

P is weakly open. Outline of the proof 4. Given ϕ, δ-close to the product f g, construct f and g, ɛ-close to f and g respectively as follows: Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 19 / 24

P is weakly open. Outline of the proof 4. Given ϕ, δ-close to the product f g, construct f and g, ɛ-close to f and g respectively as follows: f [a,x1] 0. Set f 1 = f and g 1 = ϕ f over the interval [x 0, x 1 ]. Decreasing the value of δ and applying the Extension Lemma there exists g 1 that extends g to the interval [x 1, x 2 ] so that g 1 0 on[x 1, x 2 ] and g 1 B(g, eɛ). Now define f 2 = ϕ g 1. This procedure repeats until we reach [x m 1, x m ]. 5. Now f f i m < cδ (same for g g i ), where c depends only on f, g. Choose cδ < ɛ Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 19 / 24

P is weakly open. Outline of the proof 4. Given ϕ, δ-close to the product f g, construct f and g, ɛ-close to f and g respectively as follows: f [a,x1] 0. Set f 1 = f and g 1 = ϕ f over the interval [x 0, x 1 ]. Decreasing the value of δ and applying the Extension Lemma there exists g 1 that extends g to the interval [x 1, x 2 ] so that g 1 0 on[x 1, x 2 ] and g 1 B(g, eɛ). Now define f 2 = ϕ g 1. This procedure repeats until we reach [x m 1, x m ]. 5. Now f f i m < cδ (same for g g i ), where c depends only on f, g. Choose cδ < ɛ 6. Let ϕ = f i g i for each i, and now ϕ B(f g, δ) B(f, ɛ) B(g, ɛ) Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 19 / 24

P is weakly open. Outline of the proof 4. Given ϕ, δ-close to the product f g, construct f and g, ɛ-close to f and g respectively as follows: f [a,x1] 0. Set f 1 = f and g 1 = ϕ f over the interval [x 0, x 1 ]. Decreasing the value of δ and applying the Extension Lemma there exists g 1 that extends g to the interval [x 1, x 2 ] so that g 1 0 on[x 1, x 2 ] and g 1 B(g, eɛ). Now define f 2 = ϕ g 1. This procedure repeats until we reach [x m 1, x m ]. 5. Now f f i m < cδ (same for g g i ), where c depends only on f, g. Choose cδ < ɛ 6. Let ϕ = f i g i for each i, and now ϕ B(f g, δ) B(f, ɛ) B(g, ɛ) 7. We extend f and g to π n 1 (D) following steps 2-5. Proceed until we cover [0, 1]. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 19 / 24

Behrends Theorem Theorem (E. Behrends, Linear Algebra and its Applications 517 (2017)) Let f, g C([0, 1]) be given and γ = (f, g) the associated path in R 2. Then TFAE: f g is an interior point of B(f, ɛ) B(g, ɛ), ɛ > 0 ɛ > 0, there exists a δ > 0 s.t. f g + δ and f g δ are in B(f, ɛ) B(g, ɛ) γ has no positive and no negative saddle point crossings The same result holds for (C (n) [0, 1]) D, under the same assumptions on D. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 20 / 24

Other binary maps Products on spaces of integrable functions: L p [0, 1] L q [0, 1] L r [0, 1] with 1 p + 1 q = 1 r, p, q, r [1, ] (c.f. M. Balcerzak, Mathematics Subject Classification (2013)) Densely-defined multiplication on L p (Ω) (1 p < ) Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 21 / 24

Other binary maps Products on spaces of integrable functions: L p [0, 1] L q [0, 1] L r [0, 1] with 1 p + 1 q = 1 r, p, q, r [1, ] (c.f. M. Balcerzak, Mathematics Subject Classification (2013)) Densely-defined multiplication on L p (Ω) (1 p < ) Product on C(Ω, E) with E a Banach algebra Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 21 / 24

Other binary maps Products on spaces of integrable functions: L p [0, 1] L q [0, 1] L r [0, 1] with 1 p + 1 q = 1 r, p, q, r [1, ] (c.f. M. Balcerzak, Mathematics Subject Classification (2013)) Densely-defined multiplication on L p (Ω) (1 p < ) Product on C(Ω, E) with E a Banach algebra The operation on L p ([0, 1]) The operation on L p ([0, 1]) Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 21 / 24

Appendix Definition A is a Banach algebra (with unit) if A satisfies the following conditions: 1 A is a Banach space 2 A A A satisfies the associative property of multiplication, the distributive property, and c(x y) = (c x)y = x(c y) 3 (A has a unital element e with e = 1) 4 x y x y Definition A topological vector space is a vector space such that addition and scalar multiplication are continuous. We require that the topology be Hausdorff. Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 23 / 24

Densely-defined multiplication on L p ([0, 1]) (denoted L p ) Multiplication in L p is not necessarily defined Consider f (x) = g(x) = x 1/6 L 3 : f g = x 1/3 x 1/3 / L 3 Holly Renaud (UofM) Topological Properties of Operations May 3, 2018 24 / 24