Linear structures of continuous, integrable and unbounded functions - - PowerPoint PPT Presentation
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Linear structures of continuous, integrable and unbounded functions Pablo Jos e Gerlach Mena Dpto. An alisis Matem atico Joint work with M.C.
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Pablo Jos´ e Gerlach Mena
alisis Matem´ atico Joint work with M.C. Calderon-Moreno and J.A. Prado-Bassas
6th July 2018
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Definition Let X be a topological vector space (t.v.s.), A ⊂ X. We say that A is lineable if ∃M ⊂ A ∪ {0} v.s. of infinite dimension. A is dense-lineable if M can be chosen dense in X. A is maximal-(dense)-lineable if dim(M) = dim(X).
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Definition Let X be a topological vector space (t.v.s.), A ⊂ X. We say that A is lineable if ∃M ⊂ A ∪ {0} v.s. of infinite dimension. A is dense-lineable if M can be chosen dense in X. A is maximal-(dense)-lineable if dim(M) = dim(X). Definition Let X be contained in some (linear) algebra A and B ⊂ A. We say that B is algebrable if ∃C ⊂ A so that C ⊂ B ∪ {0} and the cardinality of any system of generators of C is infinite. If in addition, A is a commutative algebra, we say that B is strongly algebrable if B ∪ {0} contains generated algebra which is isomorphic to a free algebra.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Monsters.
enez-Rodr´ ıguez, Mu˜ noz-Fern´ andez, Seoane-Sep´ ulveda, 2013: c-lineability of Weierstrass’ Monsters.
continuous surjections from R to R2.
noz, Palmberg, Puglisi, Seoane: c-lineability of Lp[0, 1]\Lq[0, 1] for 1 ≤ p < q.
ıa, Mart´ ın, Seoane, 2009: c-lineability of the set of Lebesgue integrable functions that are no Riemann integrable.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Example Consider the triangular function Tn : [0, +∞) − → R given by: Tn(x) = n(2n+1x + (1 − n2n+1)) if x ∈ [n − 1/2n+1, n), n(−2n+1x + (1 + n2n+1)) if x ∈ [n, n + 1/2n+1],
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Example Consider the triangular function Tn : [0, +∞) − → R given by: Tn(x) = n(2n+1x + (1 − n2n+1)) if x ∈ [n − 1/2n+1, n), n(−2n+1x + (1 + n2n+1)) if x ∈ [n, n + 1/2n+1],
and the function f : [0, +∞) − → R defined by the previous triangles: f(x) =
+∞
Tn(x).
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Theorem The family A of unbounded continuous integrable functions, that is, the set A =
x→+∞
|f(x)| = +∞
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Lemma Let X be a metrizable topological vector space, A ⊂ X maximal lineable and B ⊂ X dense-lineable in X with A ∩ B = ∅. If A is stronger than B then A is maximal dense-lineable.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Lemma Let X be a metrizable topological vector space, A ⊂ X maximal lineable and B ⊂ X dense-lineable in X with A ∩ B = ∅. If A is stronger than B then A is maximal dense-lineable. We define in X = C([0, +∞)) ∩ L1([0, +∞)) the metric dX(f, g) = f − gL1 +
+∞
1 2n · f − g∞,[0,n] 1 + f − g∞,[0,n] .
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Consider B the family of all the functions bn,γ(x) = p(x) if 0 ≤ x ≤ n, p(n) γ (n − x + γ) if n < x ≤ n + γ, if x > n + γ, where p(x) is a polygonal, n ∈ N and γ > 0.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Consider B the family of all the functions bn,γ(x) = p(x) if 0 ≤ x ≤ n, p(n) γ (n − x + γ) if n < x ≤ n + γ, if x > n + γ, where p(x) is a polygonal, n ∈ N and γ > 0. Theorem The family A of unbounded continuous integrable functions maximal dense-lineable.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Example Consider the “triangles” given by: Tn,p(x) = np(2n+1x + (1 − n2n+1))p if x ∈ [n − 1/2n+1, n), np(−2n+1x + (1 + n2n+1))p if x ∈ [n, n + 1/2n+1],
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Example Consider the “triangles” given by: Tn,p(x) = np(2n+1x + (1 − n2n+1))p if x ∈ [n − 1/2n+1, n), np(−2n+1x + (1 + n2n+1))p if x ∈ [n, n + 1/2n+1],
and we define the functions gp : [0, +∞) − → R as: gp(x) =
+∞
Tn,p(x).
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Theorem The family A of unbounded continuous integrable functions is strongly-algebrable.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
ujo, Bernal, Mu˜ noz, Prado and Seoane, 2017: c-lineability of sequences in MES.
ujo, Bernal, Mu˜ noz, Prado and Seoane, 2017: Maximal dense-lineability of sequences in L0([0, 1]) such that fn → 0 in measure but not pointwise a.e..
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A0 := {(fn)n : fn ∈ A, ∈ N, fn → 0 pointwise on [0, +∞), fnL1 → 0}.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A0 := {(fn)n : fn ∈ A, ∈ N, fn → 0 pointwise on [0, +∞), fnL1 → 0}. Example Consider the sequence of functions (fn)n given by: fn(x) :=
+∞
Tm(x), ∀n ∈ N, x ≥ 0.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A0 := {(fn)n : fn ∈ A, ∈ N, fn → 0 pointwise on [0, +∞), fnL1 → 0}. Example Consider the sequence of functions (fn)n given by: fn(x) :=
+∞
Tm(x), ∀n ∈ N, x ≥ 0. Theorem The family of sequences A0 on [0, +∞) is maximal lineable.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Consider now the spaces c0(X) := {F = (fn)n : fn ∈ X n ∈ N, dX(fn, 0) → 0}, c00(B) := {(bn)n : ∃n0 | bn ∈ B ∀n ≤ n0, bn = 0 ∀n > n0}.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Consider now the spaces c0(X) := {F = (fn)n : fn ∈ X n ∈ N, dX(fn, 0) → 0}, c00(B) := {(bn)n : ∃n0 | bn ∈ B ∀n ≤ n0, bn = 0 ∀n > n0}. A0 := {(fn)n : fn ∈ A, ∈ N, d((fn)n, 0) → 0}, where d((fn)n, (gn)n) = sup
n∈N
dX(fn, gn).
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Consider now the spaces c0(X) := {F = (fn)n : fn ∈ X n ∈ N, dX(fn, 0) → 0}, c00(B) := {(bn)n : ∃n0 | bn ∈ B ∀n ≤ n0, bn = 0 ∀n > n0}. A0 := {(fn)n : fn ∈ A, ∈ N, d((fn)n, 0) → 0}, where d((fn)n, (gn)n) = sup
n∈N
dX(fn, gn). Theorem The family of sequences A0 on [0, +∞) is maximal dense-lineable.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Theorem The family of sequences A0 on [0, +∞) is strongly-algebrable.
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A : function f continuous integrable unbounded
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A : function f continuous integrable unbounded = ⇒
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A : function f continuous integrable unbounded = ⇒
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A : function f continuous integrable unbounded = ⇒
A0 : sequence (fn)n continuous integrable unbounded pointwise to 0
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A : function f continuous integrable unbounded = ⇒
A0 : sequence (fn)n continuous integrable unbounded pointwise to 0 = ⇒
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A : function f continuous integrable unbounded = ⇒
A0 : sequence (fn)n continuous integrable unbounded pointwise to 0 = ⇒
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
A : function f continuous integrable unbounded = ⇒
A0 : sequence (fn)n continuous integrable unbounded almost unif. = ⇒
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces
ujo, L. Bernal-Gonz´ alez , G. A. Mu˜ n´
andez, J. A. Prado-Bassas y
ulveda, Lineability in sequence and function spaces, ArXiV http://arxiv.org/abs/1507.04477.
alez , D. M. Pellegrino y J. B. Seoane-Sep´ ulveda, Lineability: The search for linearity in mathematics, Monographs and Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2015.
. J. Garc´ ıa-Pacheco , D. P´ erez-Garc´ ıa y J. B. Seoane-Sep´ ulveda, On dense lineability of sets of functions on R, Topology 48 (2009) 149-156.
alez, Dense-lineability in spaces of continuous functions, Proc.
alez, D. Pellegrino y J. B. Seoane-Sep´ ulveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. 51 (2013), 71–130. P . Jim´ enez-Rodr´ ıguez, G. A. Mu˜ n´
andez y J. B. Seoane-Sep´ ulveda, On Weierstrass’ Monsters and lineability, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), 577–585.
Pablo Jos´ e Gerlach Mena Linear Structures