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. 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 Previous Concepts 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 Previous Concepts 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 Known Results 1. Gurariy, 1966: ℵ 0 -lineability of Weierstrass’ Monsters. 2. Fonf, Gurariy, Kadets, 1999: Spaceability of Weiertrass’ Monsters. 3. Jim´ ıguez, Mu˜ noz-Fern´ enez-Rodr´ andez, Seoane-Sep´ ulveda, 2013: c -lineability of Weierstrass’ Monsters. 4. Albuquerque, 2014: Maximal-lineability of the set of continuous surjections from R to R 2 . 5. Mu˜ noz, Palmberg, Puglisi, Seoane: c -lineability of L p [ 0 , 1 ] \ L q [ 0 , 1 ] for 1 ≤ p < q . 6. Garc´ ıa, Mart´ ın, Seoane, 2009: c -lineability of the set of Lebesgue integrable functions that are no Riemann integrable. 7. Lineability of DNM ( R ) in C ( R ) , . . . Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability Example Consider the triangular function T n : [ 0 , + ∞ ) − → R given by: n ( 2 n + 1 x + ( 1 − n 2 n + 1 )) if x ∈ [ n − 1 / 2 n + 1 , n ) , n ( − 2 n + 1 x + ( 1 + n 2 n + 1 )) if x ∈ [ n , n + 1 / 2 n + 1 ] , T n ( x ) = 0 otherwise . Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability Example Consider the triangular function T n : [ 0 , + ∞ ) − → R given by: n ( 2 n + 1 x + ( 1 − n 2 n + 1 )) if x ∈ [ n − 1 / 2 n + 1 , n ) , n ( − 2 n + 1 x + ( 1 + n 2 n + 1 )) if x ∈ [ n , n + 1 / 2 n + 1 ] , T n ( x ) = 0 otherwise . and the function f : [ 0 , + ∞ ) − → R defined by the previous triangles: + ∞ � f ( x ) = T n ( x ) . n = 1 Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability Theorem The family A of unbounded continuous integrable functions, that is, the set � � f ∈ C ([ 0 , + ∞ )) ∩ L 1 ([ 0 , + ∞ )) : lim sup A = | f ( x ) | = + ∞ x → + ∞ is maximal lineable. Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability 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 Lineability 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 , + ∞ )) ∩ L 1 ([ 0 , + ∞ )) the metric + ∞ � f − g � ∞ , [ 0 , n ] 1 � d X ( f , g ) = � f − g � L 1 + 2 n · . 1 + � f − g � ∞ , [ 0 , n ] n = 1 Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability Consider B the family of all the functions 0 ≤ x ≤ n , p ( x ) if p ( n ) b n ,γ ( x ) = ( n − x + γ ) if n < x ≤ n + γ, γ 0 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 Lineability Consider B the family of all the functions 0 ≤ x ≤ n , p ( x ) if p ( n ) b n ,γ ( x ) = ( n − x + γ ) if n < x ≤ n + γ, γ 0 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 Algebrability Example Consider the “triangles” given by: n p ( 2 n + 1 x + ( 1 − n 2 n + 1 )) p if x ∈ [ n − 1 / 2 n + 1 , n ) , n p ( − 2 n + 1 x + ( 1 + n 2 n + 1 )) p if x ∈ [ n , n + 1 / 2 n + 1 ] , T n , p ( x ) = 0 otherwise , Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Algebrability Example Consider the “triangles” given by: n p ( 2 n + 1 x + ( 1 − n 2 n + 1 )) p if x ∈ [ n − 1 / 2 n + 1 , n ) , n p ( − 2 n + 1 x + ( 1 + n 2 n + 1 )) p if x ∈ [ n , n + 1 / 2 n + 1 ] , T n , p ( x ) = 0 otherwise , and we define the functions g p : [ 0 , + ∞ ) − → R as: + ∞ � g p ( x ) = T n , p ( x ) . n = 1 Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Algebrability 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 Known Results ujo, Bernal, Mu˜ 1. Ara´ noz, Prado and Seoane, 2017: c -lineability of sequences in MES . ujo, Bernal, Mu˜ 2. Ara´ noz, Prado and Seoane, 2017: Maximal dense-lineability of sequences in L 0 ([ 0 , 1 ]) such that f n → 0 in measure but not pointwise a.e.. Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability A 0 := { ( f n ) n : f n ∈ A , ∈ N , f n → 0 pointwise on [ 0 , + ∞ ) , � f n � L 1 → 0 } . Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability A 0 := { ( f n ) n : f n ∈ A , ∈ N , f n → 0 pointwise on [ 0 , + ∞ ) , � f n � L 1 → 0 } . Example Consider the sequence of functions ( f n ) n given by: + ∞ � f n ( x ) := T m ( x ) , ∀ n ∈ N , x ≥ 0 . m = n Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability A 0 := { ( f n ) n : f n ∈ A , ∈ N , f n → 0 pointwise on [ 0 , + ∞ ) , � f n � L 1 → 0 } . Example Consider the sequence of functions ( f n ) n given by: + ∞ � f n ( x ) := T m ( x ) , ∀ n ∈ N , x ≥ 0 . m = n Theorem The family of sequences A 0 on [ 0 , + ∞ ) is maximal lineable. Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability Consider now the spaces c 0 ( X ) := { F = ( f n ) n : f n ∈ X n ∈ N , d X ( f n , 0 ) → 0 } , { ( b n ) n : ∃ n 0 | b n ∈ B ∀ n ≤ n 0 , b n = 0 ∀ n > n 0 } . c 00 ( B ) := Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability Consider now the spaces c 0 ( X ) := { F = ( f n ) n : f n ∈ X n ∈ N , d X ( f n , 0 ) → 0 } , { ( b n ) n : ∃ n 0 | b n ∈ B ∀ n ≤ n 0 , b n = 0 ∀ n > n 0 } . c 00 ( B ) := A 0 := { ( f n ) n : f n ∈ A , ∈ N , d (( f n ) n , 0 ) → 0 } , where d (( f n ) n , ( g n ) n ) = sup d X ( f n , g n ) . n ∈ N Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Lineability Consider now the spaces c 0 ( X ) := { F = ( f n ) n : f n ∈ X n ∈ N , d X ( f n , 0 ) → 0 } , { ( b n ) n : ∃ n 0 | b n ∈ B ∀ n ≤ n 0 , b n = 0 ∀ n > n 0 } . c 00 ( B ) := A 0 := { ( f n ) n : f n ∈ A , ∈ N , d (( f n ) n , 0 ) → 0 } , where d (( f n ) n , ( g n ) n ) = sup d X ( f n , g n ) . n ∈ N Theorem The family of sequences A 0 on [ 0 , + ∞ ) is maximal dense-lineable. Pablo Jos´ e Gerlach Mena Linear Structures
Previous Concepts Continuous, integrable and unbounded functions Sequence spaces Algebrability Theorem The family of sequences A 0 on [ 0 , + ∞ ) is strongly-algebrable. Pablo Jos´ e Gerlach Mena Linear Structures
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