Algebrability of Certain Subsets of C 0 (R + )

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1 Int. J. Contemp. Math. Sciences Vol no HIKARI Ltd Algebrability of Certain Subsets of C 0 (R + ) Sima Niazi Department of Mathematics Estahban Branch Islamic Azad University Estahban Iran simaniazi sn@yahoo.com Ali Farokhinia Department of Mathematics Shiraz Branch Islamic Azad University Shiraz 7955 Iran farokhinia@iaushiraz.ac.ir Copyright c 203 S. Niazi and A. Farokhinia. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Abstract. Existence of an infinitely generated algebra in a certain set is called algebrability. In this paper we will show that the space of all continuous functions on R + that are not in any L p (R + ) is algebrable. Mathematics Subject Classification: Primary 47A6 Secondary 47L0 Keywords: Lineability algebrability spaceability pathological property. Introduction The main theorem of this paper concerns the existence of an infinitely generated algebra in the space of all continuous functions on R + that are not in any L p (R + ). This is a contribution to a very new area of research in mathematical analysis that is to search for large algebraic structures (linear spaces or algebras) in space of functions that are enjoying a special property. It has become a usual notation to call a subset M of a topological vector space X μ-lineable (respectively μ-spaceable) if M {0} contains a vector space (respectively closed vector space) of dimension μ. IfM contains an infinite-dimensional (closed) vector space then M will be shortly called lineable (spaceable).

2 744 S. Niazi and A. Farokhinia The origin of lineability and spaceability is due to Gurariy ([2 22]) that showed that there exists an infinite dimensional linear space such that every non-zero element of which is a continuous nowhere differentiable function on C[0; ]. Many examples of vector spaces of functions on R or C enjoying certain special properties have been constructed in the recent years. More recently many authors got interested in this subject and gave a wide range of examples. For instance in [4] it was shown that the set of everywhere surjective functions in R is 2 c -lineable (where c denotes the cardinality of R) and that the set of differentiable functions on R which are nowhere monotone is lineable in C(R). These behaviors occur sometimes in particularly interesting ways. For example in [22] Hencl showed that any separable Banach space is isometrically isomorphic to a subspace of C[0; ] whose non-zero elements are nowhere approximately differentiable and nowhere Holder. We refer the interested reader to [ ] for a wider range of results in this topic of lineability and spaceability. Of course one could go further and not just consider linear spaces but instead larger or more complex structures. For instance in [] the authors showed that there exists an uncountably generated algebra every non-zero element of which is an everywhere surjective function on C and in [5] it was shown that if E T the unit circle is a set of measure zero and if F(T) denotes the subset of C(T) of continuous functions whose Fourier series expansion diverges at every point of E then F(T) contains an infinitely generated and dense subalgebra. One of the newest result in this area ([8]) proves the existence of uncountably generated algebras inside the following sets of special functions: Sierpinski-Zygmund functions perfectly everywhere surjective functions and nowhere continuous Darboux functions. That a space contains an infinitely generated algebra is called algebrability. It is clear that algebrability implies lineability but studying the algebrability of a space is sometimes far harder than lineability. Very recently Bartoszewicz and Glab ([0]) have proved the existence of c- generated free algebra that is dense in the space of sequences in c 0 that are not summable with any power i. e. they show that c 0 \ {l p : p } is densely strongly c-algebrable. Our main concept of this paper is to extend the theorem to continuous functions that are not in any L p space. 2. Main Theorem Theorem 2.. The set C 0 (R + )\ {L p (R + ):p } is densely strongly c algebrable. Proof. Let {r α : α<c} be a linearly independent subset of [0 ] with respect to Q. Let A be a linear algebra generated by the set S = {ζ rα : α<c}

3 where ζ : R + R + is defined by Algebrability of certain subsets of C 0 (R + ) 745 ζ(x) = ln(x +2). We will show that any non-trivial combination of elements of S is either a null function or is not in any L p (R + ) for any p. To prove this we need to prove that for any matrix (k ij i m j n) of non-negative integers with non-zero and distinct rows and any β β 2... β m R which do not vanish simultaneously the function f := β ζ k r α +...+k n r αn β m ζ k mr α +...+k mnr αn is not in any L p (R + ) space. Note that every function ζ k ir α +...+k in r αn is in C 0 (R + ) so f is also in C 0 (R + ). From calculus we know that the above function has infinite integral if and only if the series {x n } defined by x n := f(n) for all n N diverges. Since {r α : α<c} is linearly independent the elements k i r α k in r αn i m are distinct. To simplify the notation put k i := k i r α k in r αn i m. We may assume that k <k 2 <...<k m and β 0. Then β ln k (n+2) + β β m ln k 2 (n+2) ln km (n+2) β ln k (n+2) β 2... βm. ln k 2 (n+2) ln km (n+2) Since k is smaller than each k 2... k m there is a N such that β 2 β m ln k 2 (n +2) ln km (n +2) < β 2ln k (n +2) for all n N. Hence β β 2 β m ln k (n +2) ln k 2 (n +2) ln km (n +2) β 2ln k (n +2) { for all n N. Since ln q (n+2) } / l p for any p and any q>0 the result follows. Now we prove that A is dense in C 0 (R + ). Consider the subalgebra A of C(R) generated by ζ and I and A 2 generated by ζ where ζ (x) = ln( x +2). Note that A = {x + fx Rf A 2 }. Since ζ = ζ R + and is strictly decreasing on R + soa separates the points of R and doesn t vanish at any point. Therefore by Stone Weierstrass theorem A is dense in C(R). This shows that A 2 is dense in C 0 (R + ). But A 2 A soa is also dense in C 0 (R + ). This completes the proof.

4 746 S. Niazi and A. Farokhinia Note 2.2. By extending the above ζ to R we can show the densely strongly c-algebrability of C 0 (R)\ {L p (R) :p }. A question that can be asked here is the following one. What is the largest cardinal number μ that C 0 (R)\ {L p (R) : p } is densely strongly μ- algebrable? References [] R. M. Aron J. A. Conejero A. Peris and J. B. Seoane-Sepulveda Uncountably generated algebras of everywhere surjective functions Bull. Belg. Math. Soc. Simon Stevin 7 (200) no [2] R. M. Aron D. Garcia and M. Maestre Linearity in non-linear problems RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 95 (200) no [3] R. M. Aron F. J. Garcia-Pacheco D. Perez-Garcia and J. B. Seoane-Sepulveda On dense-lineability of sets of functions on R Topology 48 (2009) [4] R. M. Aron V. I. Gurariy and J. B. Seoane-Sepulveda Lineability and spaceability of sets of functions on R Proc. Amer. Math. Soc. 33 (2005) no [5] R. M. Aron D. Perez-Garcia and J. B. Seoane-Sepulveda Algebrability of the set of non-convergent Fourier series Studia Math. 75 (2006) no [6] R. M. Aron and J. B. Seoane-Sepulveda Algebrability of the set of everywhere surjective functions on C Bull. Belg. Math. Soc. Simon Stevin 4 (2007) no [7] A. Bartoszewicz and S. Glab Algebrability of conditionally convergent series with Cauchy product J. Math. Anal. Appl. 385 (202) (to appear). [8] A. Bartoszewicz S. Glab D. Pellegrino and J. B. Seoane-Sepulveda Algebrability nonlinear properties and spacial functions Electronically published Cornell University Library to appear. (arxiv: v) [9] A. Bartoszewicz S. Glab and T. Poreda On algebrability of nonabsolutely convergent series Linear Algebra Appl. 435 (20) [0] A. Bartoszewicz and S. Glab Strong abgebrability of sets of sequences and functions Proc. Amer. Math. Soc. 4 (203) [] F. Bayart and L. Quarta Algebras in sets of queer functions Israel J. Math. 58 (2007) [2] L. Bernal-Gonzalez Dense-lineability in spaces of continuous functions Proc. Amer. Math. Soc. 36 (2008) no [3] G. Botelho D. Diniz and D. Pellegrino Lineability of the set of bounded linear nonabsolutely summing operators J. Math. Anal. Appl. 357 (2009) No [4] G. Botelho M. Matos and D. Pellegrino Lineability of summing sets of homogeneous polynomials Linear Multilinear Algebra 58 (200) No [5] A. Farokhinia Lineability of Space of Quasi-Everywhere Surjective Functions J. Math. Extension 6 (202) No. 3 Ser. No [6] J. L. Gamez-Merino G. A. Mu noz-fernandez D. Pellegrino and J. B. Seoane- Sepulveda Bounded and unbounded polynomials and multilinear forms: Characterizing continuity Linear Algebra Appl. DOI 0.06/j.laa (to appear in print). [7] J. L. Gamez-Merino G. A. Mu noz-fernandez V. M. Sanchez and J. B. Seoane- Sepulveda Sierpinski-Zygmund functions and other problems on lineability Proc. Amer. Math. Soc. 38 (200) no [8] J. L. Gamez-Merino G. A. Mu noz-fernandez and J. B. Seoane-Sepulveda Lineability and additivity in RR J. Math. Anal. Appl. 369 (200) no

5 Algebrability of certain subsets of C 0 (R + ) 747 [9] D. Garcia B. C. Grecu M. Maestre and J. B. Seoane-Sepulveda Infinite dimensional Banach spaces of functions with nonlinear properties Math. Nachr. 283 (200) no [20] V. I. Gurariy Subspaces and bases in spaces of continuous functions (Russian) Dokl. Akad. Nauk SSSR 67 (966) [2] Linear spaces composed of non-differentiable functions C.R. Acad. Bulgare Sci. 44 (99) 3-6. [22] S. Hencl Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Holder functions Proc. Amer. Math. Soc. 28 (2000) no [23] H. Lebesgue Lecons sur l integration et la recherche des fonctions primitives ChaGauthier-Willars 904. Received: June 203