Hamel Bases Research Articles

Connected Hamel bases in Hilbert spaces

Connected Hamel bases in Hilbert spaces

The Point-to-Set Principle and the dimensions of Hamel bases

We prove that, for every 0 ⩽ s ⩽ 1, there is a Hamel basis of the vector space of reals over the field of rationals that has Hausdorff dimension s. The logic of our proof is of particular interest. The statement of our theorem is classical; it does not involve the theory of computing. However, our proof makes essential use of algorithmic fractal dimension–a computability-theoretic construct–and the point-to-set principle of J. Lutz and N. Lutz (2018).

Definable Hamel bases and ${\sf AC}_\omega ({\mathbb R})$

There is a model of ${\sf ZF}$ with a lightface $\Delta ^1_3$ definable Hamel bases in which ${\sf AC}_\omega ({\mathbb R})$ fails.

On finite sums of periodic functions

Abstract It is shown that any function acting from the real line ℝ {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function x → exp ⁡ ( x 2 ) {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

Hamel bases and well–ordering the continuum

Hamel bases and well–ordering the continuum

Measurability properties of certain paradoxical subsets of the real line

Abstract The paper deals with the measurability properties of some classical subsets of the real line ℝ having an extra-ordinary descriptive structure: Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpiński sets. In particular, it is shown that there exists a translation invariant measure μ on ℝ extending the Lebesgue measure and such that all Sierpiński sets are measurable with respect to μ.

Additive iterative roots of identity and Hamel bases

We describe a class of discontinuous additive functions \({a:X\to X}\) on a real topological vector space X such that \({a^n={\rm id}_X}\) and \({a({\mathcal{H}}){\setminus} {\mathcal{H}} \neq\emptyset}\) for every infinite set \({{\mathcal{H}} \subset X}\) of vectors linearly independent over \({\mathbb{Q}}\). We prove the density of the family of all such functions in the linear topological space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced on \({{\mathcal{A}}_X}\) by the Tychonoff topology of the space X X . Moreover, we consider additive functions \({a\in{\mathcal{A}}_X}\) satisfying \({a^n={\rm id}_X}\) and \({a({\mathcal{H}})= {\mathcal{H}}}\) for some Hamel basis \({{\mathcal{H}}}\) of X. We show that the class of all such functions is also dense in \({{\mathcal{A}}_X}\). The method is based on decomposition theorems for linear endomorphisms.

On some Problem of Sierpiński and Ruziewicz Concerning the Superposition of Measurable Functions. Microscopic Hamel Basis

Abstract S. Ruziewicz and W. Sierpiński proved that each function f : ℝ → ℝ can be represented as a superposition of two measurable functions. Here, a strengthening of this theorem is given. The properties of Lusin set and microscopic Hamel bases are considered

Additive involutions and Hamel bases

Our aim is to give other proofs of some slight generalizations of results from Baron (Aequat Math, 2013). We describe larger classes of discontinuous additive involutions \({a:X\to X}\) on a topological vector space X such that \({a(H)\setminus H\neq\emptyset}\) holds for a sufficiently numerous set \({H\subset X}\) of vectors linearly independent over \({{\mathbb{Q}}}\) . We also consider the topological vector space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced by the Tychonoff topology of the space XX. We prove in a simple way that some classes of discontinuous additive involutions are dense in the topological vector space \({{\mathcal{A}}_X}\) .

On additive involutions and Hamel bases

We provide an example of a discontinuous involutory additive function $${a: \mathbb{R}\to \mathbb{R}}$$ such that $${a(H) \setminus H \ne \emptyset}$$ for every Hamel basis $${H \subset \mathbb{R}}$$ and show that, in fact, the set of all such functions is dense in the topological vector space of all additive functions from $${\mathbb{R}}$$ to $${\mathbb{R}}$$ with the Tychonoff topology induced by $${\mathbb{R}^{\mathbb{R}}}$$ .

On some properties of Hamel bases and their applications to Marczewski measurable functions

AbstractWe introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.

Existence of complete vector topologies with prescribed conditions

In every infinite-dimensional vector space we build a complete, non-locally convex linear Hausdorff topology under the prescribed condition that it shall be compatible with a bounded algebraic structure for the space, that is, it admits a bounded Hamel basis whose coefficient linear functionals are continuous. As a byproduct, we obtain the existence of non-locally convex topologies having the approximate and fixed point property as prescribed conditions. The obstruction of bounded Hamel bases by complete vector topologies on linear space of countable dimension is also discussed.

On bases in Banach spaces

We investigate various kinds of bases in infinite dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain complete minimal systems in `∞ as well as in separable Banach spaces. Outline. The paper is concerned with bases in infinite dimensional Banach spaces. The first section contains the definitions of the various kinds of bases and biorthogonal systems and also summarizes some set-theoretic terminology and notation which will be used throughout the paper. The second section provides a survey of known or elementary results. The third section deals with Hamel bases and contains some consistency results proved using the forcing technique. The fourth section is devoted to complete minimal systems (including Φ-bases and Auerbach bases) and the last section contains open problems. ∗The research for this paper began during the Workshop on Set Theory, Topology, and Banach Space Theory, which took place in June 2003 at Queen’s University Belfast, whose hospitality is gratefully acknowledged. The workshop was supported by the Nuffield Foundation Grant NAL/00513/G of the third author, the EPSRC Advanced Fellowship of the second author and the grant GACR 201/03/0933 of the fourth author. 1

On the complexity of Hamel bases of infinite-dimensional Banach spaces

We call a subset S of a topological vector space V linearly Borel, if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V . It will be shown that a Hamel base of an infinite dimensional Banach space can never be linearly Borel. This answers a question of Anatolij Plichko. In the sequel, let X be any infinite dimensional Banach space. A subset S of X is called linearly Borel (w.r.t. X), if for every positive integer n, the set of all linear combinations with n vectors of S is a Borel subset of X. Since X is a complete metric space, X is a Baire space, i.e., a space in which non-empty open sets are not meager (cf. [1, Section 3.9]). Moreover, all Borel subsets of X have the Baire property, i.e., for each Borel set S, there is an open set O such that O∆S is meager, where O∆S = (O S) ∪ (S O). This is already enough to prove the following. Theorem. If X is an infinite dimension Banach space and H is a Hamel base of X, then H is not linearly Borel (w.r.t. X). I would like to thank the Swiss National Science Foundation for its support during the period in which the research for this paper has been done. On The Complexity Of Hamel Bases 2 Proof: Let X be any infinite dimensional Banach space over the field F and let H be any Hamel base of X. For a positive integer n, let [H] be the set of all n-element subsets of H and let

Vitali sets and Hamel bases that are Marczewski measurable

We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set

On elementary functions characterized by functional relationships

It is well known that any real‐valued function f continuous on the real line and satisfying the functional relationship f(a + b)= f(a) + f(b)for all real aand bmust be of the form f(x) = cxwhere cis a constant. Further, concepts such as the axiom of choice, well ordering, and Hamel bases can be employed to prove that there exists a function f discontinuous everywhere on the real line such that f(a + b) =f(a) + f(b)for all real aand b.

On quadratic functionals and some properties of Hamel bases

It is shown that any quadratic functional q : R → R whose restriction to a second category Baire subset of R is continuous must be continuous on the whole real line. For this purpose we investigate regularity properties of quadratic functionals with a continuous restriction on an analytic set containing 1 2 (H + H) , where H is a Hamel basis of R . We also prove that for any Baire set T ⊂ R of the second category there exists a Hamel basis H ⊂ R such that 1 2 (H + H) ⊂ T .

Infinite combinatorics and definability

The topic of this paper is Borel versions of infinite combinatorial theorems. For example it is shown that there cannot be a Borel subset of [ω] ω which is a maximal independent family. A Borel version of the delta systems lemma is proved. We prove a parameterized version of the Galvin-Prikry Theorem. We show that it is consistent that any ω 2 cover of reals by Borel sets has an ω 1 subcover. We show that if V \\= L, then there are π 1 1 Hamel bases, maximal almost disjoint families, and maximal independent families.

Garnir's dream spaces with Hamel bases

Garnir's dream spaces with Hamel bases

On a problem of Aczél and Erdös concerning Hamel bases

On a problem of Aczél and Erdös concerning Hamel bases