We give a brief account of some early and some recent results in the area now known as “lineability” and “spaceability.” Several examples, results, and open questions shall be presented. This article is an expanded version of a lecture given by the first author at a satellite conference of the 9ECM during June 2024, “From Classical to Modern Analysis: In memory of Professor José Carlos Petronilho.”

Let B_N be the Euclidean ball of ℂ^N . The space H^∞ (B_N) of bounded holomorphic functions on B_N is known to have a predual, denoted by G^∞ (B_N) . We study the functions in H^∞ (B_N) that attain their norm as elements of the dual of G^∞ (B_N) . We also examine similar questions for the polydisc algebra H^∞ (𝔻^N) and for the space of Dirichlet series 𝒟^∞ (ℂ_+).

Let X$X$ and Y$Y$ be complex Banach spaces with BX$B_X$ denoting the open unit ball of X$X$. This paper studies various aspects of the holomorphic Lipschitz space HL0(BX,Y)$\mathcal {H}L_0(B_X,Y)$, endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets Lip0(BX,Y)$\operatorname{Lip}_0(B_X,Y)$ of Lipschitz mappings and H infinity(BX,Y)$\mathcal {H}<^>\infty (B_X,Y)$ of bounded holomorphic mappings, from BX$B_X$ to Y$Y$. Thanks to the Dixmier-Ng theorem, HL0(BX,C)$\mathcal {H}L_0(B_X, \mathbb {C})$ is indeed a dual space, whose predual G0(BX)$\mathcal {G}_0(B_X)$ shares linearization properties with both the Lipschitz-free space and Dineen-Mujica predual of H infinity(BX)$\mathcal {H}<^>\infty (B_X)$. We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that G0(BX)$\mathcal {G}_0(B_X)$ contains a 1-complemented subspace isometric to X$X$ and that G0(X)$\mathcal {G}_0(X)$ has the (metric) approximation property whenever X$X$ has it. We also analyze when G0(BX)$\mathcal {G}_0(B_X)$ is a subspace of G0(BY)$\mathcal {G}_0(B_Y)$, and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.

We prove the following Farkas’ Lemma for simultaneously diagonalizable bilinear forms: If $$A_1,\ldots ,A_k$$ , and $$B:\mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathbb {R}$$ are bilinear forms, then one—and only one—of the following holds: We study evaluation maps over the space of bilinear forms and consequently construct examples in which Farkas’ Lemma fails in the bilinear setting.

For a complex Banach space X with open unit ball $$B_X,$$ consider the Banach algebras $$\mathcal {H}^\infty (B_X)$$ of bounded scalar-valued holomorphic functions and the subalgebra $$\mathcal {A}_u(B_X)$$ of uniformly continuous functions on $$B_X.$$ Denoting either algebra by $$\mathcal {A},$$ we study the Gleason parts of the set of scalar-valued homomorphisms $$\mathcal {M}(\mathcal {A})$$ on $$\mathcal {A}.$$ Following remarks on the general situation, we focus on the case $$X = c_0,$$ giving a complete characterization of the Gleason parts of $$\mathcal {M}(\mathcal {A}_u(B_{c_0}))$$ and, among other things, showing that every fiber in $$\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$$ over a point in $$B_{\ell _\infty }$$ contains $$2^c$$ discs lying in different Gleason parts.

We introduce and explore a new property related to reflexivity that plays an important role in the characterization of norm attaining operators. We also present an application to the theory of compact perturbations of linear operators and characterize norm attaining scalar-valued continuous $2$-homogeneous polynomials on $\ell_{2}$.

We study the notions of extendability and domain of holomorphy in the infinite-dimensional case. In this setting it is also true that the notions of domain of holomorphy and weak domain of holomorphy are equivalent. We also prove that the set of non-extendable functions belonging to some classes X(B) subset of H(B), B being the open unit ball in a separable complex Banach space, is a lineable and dense G(delta). Moreover, when Omega is H-b-holomorphically convex (defined in the text), it is shown that the set of non-extendable holomorphic functions on Omega is a lineable and dense G(delta) set.

For a complex Banach space $X$ with open unit ball $B_X,$ consider the Banach algebras $\mathcal H^\infty(B_X)$ of bounded scalar-valued holomorphic functions and the subalgebra $\mathcal A_u(B_X)$ of uniformly continuous functions on $B_X.$ Denoting either algebra by $\mathcal A,$ we study the Gleason parts of the set of scalar-valued homomorphisms $\mathcal M(\mathcal A)$ on $\mathcal A.$ Following remarks on the general situation, we focus on the case $X = c_0.$

We continue the study, started in [12], of the search for algebraic structures one can find within the sets of injective linear functions. We shall focus on the cases when the operators are considered both on finite dimensional and infinite dimensional domains. We also study the set of continuous surjective linear operators.

We study the existence of separation theorems by polynomials that are invariant under a group action. We show that if G is a finite subgroup of \(\textit{GL}(n,{\mathbb {C}})\), K is a set in \({\mathbb {C}}^{n}\) that is invariant under the action of G and z is a point in \({\mathbb {C}}^{n}\setminus K\) that can be separated from K by a polynomial Q, then z can be separated from K by a G-invariant polynomial P. Furthermore, if Q is homogeneous then P can be chosen to be homogeneous. As a particular case, if K is a symmetric polynomially convex compact set in \({\mathbb {C}}^{n}\) and \(z\notin K\) then there exists a symmetric polynomial that separates z and K.

University, vector measures, baseball, tensor products, traveling, Grothendieck's inequality, practical jokes and stories.He was even passionate about the fact that he never learned to drive.He was serious about everything and, yet, everything was fun to Joe.To Joe there was a tale behind every theorem, a person behind every example, a community behind every mathematical theory, interconnections between any two mathematical areas.Whatever the social or scientific occasion, he delighted in telling the tale to whomever would listen.Joe roamed the world as an ambassador extraordinaire, explaining to the mildly initiated the works of Grothendieck, James, Pełczyn ´ski, Pettis, and Pisier.Today we are lucky he recognized and developed his gift as a communicator and that he chose to share his many insights with us.

Given a proper holomorphic mapping \(g:\varOmega \subseteq {\mathbb {C}}^{n}\longrightarrow \varOmega ' \subseteq {\mathbb {C}}^{n}\) and an algebra of holomorphic functions \({\mathcal {B}}\) (e.g. \({\mathscr {P}}(K)\) where \(K\subset \varOmega \) is a compact set, \({\mathcal {H}}(U)\), A(U) or \({\mathcal {H}}^{\infty }(U)\) where U is an open and bounded set with \(\overline{U}\subset \varOmega \)), we study the subalgebra \({\mathcal {B}}_{g}\) of all functions compatible with the equivalence relation defined by the proper mapping g. We provide alternative representations of these algebras and describe the fibers in their spectra. Among other examples we relate the algebras of functions that are invariant under permutations and the algebras of functions defined on the symmetrized polydisk.

Let X be a complex Banach space with open unit ball B-x. We describe some recent work and a number of open problems related to the maximal ideal spaces of the Frechet algebra of holomorphic functions of bounded type on X and the Banach algebra of bounded holomorphic functions on B-X.

Given a surjective mapping $f : E \to F$ between Banach spaces, we investigate the existence of a subspace $G$ of $E$, with the same density character as $F$, such that the restriction of $f$ to $G$ remains surjective. We obtain a positive answer whenever $f$ is continuous and uniformly open. In the smooth case, we deduce a positive answer when $f$ is a $C^1$-smooth surjection whose set of critical values is countable. Finally we show that, when $f$ takes values in the Euclidean space $\mathbb R^n$, in order to obtain this result it is not sufficient to assume that the set of critical values of $f$ has zero-measure.

We show that the approximate hyperplane series property (AHSp) is stable under finite l(p)-sums (1 <= p <infinity). As a consequence, we obtain that the class of spaces Y such that the pair (l(1,) Y) has the Bishop-Phelps-Bollobas property for operators is stable under finite l(p)-sums for 1 <= p < infinity. We also deduce that every Banach space of dimension at least 2 can be equivalently renormed to have the AHSp but to fail Lindenstrauss property beta . We also show that every infinite-dimensional Banach space admitting an equivalent strictly convex norm also admits such an equivalent norm failing the AHSp.

The Hardy--Littlewood inequalities on $\ell _{p}$ spaces provide optimal exponents for some classes of inequalities for bilinear forms on $\ell _{p}$ spaces. In this paper we investigate in detail the exponents involved in Hardy--Littlewood type inequalities and provide several optimal results that were not achieved by the previous approaches. Our first main result asserts that for $q_{1},...,q_{m}>0$ and an infinite-dimensional Banach space $Y$ attaining its cotype $\cot Y$, if \begin{equation*} \frac{1}{p_{1}}+...+\frac{1}{p_{m}}<\frac{1}{\cot Y}, \end{equation*} then the following assertions are equivalent: (a) There is a constant $C_{p_{1},...,p_{m}}^{Y}\geq 1$ such that \begin{equation*} \left( \sum_{j_{1}=1}^{\infty }\left( \sum_{j_{2}=1}^{\infty }\cdots \left( \sum_{j_{m}=1}^{\infty }\left\Vert A(e_{j_{1}},...,e_{j_{m}})\right\Vert ^{q_{m}}\right) ^{\frac{q_{m-1}}{q_{m}}}\cdots \right) ^{\frac{q_{1}}{q_{2}} }\right) ^{\frac{1}{q_{1}}}\leq C_{p_{1},...,p_{m}}^{Y}\left\Vert A\right\Vert \end{equation*} for all continuous $m-$linear operators $A:\ell _{p_{1}}\times \cdots \times \ell _{p_{m}}\rightarrow Y.$ (b) The exponents $q_{1},...,q_{m}$ satisfy \begin{equation*} q_{1}\geq \lambda _{m,\cot Y}^{p_{1},...,p_{m}},q_{2}\geq \lambda _{m-1,\cot Y}^{p_{2},...,p_{m}},...,q_{m}\geq \lambda _{1,\cot Y}^{p_{m}}, \end{equation*} where, for $k=1,...,m,$ \begin{equation*} \lambda _{m-k+1,\cot Y}^{p_{k},...,p_{m}}:=\frac{\cot Y}{1-\left( \frac{1}{ p_{k}}+...+\frac{1}{p_{m}}\right) \cot Y}. \end{equation*} As an application of the above result we generalize to the $m$-linear setting one of the classical Hardy--Littlewood inequalities for bilinear forms. Our result is sharp in a very strong sense: the constants and exponents are optimal, even if we consider mixed sums.

We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical results of Runge, Mergelyan and Vitushkin. We also strengthen the notion of universal Dirichlet series.