We prove that if 𝓒 is a family of separable Banach spaces which is analytic with respect to the Effros Borel structure and no X ∈ 𝓒 is isometrically universal for all separable Banach spaces, then there exists a separable Banach space with a monotone Schauder basis which is isometrically universal for 𝓒 but not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces.

For each natural number N, we give an example of a Banach space X such that the set of norm attaining N-linear forms is dense in the space of all continuous N-linear forms on X, but there are continuous (N+1)-linear forms on X which cannot be approximated by norm attaining (N+1)-linear forms. Actually,X is the canonical predual of a suitable Lorentz sequence space. We also get the analogous result for homogeneous polynomials.

The well known Bishop-Phelps Theorem asserts that the set of norm attaining linear forms on a Banach space is dense in the dual space [3]. This note is an outline of recent results by Y. S. Choi [5] and C. Finet and the author [7], which clarify the relation between two different ways of extending this theorem.

Let be the class of Banach spaces X for which every weakly quasi-continuous mapping f: A → X defined on an α-favorable space A is norm continuous at the points of a dense $G_{\delta }$ subset of A. We will show that this class is stable under c₀-sums and ${\ell }^p$-sums of Banach spaces for 1 ≤ p < ∞.

Every Orlicz space equipped with Orlicz norm has weak sum property, therefore, it has weakly normal structure and fixed point property. A criterion of sum property also of normal structure for such spaces is given as well, which shows that every Orlicz space has isonormal structure.

Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that ${\Lambda }_{\varphi ,w}(0,\infty )$ (respectively, ${\Lambda }_{\varphi ,w}(0,1)$) is an order continuous Lorentz-Orlicz space. (1) ${\Lambda }_{\varphi ,w}$ has normal structure if and only if u₀ = 0 (respectively, ${\int }^{v₀}\varphi \left(u₀\right)·w<2andu₀<\infty ).$(2) ${\Lambda }_{\varphi ,w}$ has weakly normal structure if and only if ${\int }_0^{v₀}\varphi \left(u₀\right)·w<2$.

We give a polynomial version of Shmul'yan's Test, characterizing the polynomials that strongly attain their norm as those at which the norm is Fréchet differentiable. We also characterize the Gateaux differentiability of the norm. Finally we study those properties for some classical Banach spaces.

Let $(X,d)$, $(Y,\rho )$ be metric spaces and $f:X\to Y$ an injective mapping. We put ${\parallel f\parallel }_{Lip}=sup\{\rho (f\left(x\right),f\left(y\right))/d(x,y);x,y\in X,x\ne y\}$, and $dist\left(f\right)={\parallel f\parallel }_{Lip}.{\parallel f^{-1}\parallel }_{Lip}$ (the distortion of the mapping $f$). We investigate the minimum dimension $N$ such that every $n$-point metric space can be embedded into the space ${\ell }_{\infty }^N$ with a prescribed distortion $D$. We obtain that this is possible for $N\ge C{(logn)}^2{n}^{3/D}$, where $C$ is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into ${\ell }_p^N$ are obtained by a similar method.