This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F\left(x\right):=F̃x^{(2n+1)}$ where $F̃:X^{2n+1}\to Y$ is a continuous (2n+1)-linear operator.

We prove a uniform version of the converse Taylor theorem in infinite-dimensional spaces with an explicit relation between the moduli of continuity for mappings on a general open domain. We show that if the domain is convex and bounded, then we can extend the estimate up to the boundary.

We show that a $C^k$-smooth mapping on an open subset of ${ℝ}^n$, $k\in \mathbb{N}\cup \{0,\infty \}$, can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.

We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into c₀ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical ${\ell }_p$-spaces into c₀ and give other applications. We prove that if a Banach space embeds almost isometrically into c₀, then it embeds linearly almost isometrically into c₀. We also study Lipschitz embeddings into...

In this paper we study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by Alexandrov, Pogorelov, and Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness...

Given a compact manifold $N^n\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p\<\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s,p}(Q^m;N^n)$ when $N^n$ is $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ when $N^n$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s,p}\cap W^{1,sp}$.

Let f be a smooth function defined on a finite union U of open convex sets in a locally convex Lindelöf space E. If, for every x ∈ U, the restriction of f to a suitable neighbourhood of x admits a smooth extension to the whole of E, then the restriction of f to a union of convex sets that is strictly smaller than U also admits a smooth extension to the whole of E.

Let M be a separable $C^{\infty }$ Finsler manifold of infinite dimension. Then it is proved, amongst other results, that under suitable conditions of local extensibility the germ of a $C^{\infty }$ function, or of a $C^{\infty }$ section of a vector bundle, on the union of a closed submanifold and a closed locally compact set in M, extends to a $C^{\infty }$ function on the whole of M.

Let $X_1,\cdots ,X_n$ be Banach spaces and $f$ a real function on $X=X_1\times \cdots \times X_n$. Let $A_f$ be the set of all points $x\in X$ at which $f$ is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if $X_1,\cdots ,X_{n-1}$ are Asplund spaces and $f$ is continuous (respectively Lipschitz) on $X$, then $A_f$ is a first category set (respectively a $\sigma$-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f:X\to Y$ is a Lipschitz mapping, then there exists a $\sigma$-upper porous set $A\subset X$ such that $f$ is Fréchet differentiable at every...

We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y, then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space X has the bounded approximation...

Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and $L^2$-maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented...

In this article we formalize the definition and some facts about continuous functions from R into normed linear spaces [14].

We prove that if X is an infinite-dimensional Banach space with $C^p$ smooth partitions of unity then X and X∖ K are $C^p$ diffeomorphic for every weakly compact set K ⊂ X.

We study the set f’(X) = f’(x): x ∈ X when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a $C^p$-smooth bump b:X → ℝ such that $\left|\right|b^{\left(p\right)}{\left|\right|}_{\infty }$ is finite, then any connected open subset of X* containing 0 is the range of the derivative of a $C^p$-smooth bump. We also study the finite-dimensional case which is quite different. Finally,...