Spectral calculations on locally convex vector spaces I

We develop a holomorphic functional calculus for (multivalued linear) operators on locally convex vector spaces. This includes the case of fractional powers along Lipschitz curves.Comment: 18 page

Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\le\omega_1$. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube $I^{\omega_1}$ is homeomorphic to $I^{\omega_1}$. An analogous result is also proved for the cone over $I^{\omega_1}$. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved

Upper bounds for continuous seminorms and special properties of bilinear maps

If E is a locally convex topological vector space, let P(E) be the pre-ordered set of all continuous seminorms on E. We study, on the one hand, for g an infinite cardinal those locally convex spaces E which have the g-neighbourhood property in the sense of E. Jorda, i.e., spaces in which all sets M of continuous seminorms of cardinality up to g have an upper bound in P(E). On the other hand, we study bilinear maps b from a product of locally convex spaces E_1 and E_2 to a locally convex space F, which admit "product estimates" in the sense that for all p_{i,j} in P(F), i,j=1,2,..., there exist p_i in P(E_1) and q_j in P(E_2) such that p_{i,j}(b(x,y)) <= p_i(x)q_j(y) for all x in E_1, y in E_2. The relations between these concepts are explored, and examples given. The main applications concern spaces C^r_c(M,E)$ of vector-valued test functions on manifolds.Comment: 24 pages, LaTeX; v3: additional references, minor changes to more traditional terminolog

On some locally convex FK spaces

We provide necessary and/or sufficient conditions on vector spaces V of real sequences to be a Fréchet space such that each coordinate map is continuous, that is, to be a locally convex FK space. In particular, we show that if c00(I) ⊆ V ⊆ ∞(I) for some ideal I on ω, then V is a locally convex FK space if and only if there exists an infinite set S ⊆ ω for which every infinite subset does not belong to

Smooth norms and approximation in Banach spaces of the type C(K)

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be uniformly approximated by functions of class C^m. (ii) If C(K) admits an equivalent norm with locally uniformly convex dual norm, then C(K) admits an equivalent norm which is of class C^infty (except at 0)

Topologies related to (I)-envelopes

We investigate the question whether the (I)-envelope of any subset of a dual to a Banach space $X$ may be described as the closed convex hull in a suitable topology. If $X$ contains no copy of $\ell^1$ then the weak topology generated by functionals of the first Baire class in the weak$^*$ topology works. On the other hand, if $X$ contains a complemented copy of $\ell^1$ or $X=C([0,1])$ no locally convex topology works. If we do not require the topology to be locally convex, the problem is still open. We further introduce and compare several natural intermediate closure operators on a dual Banach space. Finally, we collect several intringuing open problems related to (I)-envelopes.Comment: 23 page