The author introduced in [3] the notion of an M-con~ected closed subset of a norrned linear space and defined the class of (BM)-spaces. An M-connected closed subset of a finite dimensional normed linear space is a sun and a sun in a space which is either of dimension two or is a finite dimensional (m>i)-space is M-connected. Theorem 1 asserts that an M-connected closed subset of a finite dimensional space is n-connected for all n = 0,1,2 .... Theorem 2 relates Tl1eorem 1 to the results of [3]. Theorem 3 is an improvement of a result of Koshcheev and asserts that a sun in a finite dimensional space is path-connected.

Two approximation theoretic properties of Hilbert spaces used in the proof of a result concerning Chebyshev sets obtained by Frerking and Westphal [4] are discussed. Investigation of one leads to a characterisation of inner product spaces of dimension at least three; it is an improvement of one due to Berens [3]. The other property is shared by all finite dimensional spaces [3] and here a topological result of which that fact is a simple consequence is proved.

A general approach is given to Lagrangian necessary conditions for a minimax problem, The necessary conditions become sufficient for a mini max under extra hypotheses, with either concave/convex or invex functions, and restrictions on the constraints. A minimax is shown to relate to a weak minimum of a vector function. The sensitivity of a minimax value to a perturbation is related to the gradient of a Lagrangian function with respect to the parameter.

Consider a non-empty proper closed subset K of a Banach space $(X, \| \cdot \|)$ where the norm is uniformly Gateaux (uniformly Frechet) differentiable. Then the associated distance function d is guaranteed to be Gateaux differentiable on a dense subset D of X\K. Furthermore, Gateaux (Frechet) differentiability of the distance function at a point $x \in$ X\K can be characterised in terms of the weak* sequences ${d' (x_n)}$ where ${x_n}$ is a sequence in D converging to x.

Recently David Preiss proved a remarkable theorem about dense differentiability of locally Lipschitz functions on Banach spaces. Using his result and adopting a technique of Petru: Kenderov used to prove generic differentiability of convex ftLnctions we establish similar generic differentiability properties for locally Lipschitz functions. We apply our results to determine further differentiability properties of distance functions.

Increasing sequences of contractive projections on a reflexive $L^p$ space share an unconditionality property similar to that exhibited sequences of self-adjoint projections on a Hilbert space. A slight variation of this property is shown to be precisely the correct condition on a reflexive Banach space to ensure that every operator with a contractive $AC$-functional calculus is scalar-type spectral.

Vl e transform the representation of monotone operators due to Krauss to get a representation of monotone operators in terms of the subdifferentials of convex functions on the product of the space and its duat The convex functions representing maximal monotone operators satisfY a minimality condition.

Given a norm-one linear functional u$*$ on a normed linear space E we give necessary and sufficient conditions for the separation by u$*$ of a convex set C from B[x,$d_c$(x)] to imply the existence, or existence and 1..miqueness, of a nearest point in C to x.

For a large class of compact topological spaces it is proved that the majority (in the Baire category sense) of the two-person zero-sum games have unique solution.

An open problem in operator theory asks whether for a complete atomic Boolean subspace lattice $\ell$ the rank one subalgebra of Alg%\ell% is strong operator dense in Alg%\ell%. A very special case of this problem turns out to be equivalent to an open problem in the Theory of Bases. Here various related questions are surveyed and some positive results are given.

We show that if $x_n$ is optimal for the problem \[ sup\left\{\sum_{x_n}{1} log x(s)ds | \sum_0^1 (x(s)- \hat{x}(s))s^i ds = 0 , i = 0, \cdots,n , 0 \leq x \in L_1[0,1]\right\}, \] then $\frac{1}{x_n} \rightarrow \frac{1}{\hat{x}}$ weakly in $L_1$ (providing $\hat{x}$ is continuous and strictly positive). This result is a special case of a theorem for more general entropic objectives and underlying spaces.

Let's say that a closed subspace $M$ of a Banach space $X$ is absolutely Chebyshev if it is Chebyshev and, for each $x \in X, \| x \|$ can be expressed as a function of only $d(x,M)$ and $\| P_m(x)\|$. A typical example is a dosed subspace of a Hilbert space. Absolutely Chebyshev subspaces are, modulo renorming, the same as semi-$L$-summands. We show that any real Banach space can be absolutely Chebyshev in some larger space, with a nonlinear metric projection. Dually, it follows that if $M$ has the 2-ball property but not the 3-ball property in $X$, no restriction exists on the quotient space $X/M$. It is not known whether such examples can be found in complex Banach spaces.

Let $F$ be a subspace lattice on a (complex) Hilbert space H . A subspace lattice $F$ on a Hilbert space K is a realization of $F$ on K if $G$ is lattice-isomorphic to $F$. In 1975 it was proved that if $F$ is completely distributive every realization $G$ of it is reflexive (that is, $F$ is the set of invariant subspaces of a family of operators). A partial converse has recently been found: If every realization of $F$ is reflexive and $F$ has a finite--dimensional realization, then $F$ is completely distributive. This is proved by showing that every non-distributive subspace lattice on a finite-dimensional space has a non-reflexive realization on the same space.

In recent years we have seen the important development in functional analysis of the study called "geometric analysis" which concerns asymptotic properties of sets in finite dimensional spaces where the dimension increases to infinity. The significance of this development is illustrated in four example areas where the author has been a contributor.

For certain m-tuples $a = {a_1, ... ,a_n)$ of elements $a_j$ in a unital Ba11ach algebra, we construct a joint spectrum $\gamma(a)$ and a functional calculus with a spectral mapping theorem. It is not assumed that the $a_j$ j commute but rather that they commute modulo the Jacobson radical of the algebra they generate. For matrices, this last condition is equivalent to their being simultaneously triangularizable. This work extends that of M.E. Taylor, R.F.V. Anderson, and A. Mcintosh and A. Pryde.

Subject to certain regularity hypotheses, the Kuhn-Tucker conditions are necessary for optimality in nonlinear programs. These conditions become sufficient under assumptions of invexity. This paper presents some known results and new observations on invexity, with extension to optimal control problems.

The concept of weak- orthogonality for a Banach lattice is examined. A proof that in such a lattice non expansive self maps of a non empty weakly compact convex set have fixed points is outlined. A geometric generalization of weak- orthogonality is introduced and related to the Opial condition.

Bellman's optimality equation of dynammic programming is examined in the context of a discrete-time, continuous-state economic development model. The main focus of the paper is on the interpretation of this functional equation as a linear programming problem in an infinite-dimensional setting. The connection between this linear programming problem and Bellman's functional equation is developed using a theory of equivalent models. The discrete-state version of the problem is discussed by usual theory for finite-dimensional linear programming. Then the abstract theory for infinite-dimensional linear progTamming is applied to the continuous-state problem in order to obtain results on existence and strong duality, The paper concludes with several simple examples of the dual pair of continuous linear progrmTmling problems.

Let X be a Banach space and C be a closed, convex subset of X such that N(C), the subset of its non support points, is non empty. We investigate differentiability properties of convex functions defined on N(C) and recover many results known to be true in the case N(C) = int(C).