Strictly convex norms and topology pdf

Finally, section 4 presents a topological frame in which these results can easily. Smith1 1university college dublin, ireland 12th 19th january 20 richard j. Convex set convex function strictly convex function concave. Dynamics of nonexpansive maps on strictly convex banach. Gruenhage compacta and strictly convex dual norms request pdf. Some examples concerning strictly convex norms on ck spaces. Strictly convex norms and topology article pdf available in proceedings of the london mathematical society 1043. To see that the converse statements are not true, observe that fx xis convex but not strictly convex and fx x4 is strictly convex but not strongly convex why. Troyanski strictly convex norms and topology july 2011 1 11. If x is a normable topological vector space, let k k be a norm on x that determines the topology. Here, by usual convex sets we mean convex sets in real linear spaces.

Dec 15, 2006 in this paper, we construct gateaux norms by utilising both strictly convex dual norms and the smooth norms of theorem 1. We present a characterization in linear topological terms. Ams proceedings of the american mathematical society. Conversely, let u be a bounded convex neighborhood of 0 in x. In mathematics, a strictly convex space is a normed vector space x, for which the closed unit ball is a strictly convex set. Introductionand preliminaries among the most wellestablishedgeometrical properties ofnorms are smoothness and strict convexity. Convex functions donald bren school of information and. Strictly convex norms and topology 199 the next signi. We present a characterization in linear topological terms of. Notes on locally convex topological vector spaces 5 ordered family of. A point uof a convex set cin a vector space is called an extreme point of. We say that is concave or concave down if 00x 0 for all x2r.

X,nlower semicontinuous norm if and only if the square of the ball b x 2 has a certain linear, topological decomposition with respect to. First papers on axiomatic convexities come from early. Cx if x is a topological space is the space of realvalued, bounded, continuous functions on x. X,nlower semicontinuous norm if and only if the square of the ball b x 2 has a certain linear, topological decomposition with respect to a given norming subspace n. A locally convex topology on a vector space is defined analytically by a family of semi norms cf. Norm mathematics from wikipedia, the free encyclopedia in linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector. The ellipsoid with the hilbert metric is the beltramiklein model for hn, and the nsimplex with the hilbert metric is isomorphic to rnwith a polygonal norm. We have the following characterization of the existence of equivalent strictly convex norms on banach spaces. It is further prove in 3 that every normed vector space can be made into s. Some examples concerning strictly convex norms onck spaces. A uniformly convex space is a normed vector space such that, for every. This means that gruenhage spaces are not necessary for the construction of strictly convex dual norms on dual banach spaces, answering a question posed by orihuela, troyanski and the author.

Among others, we show that if rn, is a strictly convex normed space that. Seminorm as the topology with basis of neighbourhoods of zero consisting of the sets of the form, where runs through the natural numbers and is the family of all finite intersections of the sets of the form. It is also strongly convex and hence strictly convex too, with strong convexity constant 2. Locally convex topology encyclopedia of mathematics.

We demonstrate that the space of stable norms associated to metrics on t2 forms a proper dense subset of the space of strictly convex norms on r2. The fixed point property in banach spaces via the strict. By using results of edelstein and lyubich, we show that if x rn,k k is strictly convex and xhas no 1complemented euclidean plane, then every bounded orbit of a nonexpansive map f. A topological characterization of dual strict convexity in. Smith and stanimir troyanski abstract we introduce a new topological property called. Radstrom, convexity and norm in topological groups in the example of figure 1, it is clear that the sets 1 a fill up more and n more of the convex hull of a as n increases.

Trees, gateaux norms and a problem of haydon bulletin of. Our aim in this paper is the study of those normed spaces which admit an equivalent strictly convex norm. T1, locally convex, but not locally bounded, showing tw to be strictly weaker than. Convexity, inequalities, and norms convex functions you are probably familiar with the notion of concavity of functions. Hot network questions could byzantines bridge a deep strait separating 2 continents. R, we say that is convex or concave up if 00x 0 for all x2r. Let x be a strictly convex metric space with convex round balls and k be a convex and compact subset of x and y 2 x. Gruenhage compacta and strictly convex dual norms sciencedirect. Topology since halfspaces are convex and intersections of collections of convex sets are convex, we have the following consequence of the theorem. The weak topology of locally convex spaces and the weak.

The vector 2norm square kxk2 euclidean norm square and frobenius norm square kxk2 f, for example, are strictly convex functions of their respective argument. Strictly convex norms geometry of on amenable groups. Smith ucd strictly convex norms and topology january 20 1 14. Dynamics of nonexpansive maps on strictly convex banach spaces. Strictly convex norms and topology by jose orihuela, richard j. Convex functions basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions logconcave and logconvex functions convexity with respect to generalized inequalities 31. It can be used to determine the derivative of a norm in an arbitrary banach space. Then b 1 is clearly a bounded convex neighborhood of 0. In recent years, trees have assumed an important role in renorming theory, largely through the work of haydon see, for example, 3, 4 and 6.

Show that the sum of strictly concave function is strictly concave. In this sence, a unit ball is a strictly convex set, however, a norm as a function is not strictly convex. Strictly convex norms and topology 41st winter school in abstract analysis, kacov richard j. Using, we characterize those banach spaces which admit equivalent strictly convex norms, and give an internal topological characterization of those scattered compact spaces k, for which the dual banach space c k admits an equivalent strictly convex dual norm. In particular, given a strictly convex norm kk 1 on r 2 we construct a sequence hkk j i 1 1 of stable norms that converge to kk 1 in the topology of. A locally convex topology on a vector space is defined analytically by a family of seminorms cf. Maximiliansuniversitat, germany, 20152016, available in pdf format at. A topological convex structure consists of a set, equipped with both a convexity and a topology, such that all polytopes are closed. Using, we characterise those banach spaces which admit equivalent strictly convex norms. In fact, for sufficiently large n any given interior point of the convex hull of a will be an element of. Strictly convex norms and topology richard smith1 with jose orihuela and stanimir troyanski2 1university college dublin, ireland 2university of murcia, spain new york topology conference, summer 2011 j. The weak topology of locally convex spaces and the weak topology of their duals jordan bell jordan. Strictly convex metric spaces with convex round balls inherent some good properties that we formulate as lemmas.

By putting extra assumptions on the derivatives of the norm, we also. We establish some relationships between and other topological concepts. Does there exist a norm g r, such that balls are strictly convex. On strictly convex and strictly convex according to an index. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere. In addition, we show that the banach space of continuous functions admits a smooth bump function. Request pdf gruenhage compacta and strictly convex dual norms we prove that if k is a gruenhage compact space then ck admits an equivalent, strictly convex dual norm. If, in addition, the closure of each convex set is convex, then the structure is called closure stable. Is the convex combination of a convex and strictly convex set, strictly convex. It is strictly convex, even though the second derivative is not strictly positive at all points.

Regarding strictly convex norms, the authors of 6 recently showed that x, which can be a dual space, admits an equivalent, strictly convex. Prove that a strictly concave function of a strictly concave function maybe strictly convex. Sorry, we are unable to provide the full text but you may find it at the following locations. Pdf strictly convex norms and topology researchgate. Using, we characterise those banach spaces which admit equivalent strictly convex norms, and give an internal topological characterisation of those. Topological dynamics of nonstrictly convex hilbert geometries 3 y x a b d x. Simmons, \introduction to topology and modern analysis.