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Continuous operators on asymmetric normed spaces. (English) Zbl 1199.54165

For a real linear space, a function $p:X\to {ℝ}^{+}$ is called an asymmetric norm on $X$ if for all $x,y\in X$ and $r\in {ℝ}^{+}$, (i) $p\left(x\right)=p\left(-x\right)=0$; (ii) $p\left(rx\right)=rp\left(x\right)$; (iii) $p\left(x+y\right)\le p\left(x\right)+p\left(y\right)$. For an asymmetric norm $p$ on $X$, ${p}^{-1}$, defined on $X$ by ${p}^{-1}\left(x\right)=p\left(-x\right)$ is also an asymmetric norm on $X$; the function ${p}^{s}$ defined on $X$ by ${p}^{s}\left(x\right)=max\left\{p\left(x\right),{p}^{-1}\left(x\right)\right\}$ is obviously a norm on $X$; also, for a normed lattice $\left(X,\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel \right)$, $p\left(x\right)=\parallel {x}^{+}\parallel$ with ${x}^{+}=sup\left\{x,0\right\}$ is an asymmetric norm on $X$.

The author uses the symbol $LC\left(X,Y\right)$ to denote the set of all continuous linear mappings from $\left(X,p\right)$ to $\left(Y,q\right)$ where $p$, $q$ are asymmetric norms whereas $L{C}^{s}\left(X,Y\right)$ is used to denote the set of all continuous linear mappings from $\left(X,{p}^{s}\right)$ to $\left(Y,{q}^{s}\right)$; $LC\left(X,Y\right)$ is not a linear space but a cone which is included in $L{C}^{s}\left(X,Y\right)$.

If $\left(Y,q\right)$ is $\left(ℝ,u\right)$ where $u$ is the asymmetric norm on $ℝ$ given by $u\left(x\right)={x}^{+}$, then $LC\left(X,ℝ\right)$ and $L{C}^{s}\left(X,ℝ\right)$ are denoted by ${X}^{*}$ and ${X}^{s*}$, respectively. It is proved that, with $\left(X,\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel \right)$ and $\left(Y,\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel \right)$ two normed lattices, $p\left(x\right)=\parallel {x}^{+}\parallel$ if $x\in X$ and $q\left(y\right)=\parallel {y}^{+}\parallel$ if $y\in Y$, $f\in LC\left(X,Y\right)$ iff $f\in L{C}^{s}\left(X,Y\right)$ and $f\ge 0$; also ${p}_{q}^{*}\left(f\right)\le \parallel f\parallel \le 2{p}_{q}^{*}\left(f\right)$, for all $f\in LC\left(X,Y\right)$ where ${p}_{q}^{*}\left(f\right)=sup\left\{q\left(f\left(x\right)\right):p\left(x\right)\le 1\right\}$, it is further proved that, if $\left(X,\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel \right)$ is a real normed lattice and $q\left(x\right)=\parallel {x}^{+}\parallel$, then ${X}^{s*}={X}^{*}-{X}^{*}$.

In the last section, open mapping and closed graph theorems are given, in a suitable manner, in the new setting.

##### MSC:
 54E35 Metric spaces, metrizability 54A05 Topological spaces and generalizations 46A03 General theory of locally convex spaces
##### References:
 [1] C. Alegre, Estructuras Topológicas no Simétricas y Espacios Bitopológicos, Thesis, Universidad Politécnica de Valencia (1994). [2] C. Alegre, I. Ferrando, L. M. García-Raffi and E. A. Sánchez-Pérez, Compactness in asymmetric normed spaces, Topology Appl., 155 (2008), 527–539. · Zbl 1142.46004 · doi:10.1016/j.topol.2007.11.004 [3] C. Alegre, J. Ferrer and V. Gregori, On the Hahn–Banach theorem in certain linear quasi-uniform structures, Acta Math. Hungar., 82 (1999), 315–320. · Zbl 0930.46004 · doi:10.1023/A:1006692309917 [4] C. Alegre, J. Ferrer and V. Gregori, On a class of real normed lattices, Czech. Math. J., 48 (1998), 785–791. · Zbl 0949.54045 · doi:10.1023/A:1022499925483 [5] A. Alimov, On the structure of the complements of Chebyshev sets, Funct. Anal. Appl., 35 (2001), 176–182. · Zbl 1099.41501 · doi:10.1023/A:1012370610709 [6] A. Alimov, The Banach–Mazur theorem for spaces with nonsymmetric distances, Uspekhi Mat. Nauk, 58 (2003), 159–160. [7] C. Aliprantis and K. Border, Infinite Dimensional Analysis, Springer ( New York, 2006). [8] S. Cobzas, Separation of convex sets and best aproximation in spaces with asymmetric norms, Quaest. Math., 27 (2004), 275–296. · Zbl 1082.41024 · doi:10.2989/16073600409486100 [9] J. Ferrer, V. Gregori and C. Alegre, Quasi-uniform structures in linear lattices, Rocky Mountain J. Math., 23 (1993), 877–884. · Zbl 0803.46007 · doi:10.1216/rmjm/1181072529 [10] P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker ( New York, 1982). [11] L. M. García-Raffi, Asymmetric Norms and the Dual Complexity Spaces. Thesis, Universidad Politécnica de Valencia (2003). [12] L. M. García-Raffi, S. Romaguera and E. A. Sánchez-Pérez, The bicompletion of an asymmetric normed linear space, Acta Math. Hungar., 97 (2002), 183–191. · Zbl 1012.54031 · doi:10.1023/A:1020823326919 [13] L. M. García-Raffi, S. Romaguera and E. A. Sánchez-Pérez, Sequence spaces and asymmetric norms in the theory of computational complexity, Math. Comp. Model., 36 (2002), 1–11. · Zbl 1063.68057 · doi:10.1016/S0895-7177(02)00100-0 [14] L. M. García Raffi, S. Romaguera and E. A. Sánchez Pérez, On Hausdorff asymmetric normed linear spaces, Houston J. Math., 29 (2003), 717–728. [15] L. M. García Raffi, S. Romaguera and E. A. Sánchez-Pérez, The dual space of an asymmetric normed linear space, Quaestiones Math., 26 (2003), 83–96. · Zbl 1043.46021 · doi:10.2989/16073600309486046 [16] L. M. García Raffi, S. Romaguera and E. A. Sánchez Pérez, Weak topologies on asymmetric normed linear spaces and non-asymptotic criteria in the theory of complexity analysis of algorithms, J. Anal. Appl., 2 (2004), 125–138. [17] J. C. Kelly, Bitopological spaces, Proc. London Math. Soc., 13 (1963), 71–89. · Zbl 0107.16401 · doi:10.1112/plms/s3-13.1.71 [18] H. P. A. Künzi, Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, in: Handbook of the History of General Topology, C. E. Aull and R. Lowen (eds), vol. 3, Kluwer (Dordrecht, 2001), pp. 853–968. [19] I. L. Reilly, P. V. Subrahmanyam and M. K. Vamanamurthy, Cauchy sequences in quasi-pseudometric spaces, Monatsch. Math., 93 (1982), 127–140. · Zbl 0472.54018 · doi:10.1007/BF01301400 [20] A. Robertson and W. Robertson, Topological Vector Spaces, Cambridge University Press (1980). [21] S. Romaguera, E. A. Sánchez Pérez and O. Valero, Computing complexity distances between algorithms, Kybernetika, 36 (2003), 369–382. [22] S. Romaguera and M. Sanchis, Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103 (2000), 292–301. · Zbl 0980.41029 · doi:10.1006/jath.1999.3439 [23] S. Romaguera and M. Schellekens, Duality and quasi-normability of complexity space spaces, Appl. Gen. Topology, 3 (2002), 91–112. [24] M. Schellekens, The Smyth completion: a common foundation for denotational semantics and complexity analysis, in: Proc. MFPS 11, Electronic Notes in Theoretical Computer Science, 1 (1995), pp. 211–232. [25] R. Tix, Continuous D-Cones: Convexity and Powerdomain Constructions, Thesis, Darmstadt University (1999).