10.30495/maca.2025.2057641.1134

Norm-attainment in locally convex spaces: Weak-* topology, inductive limits, and reflexivity

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  • Evans N. Mogoi1,
  • Evans N. Mogi1,
  • Priscah Moraa Ohuru*1,
  1. Jaramogi Oginga Odinga University of Science and Technology, Kenya

Published in Issue 2025-11-10

How to Cite

Norm-attainment in locally convex spaces: Weak-* topology, inductive limits, and reflexivity. (2025). Mathematical Analysis and Its Contemporary Applications, 7(4). https://doi.org/10.30495/maca.2025.2057641.1134

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Abstract

We characterize norm-attaining functionals in locally convex spaces (LCS), with particular focus on three fundamental aspects: the weak-* (weak-star) topology in dual spaces, inductive limits (including LF-spaces and DF-spaces), and reflexivity conditions. Our main results establish that (1) norm-attainment in the weak-* dual coincides precisely with the canonical embedding X into X**; (2) strict inductive limits (such as D(R)) permit non-attaining functionals, whereas Montel spaces ensure universal attainment; and (3) both barrelledness and reflexivity conditions recover norm-attainment through weak-* continuity. This work extends classical Banach space techniques to general LCS settings, revealing the crucial interplay between compactness properties and approximation methods in determining norm-attainment behavior.

Keywords

  • Norm-attainment,
  • Weak-* topology,
  • Locally convex spaces,
  • Inductive limits,
  • Montel spaces,
  • Reflexivity,
  • Dual spaces
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References

  1. [1] N. Bourbaki, Topological Vector Spaces, Springer-Verlag, 1987.
  2. [2] J. B. Conway, A Course in Functional Analysis, Springer-Verlag, 1990.
  3. [3] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, 1984.
  4. [4] R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, 1965.
  5. [5] M. N. Evans and R. Obogi, Characterizing norm-attainability in operator ideals: Necessary and sufficient conditions for operators in compact, Hilbert-Schmidt, and Schatten classes, Ann. Pure Appl. Math., 31(1) (2025), 1–8.
  6. [6] M. N. Evans and R. Obogi, Geometry of norm attainability in Orlicz spaces, Math. Anal. Contemp. Appl., 7(1) (2025), 71–78.
  7. [7] M. N. Evans and P. Moraa, A note on norm-attaining properties for frame operators, Asian J. Adv. Res. Rep., 19(3) (2025), 337–343.
  8. [8] M. N. Evans and A. Samwel O., Norm attainability of compact operators: Spectral, geometric, and perturbation insights, Arch. Current Res. Int., 25(4) (2025), 287–294.
  9. [9] R. C. James, Weakly compact sets, Trans. Amer. Math. Soc., 113 (1964), 129-140.
  10. [10] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.
  11. [11] G. Kothe, Topological Vector Spaces I, Springer-Verlag, 1969.
  12. [12] R. E. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, 1998.
  13. [13] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Springer-Verlag, 1993.
  14. [14] W. Rudin, Functional Analysis, McGraw-Hill, 1991.
  15. [15] F. Treves, Topological Vector Spaces, Distributions and Kernels, Dover Publications, 2006.
  16. [16] M. Valdivia, Topics in Locally Convex Spaces, North-Holland, 1982.