FUNGSIONAL-N LINEAR TERBATAS DAN ORTOGONALITAS DI RUANG NORM-N

• Main Author: MUHTAR GOZALI , SUMANANG
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/17325
• Institution: Institut Teknologi Bandung

The theory of normed spaces is extended to the so-called n-normed spaces. Geometrically, an n-norm can be interpreted as the volume of an n-dimensional parallelepiped. There is a close relation between n-normed and n-inner product spaces, in the sense that the later is a special case of the former. Many formulas of n-norms can be defined in a vector space. Especially, in a Hilbert space there are at least four formulas, namely the standard one, the G¨ahler’s, the Gunawan’s and another one which will be introduced in this disertation. Although the formulas are different one from another, it is shown that they are mutually identical. In addition, it is pointed out that an n-norm in a normed space induces one in its dual. In an n-normed space we can define the so-called bounded linear n-functionals. The set of such n-functionals forms a normed space. The main properties of n- functionals are investigated as well as supported by examples. The results could be a reference to inquire the suitable Riesz’s theorem. Several concepts of orthogonality in n-normed spaces have been formulated, including those of Khan-Siddiqui, Cho-Kim, Gunawan et al, and Mazaheri-Nezhad. The four concepts are reviewed as well as compared one to another. In particular, current results on b-orthogonality which was introduced by Mazaheri-Nezhad will be presented. One of the results states that b-orthogonality is equivalent to linear independence.