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All Journal Journal of Mathematics Education and Science Trigonometri : Jurnal Matematika
Helmi Firdaus
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Education Mathematics Other

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Kelengkapan Pada Ruang Bernorma Kuasi Helmi Firdaus
Trigonometri: Jurnal Matematika Vol. 1 No. 2 (2024): Edisi Juli
Publisher : Lppm Universitas Nurul Hud

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30599/trigonometri.v1i2.3458

Abstract

Konsep norma kuasi merupakan generalisasi dari konsep norma pada suatu ruang vektor. Norma kuasi muncul karena terdapat fungsi yang tidak bisa dinyatakan sebagai norma akibat tidak terpenuhi salah satu aksioma norma pada ruang vektor. Berdasarkan hal tersebut, tulisan ini bertujuan untuk membahas tentang sifat-sifat yang ada pada ruang bernorma, seperti konvergen, Cauchy, dan terbatas berlaku pada suatu ruang bernorma kuasi. Lebih lanjut dapat diteliti sifat kelengkapan dari ruang vektor terhadap norma kuasinya. Penelitian menggunakan metode studi literatur dari beberapa buku dan jurnal yang terkait dengan norma dan norma kuasi pada ruang vektor dengan membuktikan beberapa teorema dan memberikan contoh terhadap definisi yang ada di dalam ruang bernorma kuasi. Hasil dari penelitian disimpulkan bahwa ruang bernorma kuasi merupakan ruang yang lengkap terhadap norma kuasinya. Pada akhir tulisan ini diberikan suatu lema yang menjeleaskan tentang kombinasi linear pada ruang bernorma kuasi.
Keberadaan dan Ketunggalan Titik Tetap Pada Pemetaan Non Ekspansif Dalam Ruang Bernorma Kuasi Linear Helmi Firdaus
Journal of Mathematics Education and Science Vol. 8 No. 2 (2025): Journal of Mathematics Education and Science
Publisher : Universitas Nahdlatul Ulama Sunan Giri Bojonegoro

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.32665/james.v8i2.5236

Abstract

This study examines the existence and uniqueness of fixed points of non-expansive mappings in quasi-normed spaces, to establish the existence of a solution to a non-expansive function in a quasi-normed space. The research method employed is a literature review, which provides some theorems with proofs and formal examples. The research began by outlining fundamental notions, such as convergence, Cauchy sequences, boundedness, and completeness, in the context of quasi-norms. Furthermore, the properties of compactness and their implications were elaborated as part of a theoretical framework. In the section on mappings, the characteristics of operators in quasi-normed spaces were first explained, including continuous and bounded mappings, along with their equivalence. Non-expansive and contraction mappings were then formally defined, serving as the basis for demonstrating the existence and uniqueness of fixed points. By applying a sequence approach and the completeness property, it was proven that every non-expansive mapping on a quasi-Banach space possesses a unique fixed point. Finally, it was shown that a quasi-normed space that is both compact and convex guarantees the existence of fixed points for non-expansive mappings defined on such spaces.
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