NUR HALIMAH , SA’DIAH (2026) TEOREMA ROLLE DAN TEOREMA NILAI RATA-RATA PADA FUNGSI BERNILAI BARISAN ℓ1. FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM, UNIVERSITAS LAMPUNG .
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Abstract
Rolle’s Theorem and the Mean Value Theorem are studied for sequence-valued functions in the Banach space ℓ , which is the space of sequences equipped with the norm The space ℓ 1 1 ∥p∥ 1 = ∞ X d=1 |p d |. allows the extension of fundamental concepts of real analysis, such as limits, continuity, and derivatives, to infinite-dimensional spaces. The discussion is carried out using a mathematical analysis approach based on literature study, employing the definitions of continuity, differentiability, and the completeness property of the spaceℓ 1 . The results show that Rolle’s Theorem and the Mean Value Theorem remain valid for sequence-valued functions in the space ℓ , provided that the function is continuous on a closed interval and differentiable on an open interval. The proof is obtained by examining the derivatives of each sequence component and utilizing the properties of the ℓ spaces, particularly the spaceℓ 1 1 . Keywords:Rolle’s Theorem, Mean Value Theorem, ℓ 1 norm, ensuring the existence of a point within the interval that satisfies the conditions of both theorems. These results confirm that fundamental theorems of differential analysis on R can be generalized to Banach space, Banach space, sequence-valued functions, derivative. Teorema Rolle dan Teorema Nilai Rata-Rata dibahas pada fungsi bernilai barisan dalam ruang Banachℓ Ruang ℓ 1 1 , yaitu ruang barisan yang memiliki norma ∥p∥ 1 = ∞ X d=1 |p d |. memungkinkan pengembangan konsep dasar analisis real, seperti limit, kekontinuan, dan turunan, ke ruang berdimensi tak hingga. Pembahasan dilakukan melalui pendekatan analisis matematis berbasis studi literatur dengan memanfaatkan definisi fungsi kontinu, fungsi terdiferensialkan, serta sifat kelengkapan ruangℓ . Hasil pembahasan menunjukkan bahwa Teorema Rolle dan Teorema Nilai Rata-Rata tetap berlaku pada fungsi bernilai barisan di ruang ℓ apabila fungsi kontinu pada interval tertutup dan terdiferensialkan pada interval terbuka. Pembuktian diperoleh dengan meninjau turunan setiap komponen barisan serta menggunakan sifat norma pada ruang ℓ 1 1 , sehingga diperoleh suatu titik pada interval yang memenuhi kondisi kedua teorema tersebut. Hasil ini menegaskan bahwa teorema dasar analisis diferensial padaRdapat diperluas ke ruang Banach, khususnya ruangℓ Kata-kata kunci:Teorema Rolle, Teorema Nilai Rata-Rata, ruang ℓ , ruang Banach, fungsi bernilai barisan, turunan. 1 1 . 1
| Item Type: | Other |
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| Subjects: | ?? 500 ?? ?? 510 ?? |
| Divisions: | Fakultas MIPA > Prodi Matematika |
| Depositing User: | 2602230892 Digilib |
| Date Deposited: | 19 Feb 2026 01:40 |
| Last Modified: | 19 Feb 2026 01:40 |
| URI: | http://digilib.unila.ac.id/id/eprint/96475 |
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