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Analisis fungsional adalah suatu cabang matematika (khususnya analisis) yang mengkaji ruang vektor topologis dan pemetaannya. Dalam kajian ini, analisis, topologi, dan aljabar saling bertaut. Tujuan penyelidikan analisis fungsional adalah untuk menemukan aturan abstrak dari bidang-bidang tersebut, yang kemudian dapat diterapkan untuk berbagai masalah konkrit.

Akar sejarah analisis fungsional dapat dilacak dari kajian ruang fungsi khususnya transformasi fungsi seperti transformasi Fourier, dan kajian persamaan diferensial. Penggunaan istilah fungsional berasal dari kalkulus variasi, untuk menandakan fungsi yang argumennya adalah fungsi. Konsep umum fungsional diperkenalkan pada tahun 1887 oleh Vito Volterra,^[1]^[2] seorang matematikawan dan fisikawan asal Italia. Stefan Banach, matematikawan asal Polandia, sering kali dianggap sebagai pengasas analisis fungsional modern.^[3]

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