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- W2487788349 abstract "Banach contributed to the theory of orthogonal series and made innovations in the theory of measure and integration, but his most important contribution was in functional analysis. Of his published works, his Theorie des operations lineaires (1932; “Theory of Linear Operations”) is the most important. Banach and his coworkers summarized the previously developed concepts and theorems of functional analysis and integrated them into a comprehensive system. Banach himself introduced the concept of [complete] normed linear spaces, which are now known as Banach spaces. He also proved several fundamental theorems in the field, and his applications of theory inspired much of the work in functional analysis for the next few decades . (Encyclopedia Britannica) Continuity and subdifferentials Throughout the rest of the book X denotes a normed or a Banach space. We assume familiarity with basic results and language from functional analysis, and we will often draw upon the classical spaces l p , c 0 and others for examples. Notwithstanding, we will sketch some of the classic convexity theorems in our presentation. We write X * for the real dual space of continuous linear functionals, and denote the unit ball and sphere by B X ≔ { x ∈ X : ∥ x ∥ ≤ 1} and S X ≔ { x ∈ X : ∥ x ∥ = 1}, respectively. We say that a subset C of X is convex if, for any x, y ∈ C and any λ ∈ [0, 1], λ x + (1 - λ) y ∈ C ." @default.
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- W2487788349 date "2011-10-06" @default.
- W2487788349 modified "2023-09-23" @default.
- W2487788349 title "Convex functions on Banach spaces" @default.
- W2487788349 doi "https://doi.org/10.1017/cbo9781139087322.005" @default.
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