Elements of the Theory of Functions and Functional Analysis, Band 1

$29 Theorem A₁ adjoint operator arbitrary number assume Banach space belong bounded linear operator bounded set cardinal number characteristic value closed interval closed sets compact compactum completely continuous operator concept consequently contains continuous functions contraction mapping convergent subsequence convex set COROLLARY corresponding countable curves d-function definition denote denumerable set derivative equal equivalent everywhere dense example exists f(xn f(xo fact finite number func functional f(x functional ƒ functions defined inequality infinite sets infinite-dimensional space intersection introduce inverse image Let us consider limit point linear functional linear operator M₁ metric space natural numbers necessary and sufficient neighborhood O(x normed linear space obtain one-to-one open set p(xn point xo prove real line real numbers satisfy the condition scalar semicontinuous solution solvable space C[a space Ē space E₁ spectrum sphere subset subspace vector verify weak convergence x₁ y₁ zero