By Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, Václav Zizler

Banach areas offer a framework for linear and nonlinear sensible research, operator idea, summary research, likelihood, optimization and different branches of arithmetic. This booklet introduces the reader to linear practical research and to comparable elements of infinite-dimensional Banach area idea. Key positive aspects: - Develops classical conception, together with vulnerable topologies, in the community convex house, Schauder bases and compact operator conception - Covers Radon-Nikodým estate, finite-dimensional areas and native conception on tensor items - comprises sections on uniform homeomorphisms and non-linear idea, Rosenthal's L1 theorem, mounted issues, and extra - comprises information regarding extra themes and instructions of study and a few open difficulties on the finish of every bankruptcy - offers a variety of workouts for perform The textual content is appropriate for graduate classes or for autonomous learn. must haves comprise easy classes in calculus and linear. Researchers in sensible research also will profit for this booklet because it can function a reference book.

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**Example text**

We will show that it is closed there. Consider Tn ∈ K(X, Y ) such that lim Tn = T in B(X, Y ). 48). First note that Tn → T in B(X, Y ) means that lim Tn (x) = T (x) uniformly for x ∈ B X . Thus there exists n 0 such that Tn (x) − T (x) < ε/2 for x ∈ B X and n ≥ n 0 . Since Tn 0 (B X ) is totally bounded in Y , there is a finite ε/2-net F for Tn 0 (B X ). We claim that F is a finite ε-net for T (B X ). Indeed, given x ∈ B X , we find y ∈ F such that Tn 0 (x) − y < ε/2. Then T (x) − y ≤ T (x) − Tn 0 (x) + Tn 0 (x) − y < ε.

Hint. If ϕ is affine, we have x0 = λi x0 and thus n ϕ n λi xi = i=1 n i=1 n λi xi = λi x0 + T i=1 n λi (x0 + T (xi )) = i=1 λi ϕ(xi ). i=1 If ϕ has the stated property, it is enough to show that ψ(x) = ϕ(x) − ϕ(0) is a linear mapping. For x, y ∈ V and scalars α, β we have ψ(αx + βy) = ψ(αx + βy + (1 − α − β)0) = ϕ(αx + βy + (1 − α − β)0) − ϕ(0) = αϕ(x) + βϕ(y) + (1 − α − β)ϕ(0) − ϕ(0) = αϕ(x) + βϕ(y) − αϕ(0) − βϕ(0) = αψ(x) + βψ(y). 6 Let X be a normed linear space. Prove that for any x, y ∈ X we have y ≤ x−y .

P1 Thus 1≤ x λp p · C p2 (1−λ) · x −λp p1 . Hence x λp p1 ≤ C p2 (1−λ) · x λp p . Thus x p1 ≤C p2 (1−λ) λp · x p ≤C p2 (1−λ) λp p1 . 18 Show that for every p ≥ 1, p is linearly isometric to a subspace of L p [0, 1]. 1/ p Hint. Consider span{ f n }, where f n := n(n + 1) χ[ 1 , 1 ] . 19 Let 1 ≤ p ≤ ∞ and let μ be a σ -finite measure. Show that L p (μ) is isometric to L p (ν), where ν is a probability measure. Hint. e. positive function such that f dμ = 1, and define a measure ν by dν = gdμ. Then the mapping f → f · g −1/ p defines an isometry from L p (μ) onto L p (ν).