By Mingjun Chen; Zhongying Chen; G Chen

Those chosen papers of S.S. Chern talk about subject matters similar to vital geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles Ch. 1. creation -- Ch. 2. Operator Equations and Their Approximate suggestions (I): Compact Linear Operators -- Ch. three. Operator Equations and Their Approximate suggestions (II): different Linear Operators -- Ch. four. Topological levels and stuck element Equations -- Ch. five. Nonlinear Monotone Operator Equations and Their Approximate ideas -- Ch. 6. Operator Evolution Equations and Their Projective Approximate ideas -- App. A. primary useful research -- App. B. creation to Sobolev areas

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6. GF; (ii) Pn : V —► X is a bounded (projective) linear operator, n = 1,2, • • •, and {Pn} converges pointwise to the continuous embedding operator r : V —► X, namely, lim \\Pnv - v\\x = 0, VveV. n—KX> Then for any compact linear operator K : X —> V, we have lim \\PnK-rK\\=0, Chapter 2 40 where 11 • 11 is the operator norm. 3. 6, and K : X —► V be a compact linear operator. If A ^ 0 is a regular value of operator TK : X —► X, then there is an integer N > 0 such that for all n>N, the approximate equation \un = PnKun + Pnj, feV, always has a unique solution un € Vn = PnV, constant c > 0 such that \\un -u\\x where u eV and moreover there is a < c\\Pnu-u\\x, n>N, is the unique solution of the equation At* = if« + / , feV.

K is said to be a compact operator if it is continuous and relatively compact. K is said to be a completely continuous operator if for any sequence {xn} in Q such that € ft, it always follows that \\Kxn - Kx||y —» 0 as n —► oo. We first point out a subtle difference between a compact operator and a completely continuous operator, both map from the Banach space X to the Banach space Y. The following results are well known [13, 40]. 1. Let X and Y be Banach spaces and K : X —► Y be an operator. (i) If K is a compact linear operator, then K is completely continuous.

N, qj (t) = polynomial of degree m — 1, j = 1, • • •, n , 4(^j)=0, j = l, — , n ; fc = l, • - , m . The collocation problem, in this case, is to find xn(t) e Xn such that ( Dmxn + j ^ ajDjxn - / J (tfc) = 0, k = 1, • • •, n. 21) where P n : C[—1,1] —» V^ is the Lagrange interpolation operator defined as usual by P n (-)(t) = £ L i ( 0 ( * ) M t ) , w i t h L * W = n ? 2. 19) has a unique solution xn(t) € Xn. 14). 4 Projection Algorithms: The Hilbert Space Setting We are concerned with Hilbert spaces in this section.