A course in simple-homotopy theory by M.M. Cohen

By M.M. Cohen

Cohen M.M. A path in simple-homotopy idea (Springer, [1973)(ISBN 3540900551)

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4) From part (a), there exists a δ = δ(ε/r) > 0 such that ε x+y ≤r 1−δ 2 r . 4), we have tx + (1 − t)y ≤ 2t 1 − δ ≤ r 1 − 2tδ ε r r + (1 − 2t)r ε r (as y ≤ r) . Now by the choice of t ∈ [1/2, 1), we have tx + (1 − t)y (2t − 1)x + (1 − t)(x + y) x+y ≤ (2t − 1) x + 2(1 − t) 2 = ≤ (2t − 1)r + 2(1 − t)r 1 − δ = r 1 − 2(1 − t)δ ε r ε r . Therefore, tx + (1 − t)y ≤ r 1 − 2 min{t, 1 − t}δ ε r . 7 Let X be a Banach space. Then the following are equivalent: (a) X is uniformly convex. (b) For two sequences {xn } and {yn } in X, xn ≤ 1, yn ≤ 1 and lim n→∞ xn + yn = 2 ⇒ lim n→∞ xn − yn = 0.

9), we have fx ∗∗ ≤ x . 5, there exists a nonzero functional j ∈ X ∗ such that x, j = x j This implies that fx ∗∗ ∗ and j ∗ = x . = x . Define a mapping ϕ : X → X ∗∗ by ϕ(x) = fx , x ∈ X. Then ϕ is called the natural embedding mapping from X into X ∗∗ . It has the following properties: (i) ϕ is linear: ϕ(αx + βy) = αϕ(x) + βϕ(y) for all x, y ∈ X, α, β ∈ K; (ii) ϕ(x) is isometry: ϕ(x) = x for all x ∈ X. Generally, however, the natural embedding mapping ϕ from X into X ∗∗ is not onto. It means that there may be elements in X ∗∗ that cannot be represented by elements in X.

12 Every compact subset of a normed space is bounded, but the converse may not be true. 13 Every compact subset of a normed space is separable. 14 A closed and bounded subset of a normed space need not be compact. 34 1. Fundamentals ∞ Proof. Let X = 2 . Then the unit ball BX = {x ∈ 2 : x 2 = ( i=1 |xi |2 )1/2 ≤ 1} is closed and bounded. We now show that BX is not compact. Let {xn } be a sequence in BX defined by xn = (0, 0, · · · , 1, 0, · · · ), n ∈ N. , there is no convergent subsequence of {xn }.

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