By H Takayasu
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Extra resources for Fractals in physical sciences
Thus, ||P − Pk ||∞ = sup x∈[0,1/2] ∞ ≤ n=k+1 Since ∞ 1 1 n=0 n+1 2n |P (x) − Pk (x)| 1 n+1 1 2 n . is convergent, we infer that ||P − Pk ||∞ → 0 as k → ∞. 6 it follows that P is a continuous function. Thus, the sequence Pk converges in C[0, 12 ] with respect to the norm || · ||∞ . But the limit P is not a polynomial! 3 this proves that W does not form a closed subset of C[0, 12 ]. 8). 3 Approximation in normed vector spaces In the technical sense, normed vector spaces can contain elements that are very complicated to deal with (concrete instances will occur in the context of the Lp -spaces discussed in Chapter 5).
Assume that the hypotheses are satisﬁed, and let x0 ∈ I and > 0 be given. (i) Argue that there exists an N ∈ N such that |f (x) − fN (x)| ≤ /3, ∀x ∈ I. (ii) Argue that we can choose δ > 0 such that |fN (x) − fN (x0 )| ≤ /3 whenever x ∈ I and |x − x0 | ≤ δ. (iii) Use the triangle inequality to show that for x ∈ I with |x − x0 | ≤ δ, |f (x) − f (x0 )| ≤ . 1(i). (i) Prove Young’s inequality: for any a, b > 0 and any p, q > 1 with p−1 + q −1 = 1, ab ≤ 1 p 1 q a + b . p q Hint: consider the xy-plane, the graph of the function y = xp−1 , and the lines x = a, y = b.
Ii) Use (i) to conclude that W cannot be an open subset of C[0, 12 ]. 9 Let W be a subspace of a normed vector space V. Show that the closure W is the smallest closed subspace of V that contains W. 10 Consider the linear map T : R2 → R2 , T x1 x2 = 2x1 − x2 x1 + x2 . Equip R2 with the canonical norm, and answer the following: (i) Is T injective? (ii) Is T surjective? (iii) Is T an isometry? 11 Consider the linear map x1 x2 T : R2 → R2 , T ⎛ =⎝ 1 3 x1 − + 2 3 x1 + 2 3 x2 1 3 x2 ⎞ ⎠. Equip R2 with the canonical inner product and check the following: (i) Is T injective?