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An Introduction to Some Aspects of Functional Analysis

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An Introduction to Some Aspects of Functional Analysis An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms on vector spaces, bounded linear operators, and dual spaces of bounded linear functionals in particular. Contents 1 Norms on vector spaces 3 2 Convexity 4 3 Inner product spaces 5 4 A few simple estimates 7 5 Summable functions 8 6 p-Summable functions 9 7 p =2 10 8 Bounded continuous functions 11 9 Integral norms 12 10 Completeness 13 11 Bounded linear mappings 14 12 Continuous extensions 15 13 Bounded linear mappings, 2 15 14 Bounded linear functionals 16 1 15 ℓ1(E)∗ 18 ∗ 16 c0(E) 18 17 H¨older’s inequality 20 18 ℓp(E)∗ 20 19 Separability 22 20 An extension lemma 22 21 The Hahn–Banach theorem 23 22 Complex vector spaces 24 23 The weak∗ topology 24 24 Seminorms 25 25 The weak∗ topology, 2 26 26 The Banach–Alaoglu theorem 27 27 The weak topology 28 28 Seminorms, 2 28 29 Convergent sequences 30 30 Uniform boundedness 31 31 Examples 32 32 An embedding 33 33 Induced mappings 33 34 Continuity of norms 34 35 Infinite series 35 36 Some examples 36 37 Quotient spaces 37 38 Quotients and duality 38 39 Dual mappings 39 2 40 Second duals 39 41 Continuous functions 41 42 Bounded sets 42 43 Hahn–Banach, revisited 43 References 44 1 Norms on vector spaces Let V be a vector space over the real numbers R or the complex numbers C. A norm on V is a nonnegative real-valued function kvk defined for v ∈ V such that (1.1) kvk =0 if and only if v = 0, (1.2) kt vk = |t|kvk for every v ∈ V and t ∈ R or C, as appropriate, and (1.3) kv + wk≤kvk + kwk for every v, w ∈ V . Here |t| is the absolute value of t when t ∈ R, and the modulus of t when t ∈ C. For example, if V is the one-dimensional vector space R or C, then the absolute value or modulus defines a norm on V , and every norm on V is a positive multiple of this norm. More generally, let n be a positive integer, and let V be the space Rn or n C of n-tuples v = (v1,...,vn) of real or complex numbers. As usual, this is a vector space with respect to coordinatewise addition and scalar multiplication. The standard Euclidean norm on V is given by n 1/2 2 (1.4) kvk = |vj | . Xj=1 It is well-known that this is a norm, although the triangle inequality (1.3) is not immediately obvious. A couple of ways to show the triangle inequality in this case will be reviewed in the next sections. It is easy to see directly that n (1.5) kvk1 = |vj | Xj=1 n n satisfies the triangle inequality. Hence kvk1 defines a norm on R , C . Similarly, (1.6) kvk∞ = max(|v1|,..., |vn|) 3 defines a norm on Rn, Cn. The standard Euclidean norm on Rn, Cn is some- times denoted kvk2. If kvk is a norm on a real or complex vector space V , then (1.7) d(v, w)= kv − wk defines a metric on V . For instance, the standard Euclidean metric on Rn or Cn is the same as the metric associated to the standard Euclidean norm in this way. n n The metrics on R or C associated to the norms kvk1, kvk∞ determine the same topology as the standard Euclidean metric, and we shall say more about this soon. The main point is that kvk1, kvk2, and kvk∞ are equivalent norms on Rn and Cn for each positive integer n, in the sense that each is bounded by constant multiples of the others. 2 Convexity Let V be a real or complex vector space. As usual, a set E ⊆ V is said to be convex if for every x, y ∈ E and real number t with 0 <t< 1, we have that (2.1) t x + (1 − t) y is also an element of E. If kvk is a norm on V , then the closed unit ball (2.2) B1 = {v ∈ V : kvk≤ 1} is a convex set in V . Similarly, the open unit ball (2.3) {v ∈ V : kvk < 1} is also a convex set in V . Conversely, suppose that kvk is a nonnegative real-valued function on V which is positive when v 6= 0 and satisfies the homogeneity condition (1.2). If the corresponding closed unit ball B1 is convex, then one can check that kvk satisfies the triangle inequality, and is thus a norm. For if v, w are nonzero vectors in V , then v w (2.4) v′ = , w′ = kvk kwk have norm 1, and the convexity of B1 implies that kvk kwk (2.5) v′ + w′ kvk + kwk kvk + kwk has norm less than or equal to 1. Equivalently, v + w (2.6) ∈ B , kvk + kwk 1 which implies (1.3). 4 Let n be a positive integer, let p be a positive real number, and consider n 1/p p (2.7) kvkp = |vj | Xj=1 for v ∈ Rn or Cn. This is the same as the standard Euclidean norm when p = 2, and is easily seen to be a norm when p = 1, as in the previous section. Let us check that the closed unit ball associated to kvkp is convex when p> 1, so that kvkp defines a norm in this case. This does not work for p < 1, even when n = 2. n n Suppose that v, w ∈ R or C satisfy kvkp, kwkp ≤ 1, which is the same as n n p p (2.8) |vj | , |wj | ≤ 1. Xj=1 Xj=1 If p ≥ 1 and 0 <t< 1, then p p p p (2.9) |t vj + (1 − t) wj | ≤ (t |vj | + (1 − t) |wj |) ≤ t |vj | + (1 − t) |wj | for each j, because of the convexity of the function f(x) = xp on the positive real numbers. Summing over j, we get that n n n p p p (2.10) |t vj + (1 − t) wj | ≤ t |vj | + (1 − t) |wj | ≤ 1. Xj=1 Xj=1 Xj=1 This implies that (2.11) kt v + (1 − t) wkp ≤ 1, as desired. 3 Inner product spaces An inner product on a real or complex vector space V is a real or complex-valued function hv, wi defined for v, w ∈ V , as appropriate, which satisfies the following conditions. First, hv, wi is linear as a function of v for each w ∈ V . Second, (3.1) hw, vi = hv, wi for every v, w ∈ V in the real case, while (3.2) hw, vi = hv, wi in the complex case. Here z denotes the complex conjugate of z ∈ C, defined by (3.3) z = x − yi 5 when z = x + yi and x, y ∈ R. It follows from this symmetry condition that hv, wi is linear in w in the real case, and conjugate-linear in w in the complex case. It also follows that (3.4) hv, vi ∈ R for every v ∈ V , even in the complex case. The third requirement of an inner product is that (3.5) hv, vi > 0 for every v ∈ V with v 6= 0, and of course the inner product is equal to 0 when v = 0. If hv, wi is an inner product on V , then the Cauchy–Schwarz inequality states that (3.6) |hv, wi| ≤ hv, vi1/2 hw, wi1/2 for every v, w ∈ V . Because of homogeneity, it suffices to show that (3.7) |hv, wi| ≤ 1 when (3.8) hv, vi = hw, wi =1, since the inequality is trivial when v or w is 0. To do this, one can use the fact that (3.9) hv + tw,v + t wi≥ 0 for every t ∈ R or C, and expand the inner product to estimate hv, wi. The norm on V associated to the inner product is defined by (3.10) kvk = hv, vi1/2. This clearly satisfies the positivity and homogeneity requirements of a norm. In order to verify the triangle inequality, one can use the Cauchy–Schwarz inequal- ity to get that (3.11) kv + wk2 = hv + w, v + wi = hv, vi + hv, wi + hw, vi + hw, wi ≤ kvk2 +2 kvkkwk + kwk2 = (kvk + kwk)2 for every v, w ∈ V . The standard inner product on Rn is defined by n (3.12) hv, wi = vj wj . Xj=1 Similarly, the standard inner product on Cn is defined by n (3.13) hv, wi = vj wj . Xj=1 The corresponding norms are the standard Euclidean norms. 6 4 A few simple estimates Observe that (4.1) kvk∞ ≤kvkp for each v ∈ Rn or Cn and p> 0. If 0 <p<q< ∞, then n n q q−p p (4.2) |vj | ≤kvk∞ |vj | Xj=1 Xj=1 for every v ∈ Rn or Cn, and hence 1−(p/q) p/q (4.3) kvkq ≤kvk∞ kvkp ≤kvkp. Thus kvkp is monotone decreasing in p. In the other direction, 1/p (4.4) kvkp ≤ n kvk∞ for each v ∈ Rn or Cn and p> 0. If 0 <p<q< ∞, then (1/p)−(1/q) (4.5) kvkp ≤ n kvkq. To see this, remember that f(x)= xr is a convex function of x ≥ 0 when r ≥ 1, so that n n 1 r 1 (4.6) a ≤ ar n j n j Xj=1 Xj=1 for any nonnegative real numbers a1,...,an.
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