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Review of Hilbert and Banach Spaces Review of Hilbert and Banach Spaces Definition 1 (Vector Space) A vector space over C is a set V equipped with two operations, (v, w) ∈ V × V 7→ v + w ∈V (α, v) ∈ C × V 7→ αv ∈V called addition and scalar multiplication, respectively, that obey the following axioms. Additive Axioms. There is an element 0 ∈ V and, for each x ∈ V there is an element −x ∈V such that, for all x, y, z ∈V, (i) x + y = y + x (ii) (x + y) + z = x +(y + z) (iii) 0 + x = x + 0 = x (iv) (−x) + x = x +(−x) = 0 Multiplicative Axioms. For every x ∈V and α, β ∈ C, (v) 0x = 0 (vi) 1x = x (vii) (αβ)x = α(βx) Distributive Axioms. For every x, y ∈V and α, β ∈ C, (viii) α(x + y) = αx + αy (ix) (α + β)x = αx + βx Definition 2 (Subspace) A subset W of a vector space V is called a linear subspace of V if it is closed under addition and scalar multiplication. That is, if x + y ∈ W and αx ∈ W for all x, y ∈ W and all α ∈ C. Then W is itself a vector space over C. Definition 3 (Inner Product) (a) A inner product on a vector space V is a function (x, y) ∈ V × V 7→ hx, yi ∈ C that obeys (i) (Linearity in the second argument) hx,αyi = α hx, yi, hx, y + zi = hx, yi+hx, zi (ii) (Conjugate symmetry) hx, yi = hy, xi (iii) (Positive–definiteness) hx, xi > 0 if x 6= 0 for all x, y, z ∈V and α ∈ C. (b) Two vectors x and y are said to be orthogonal with respect to the inner product h· , ·i if hx, yi = 0. (c) We’ll use the terms “inner product space” or “pre–Hilbert space” to mean a vector space over C equipped with an inner product. September25,2018 ReviewofHilbertandBanachSpaces 1 Definition 4 (Norm) (a) A norm on a vector space V is a function x ∈ V 7→ kxk ∈ [0, ∞) that obeys (i) kxk = 0 if and only if x = 0. (ii) kαxk = |α|kxk (iii) kx + yk≤kxk + kyk for all x, y ∈V and α ∈ C. (b) A sequence vn n IN ⊂V is said to be Cauchy with respect to the norm k·k if ∈ ∀ε > 0 ∃N ∈ IN s.t. m, n > N =⇒ kvn − vmk < ε (c) A sequence vn n IN ⊂V is said to converge to v in the norm k·k if ∈ ∀ε > 0 ∃N ∈ IN s.t. n > N =⇒ kvn − vk < ε (d) A normed vector space is said to be complete if every Cauchy sequence converges. (e) A subset D of a normed vector space V is said to be dense in V if D = V, where D is the closure of D. That is, if every element of V is a limit of a sequence of elements of D. Theorem 5 Let h · , ·i be an inner product on a vector space V and set kxk = hx, xi for all x ∈V. p (a) The inner product is sesquilinear. That is, hx,αy + βzi = α hx, yi + β hx, zi hαx + βy, zi = α hx, zi + β hy, zi for all x, y, z ∈V and α, β ∈ C.(1) (b) kxk is a norm. (c) The inner product and associated norm obeys (i) (Cauchy–Schwarz inequality) hx, yi ≤kxk kyk 2 2 2 2 (ii) (Parallelogram law) kx + yk + kx − yk =2kxk +2kyk (iii) (Polarization identities) 1 2 2 2 1 2 2 2 hx, yi = 2 kx + yk −kxk −kyk + 2i kx + iyk −kxk −kyk 1 2 2 1 2 2 = 4 kx + yk −kx − yk + 4i kx + iyk −kx − iyk for all x, y ∈V (1) Physicists and mathematical physicists generally use the convention that inner products are linear in the second argument and conjugate linear in the first. Some mathematicians use the convention that inner products are linear in the first argument and conjugate linear in the second. September25,2018 ReviewofHilbertandBanachSpaces 2 Lemma 6 Let k·k be a norm on a vector space V. There exists an inner product h · , · i on V such that hx, xi = kxk2 for all x ∈V if and only if k · k obeys the parallelogram law. Definition 7 (Banach Space) (a) A Banach space is a complete normed vector space. (b) Two Banach spaces B1 and B2 are said to be isometric if there exists a map U : B1 → B2 that is (i) linear (meaning that U(αx+βy) = αU(x)+βU(y) for all x, y ∈ B1 and α, β ∈ C) (ii) onto (a.k.a. surjective) (iii) isometric (meaning that kUxk 2 = kxk 1 for all x ∈ B1). This implies that U is B B 1–1 (a.k.a. injective). Definition 8 (Hilbert Space) (a) A Hilbert space H is a complex inner product space that is complete under the asso- ciated norm. (b) Two Hilbert spaces H1 and H2 are said to be isomorphic (denoted H1 =∼ H2) if there exists a map U : H1 → H2 that is (i) linear (ii) onto (iii) inner product preserving (meaning that hUx,Uyi 2 = hx, yi 1 for all x, y ∈ H1) H H Such a map is called unitary. Lemma. If H1 and H2 are two Hilbert spaces and if U : H1 → H2 is linear, onto and isometric, then U is unitary. Theorem 9 (Completion) If V, h· , ·i is any inner product space, then there exists V a Hilbert space H, h·, , ·i and a map U : V → H such that H (i) U is 1–1 (ii) U is linear (iii) hUx,Uyi = hx, yi for all x, y ∈V H V (iv) U(V) = Ux x ∈V is dense in H. If V is complete, then U(V) = H. H is called the completion of V. September25,2018 ReviewofHilbertandBanachSpaces 3 Example 10 n (a) C = x = (x1, ··· xn) x1, ··· xn ∈ C together with the inner product hx, yi = n x¯ℓyℓ is a Hilbert space. ℓP=1 p p (b) If 1 ≤ p < ∞, then ℓ = (xn)n IN {xn}n IN ⊂ C, n∞=1 |xn| < ∞ together with ∈ 1/p ∈ p P the norm (xn)n IN = ∞ |xn| is a Banach space. ∈ p n=1 h P i 2 2 (c) ℓ = (xn)n IN ∞ |xn| < ∞ is a Hilbert space with the inner product ∈ n=1 (xn)n IN,(yn)n IN = P∞ xnyn. ∈ ∈ n=1 P (d) ℓ∞ = (xn)n IN sup|xn| < ∞ and c0 = (xn)n IN lim xn =0 are both Banach ∈ ∈ n n →∞ spaces with the norm (xn)n IN = sup|xn|. ∈ ∞ n Theorem Let 1 ≤ p,q,r ≤∞ and let x =(xn)n IN, y =(yn)n IN. Then ∈ ∈ p p (a) (Minkowski inequality) If x, y ∈ ℓ , then x+y ∈ ℓ and kx+ykp ≤kxkp +kykp p q r (b) (H¨older inequality) If x ∈ ℓ and y ∈ ℓ , then (xnyn)n IN ∈ ℓ and we have ∈ k(xnyn)n INkr ≤kxkpkykq ∈ (e) Let X be a metric space (or more generally a topological space) and C(X ) = f : X → C f continuous, bounded C0(X ) = f : X → C f continuous, compact support If X is a subset of IRn or Cn for some n ∈ IN, let C (X ) = f : X → C f continuous, lim f(x)=0 ∞ x | |→∞ Then C(X ) and C (X ) are Banach spaces with the norm kfk = sup |f(x)|. C0(X ) is a ∞ x normed vector space, but need not be complete. ∈X (f) Let 1 ≤ p ≤∞. Let (X, M,µ) be a measure space, with X a set, M a σ–algebra and µ a measure. For p < ∞, set Lp(X, M,µ) = ϕ : X → C ϕ is M–measurable and |ϕ(x)|p dµ(x) < ∞ 1/p R p kϕkp = |ϕ(x)| dµ(x) h Z i For p = ∞, set(2) L∞(X, M,µ) = ϕ : X → C ϕ is M–measurable and ess sup |ϕ(x)| < ∞ kϕk = ess sup |ϕ(x)| ∞ (2) The essential supremum of |ϕ|, with respect to the measure µ, is denoted ess supx∈X |ϕ(x)| and is defined by inf{ a ≥ 0 | |ϕ(x)|≤ a almost everywhere, with respect to µ }. September25,2018 ReviewofHilbertandBanachSpaces 4 This is not quite a Banach space because any function ϕ that is zero almost everywhere has “norm” zero. So we define an equivalence relation on Lp(X, M,µ) by ϕ ∼ ψ ⇐⇒ ϕ = ψ a.e. As usual, the equivalence class of ϕ ∈Lp(X, M,µ) is [ϕ] = ψ ∈Lp(X, M,µ) ψ ∼ ϕ Then Lp(X, M,µ) = [ϕ] ϕ ∈Lp(X, M,µ) is a Banach space with [ϕ]+[ψ]=[ϕ + ψ] a[ϕ]=[aϕ] [ϕ] p = kϕkp for all ϕ, ψ ∈ Lp(X, M,µ) and a ∈ C, and L2(X, M,µ) is a Hilbert space with inner product h[ϕ], [ψ]i = ϕ(x) ψ(x) dµ(x) Z for all ϕ, ψ ∈L2(X, M,µ). It is standard to write ϕ in place of [ϕ]. If X is a Lebesgue measurable subset of IR, then Lp(X) denotes the Lp(X, M,µ) with (M,µ) being Lebesgue measure. Theorem Let 1 ≤ p,q,r ≤∞. (a) (Minkowski inequality) If f,g ∈ Lp(X, M,µ), then f + g ∈ Lp(X, M,µ) and kf + gkLp ≤kfkLp + kgkLp (b) (H¨older inequality) If f ∈ Lp(X, M,µ) and g ∈ Lq(X, M,µ), then we have r fg ∈ L (X, M,µ) and kfgkLr ≤kfkLp kgkLq (g) Let D be an open subset of C.
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