June 15, 2020 Locally Convex Spaces and the Tempered Distributions Níckolas Alves Department of Mathematical Physics, Institute of Physics - University of São Paulo, R. do Matão 1371, Cidade Universitária, São Paulo, Brazil NUSP: 10300894 E-mail: [email protected] Abstract: This text is a brief discussion about some elementary topics on the theory of locally convex spaces and tempered distributions as continuous linear functionals on a particular LCS: the Schwartz space of functions of rapid decrease. It is part of the evaluation of the course “Mathematical Physics III”, taught by Prof. Walter de S. Pedra at IFUSP on the first semester of 2020. Familiarity with general topology and linear topological spaces is assumed. Keywords: Distribution Theory, Locally Convex Spaces, Functional Analysis Contents 1 Tempered Distributions1 2 Formulation in Locally Convex Spaces4 1 Tempered Distributions The goal of this text is to understand tempered distributions under the light of the theory of locally convex spaces. In order to do so, let us begin by defining what we mean by a tempered distribution. Definition 1 [Functions of Rapid Decrease and Schwartz Space]: n Let f: R Ñ R be a smooth function. f is said to be a function of rapid decrease if, and only if, it holds that sup jppxqDαfpxqj ă ` (1.1) n xPR n for every polynomial ppxq “ ppx1; : : : ; xnq and every multiindex1 α P N0 . n The collection of all functions f: R Ñ R of rapid decrease is named Schwartz space n and usually denoted by S pR q - or simply S . ♠ This definition is often stated in a different way. [1,2], for example, require lim ppxqDαfpxq “ 0; (1.2) kxkÑ` which is equivalent to the condition we1 required on Eq. (1.1). Since f is smooth, all of its derivatives are continuous. The image of a compact set through a continuous function is compact, and by the Heine-Borel Theorem this implies n α the image of any closed, bounded set K Ď R under ppxqD fpxq is a closed, bounded subset α of R. As a consequence, ppxqD fpxq can only escape to infinity when kxk Ñ ` . Non- constant polynomials always escape to infinity for kxk Ñ ` . Therefore, if Dαfpxq Ñ λ for some non-vanishing λ, then ppxqDαfpxq will escape to infinity at kxk Ñ1 ` . If ppxqDαfpxq Ñ λ ‰ 0 for kxk Ñ ` , then x ¨ ppxqDαfpxq will1 escape to infinity. Thus, Eq. (1.2) is implied by Eq. (1.1). If Eq. (1.2) holds, then the fact that ppxqDαfpxq can1 only escape to infinity when kxk Ñ ` 1ensures Eq. (1.1). Proposition 2: n 1 Let f: R Ñ R be a smooth function. f is of rapid decrease if, and only if, α β n kfkα,β “ sup x D fpxq ă ` ; @ α, β P N0 ; (1.3) n xPR n α n αi where k¨k denotes the Euclidean norm in R and x ” 1i“1 xi . ± – 1 – Proof: n β Assume f P S and let α P . Then sup n ppxqD fpxq ă ` for every polyno- N0 xPR n α mial ppxq “ ppx1; : : : ; xnq and every multiindex β P N0 . Pick the polynomial ppxq “ x . It follows that kfkα,β ă ` . 1 n Let us assume now that kfkα,β ă ` ; @ α, β P N0 . Notice that a polynomial in n m αi variables ppxq can always1 be written in the form ppxq “ i“1 aix for some m P N, n n appropriate multiindices αi P N0 and coefficients1 ai P R. Notice that given β P N0 , it n ř holds @ x P R that m β αi β ppxqD fpxq “ aix D fpxq ; i“1 m¸ αi β ¤ jaij x D fpxq : (1.4) i“1 ¸ If we take the supremum on each side, it follows that m β αi β sup ppxqD fpxq ¤ sup jaij x D fpxq ; n n xPR xPR #i“1 + m ¸ αi β ¤ jaij sup x D fpxq : (1.5) n i“1 xPR ¸ β Hence, sup n ppxqD fpxq is less than or equal to a finite sum of finite terms. xPR β Therefore, sup n ppxqD fpxq ă ` , and we conclude f P S . xPR Theorem 3: 1 Consider the space S of functions of rapid decrease. Let f; g P S , λ P R. The following statements hold n i. kλ ¨ fkα,β “ jλj ¨ kfkα,β; @ α, β P N0 ; n ii. kf ` gkα,β ¤ kfkα,β ` kgkα,β; @ α, β P N0 ; iii. S is a real vector space; n iv. kfkα,β “ 0; @ α, β P N0 ñ f “ 0. Proof: Let λ P R. Notice that, for any f P S , α β kλ ¨ fkα,β “ sup x D pλ ¨ fqpxq ; n xPR α β “ sup λ ¨ x D fpxq ; n xPR α β “ sup jλj x D fpxq ; n xPR α β “ jλj ¨ sup x D fpxq ; n xPR “ jλj ¨ kfkα,β: (1.6) – 2 – Let now g P S . With f defined as before, we see that α β kf ` gkα,β “ sup x D pf ` gqpxq ; n xPR α β β “ sup x D fpxq ` D gpxq : (1.7) n xPR ` ˘ n Notice that, @ x P R , it holds that α β α β α β α β x D fpxq ` x D gpxq ¤ x D fpxq ` x D gpxq : (1.8) Thus, we may take the supremum on both sides and see that α β α β α β α β sup x D fpxq ` x D gpxq ¤ sup x D fpxq ` x D gpxq ; n n xPR xPR “ α β α β ‰ kf ` gkα,β ¤ sup x D fpxq ` x D gpxq ; n xPR α β α β ¤ sup “x D fpxq ` sup x D g‰pxq ; n n xPR xPR “ kfkα,β ` kgkα,β: (1.9) This not only proves that kλ ¨ fkα,β “ jλj ¨ kfkα,β and kf ` gkα,β ¤ kfkα,β ` kgkα,β, but also that S is closed under pointwise addition and multiplication by scalar, since finiteness of kfkα,β and kgkα,β implies finiteness of kf ` λ ¨ gkα,β. Since 0 P S - for n constant functions are smooth and k0kα,β “ 0 - and the space C pR q of smooth functions n n f: R Ñ R is a real vector space, we see S is a linear subspace1 of C pR q. n Suppose f P S is such that kfkα,β “ 0; @ α, β P N0 . Then, in particular,1 it holds that kfk0;0 “ 0, where the multiindex 0 should be understood as the n-uple p0; : : : ; 0q. This n means that sup n jfpxqj “ 0. Since sup n jfpxqj ¥ jfpyqj ¥ 0; @ y P , we conclude xPR xPR R n fpyq “ 0; @ y P R . Definition 4 [Tempered Distribution]: Consider the space S of functions of rapid decrease and let ': S Ñ R be a linear functional. ' is said to be a tempered distribution if, and only if, it holds for every function f P S that lim 'pfnq “ 'pfq (1.10) nÑ` pf q P N 1 for every sequence n nPN S with the property that lim kfn ´ fkα,β “ 0: (1.11) nÑ ♠ 1 This definition is equivalent to the one found in [1], but we chose a slightly different collection of seminorms. [1] uses the functions m α kfkm,α “ sup p1 ` kxkq jD fpxqj (1.12) n xPR – 3 – as seminorms, which can make the proof for Proposition 2 a bit more complicated, since kxk is not a n-variable polynomial due to the square root in n kxk “ x2: (1.13) g i fi“1 f ¸ e [3] uses the same choice of seminorms we made, which uses a multiindex instead of a non-negative integer, but uses the polynomial xα instead of p1 ` kxkqm. It is common to denote the action of a tempered distribution ' on a function f through x'; fy. 2 Formulation in Locally Convex Spaces Let us now turn our attention to the theory of locally convex spaces. Theorem 3 motivates us to define the concept of a seminorm. Definition 5 [Seminorm]: Let pV; F; `; ¨q be a vector space, with F being either the real line or the complex plane. A function k¨k: V Ñ R` is said to be seminorm on pV; F; `; ¨q whenever the following conditions hold, @ x; y P V: i. kx ` yk ¤ kxk ` kyk (triangle inequality); ii. kα ¨ xk “ jαjkxk; @ α P F. ♠ One should notice the functions k¨kα,β defined on the Schwartz space are seminorms. From a topological point of view, the notion of seminorm is closely related to that of a pseudometric. Norms are known to induce metrics. Seminorms are almost norms, but they fail to ensure that kxk “ 0 ñ x “ 0. Thus, the “not-a-metric” induced by them fails to ensure dpx; yq ñ x “ y. Definition 6 [Pseudometric Space]: Let M be a set and d: M ˆ M Ñ R` be a function satisfying the following axioms, @ x; y; z P M, i. dpx; xq “ 0; ii. dpx; yq “ dpy; xq; iii. dpx; yq ¤ dpx; zq ` dpy; zq. pM; dq is said to be a pseudometric space and d is said to be a pseudometric on M. ♠ Proposition 7: Let V be a vector space over F, with F being either the real line or the complex plane, and let k¨k be a seminorm on V. If we define d: V Ñ R` by dpx; yq :“ kx ´ yk, pV; dq is a pseudometric space. pV; dq is a metric space if, and only if, k¨k is a norm, id est, if it holds that kxk “ 0 ñ x “ 0.