François Treves

Existence and Approximation of Solutions to a Linear Partial Differential Equation

We shall denote by P n the vector space of polynomials in n indeterminates (or “variables”) X1,...,X n , with complex coefficients, by P n d (d=0, 1,...) the subspace of P n consisting of the polynomials of degree ≤d. We provide P n with the locally convex topology, inductive limit of the finite-dimensional Hausdorff TVS P n d : a seminorm 𝖕 on P n is continuous if and only if its restriction to every P n d for d=0, 1,..., is continuous. But every seminorm on a finite dimensional Hausdorff TVS is continuous (cf. Treves [2], Theorem 9.1), therefore this is also true in P n Spec P n is the total spectrum of P n in other words, P n carries the finest locally convex topology. This is equivalent with saying that every linear functional on P n is continuous: the algebraic dual of P n is equal to its dual.

Applications of the Epimorphism Theorem

We denote by R n the n-dimensional Euclidean space, by x=(x1,..., x n ) the variable point in R n , by N n the subset of R n consisting of the n-tuples p=(p1,...,p n ) where the p j are integers ≥0. We set |p|=p1+...+p n (whereas \( \left| x \right| = {(x_{1}^{2} + ... + x_{n}^{2})^{{1/2}}} \). If ƒis a C∞ functions in R n , say with complex values, we write

The Spectrum of a Locally Convex Space

{f^{{(p)}}}for{(\partial /\partial {x_{1}})^{{p1}}}...{(\partial /\partial {x_{n}})^{{Pn}}}f

Translation into Duality

We denote by E’ the space of distributions in R n having a compact support. We recall that the space of all the distributions in R n , D’, is a convolution module over E’: that is to say, (µ, v) → µ*v is a bilinear mapping of D’ × E’ into D’.

Existence and Approximation of Solutions to a Functional Equation

Throughout the forthcoming, E will denote a vector space over a field K which will always be either the field of real numbers, R, or the one of complex numbers, C.

On local solvability of linear partial differential equations part I: Necessary conditions

Let S be a set of seminorms on E; S is said to be equicontinuous at the point x° of E if to every e>0 there is a neighborhood x° + U of x° (U:neighborhood of О in E) such that, for all 𝖕∈S and all x∈x° + U,