Reproducing kernel Banach space defined by the minimal norm property and applications to partial differential equation theory

It it well known that a Hilbert space V of functions defined on U is a reproducing kernel Hilbert space if and only if for any z ∈ U {z\in U} , in the set V z := { f ∈ V ∣ f ⁢ ( z ) = 1 } {V_{z}:=\{f\in V\mid f(z)=1\}} , if non-empty, there is exactly one element with minimal norm and there is a direct connection between the reproducing kernel and such an element. In this paper, we define reproducing kernel Banach space as a space which satisfies this property and the reproducing kernel of it using this relation. We show that this reproducing kernel share a lot of basic properties with the classical one. The notable exception is that in Banach spaces the equality K ⁢ ( z , w ) = K ⁢ ( w , z ) ¯ {K(z,w)=\overline{K(w,z)}} does not have to be true without assumptions that K ⁢ ( z , w ) ≠ 0 , K ⁢ ( w , z ) ≠ 0 {K(z,w)\neq 0,K(w,z)\neq 0} . We give sufficient and necessary conditions for a Banach space of functions to be a reproducing kernel Banach space. At the end, we give some examples including ones which show how reproducing kernel Banach spaces can be used to solve extremal problems of Partial Differential Equations Theory.