Normed Spaces

Abstract

The topic of this book is linear integral equations of which

$$ \int_a^b {K(x,y)\rho (y)dy = f(x),x \in \left[ {a,b} \right]} , $$

and

$$ rho (x) - \int_a^b {K(x,y)\rho (y)dy = f(x),x \in \left[ {a,b} \right]} , $$

are typical examples. In these equations the function ϕ is the unknown, and the so-called kernel K and the right-hand side f are given functions. The above equations are called Fredholm integral equationsof the firstand second kind,respectively. We will regard them as operator equations of the firstand second kindin appropriate normed function spaces. The symbol A: X→Ywill mean a single-valued mapping whose domain of definition is a set Xand whose range is contained in a set Y,i.e., for every ϕ ∈ Xthe mapping Aassigns a unique element Aϕ ∈ Y. The range A(X)is the set A(X):= Aϕ: ϕ ∈ X of all image elements. We will use the terms mapping, function,and operatorsynonymously.