A property of L − L integral transformations

The main result of this paper is the result that the collection of all integral transformations of the form F ( x ) = ∫ 0 ∞ G ( x , y ) f ( y ) d y for all x ≥ 0 , where f ( y ) is defined on [ 0 , ∞ ) and G ( x , y ) defined on D = { ( x , y ) : x ≥ 0 , y ≥ 0 } has no identity transformation on L , where L is the space of functions that are Lebesgue integrable on [ 0 , ∞ ) with norm ‖ f ‖ = ∫ 0 ∞ | f ( x ) | d x . That is to say, there is no G ( x , y ) defined on D such that for every f ∈ L , f ( x ) = ∫ 0 ∞ G ( x , y ) f ( y ) d y for almost all x ≥ 0 . In addition, this paper gives a theorem that is an improvement of a theorem that is proved by J. B. Tatchell (1953) and Sunonchi and Tsuchikura (1952).