Linear continuous surjections of Cp-spaces over compacta Academic Article uri icon

abstract

  • Let X and Y be compact Hausdorff spaces and suppose that there exists a linear continuous surjection T : C p ( X ) → C p ( Y ) , where C p ( X ) denotes the space of all real-valued continuous functions on X endowed with the pointwise convergence topology. We prove that dim ⁡ X = 0 implies dim ⁡ Y = 0 . This generalizes a previous theorem [7, Theorem 3.4] for compact metrizable spaces. Also we point out that the function space C p ( P ) over the pseudo-arc P admits no densely defined linear continuous operator C p ( P ) → C p ( [ 0 , 1 ] ) with a dense image.

publication date

  • January 1, 2017