This thesis is devoted to a study of various geometric properties of Banach function spaces, relations with each other and its applications.
First, we study the effect of M-ideal properties and complex convexity to extension of polynomials. In particular, we show when order continuous subspace of Marcinkiewicz spaces has M-ideal properties in its bidual and we find the relation between the complex extreme points and extension of 2-homogeneous polynomials in order continuous subspace of complex Marcinkiewicz sequence spaces.
Secondly, we find the necessary and sufficient conditions for complex convexity of Orlicz-Lorentz spaces. Using complex convexity, we show that the norm-attaining subspace $NA(d_*(w,1), d(w,1))$ is not dense in $L(d_*(w,1), d(w,1))$ if and only if $w ∈ ℓ_2$ in the complex case.
Finally, we deal with complex convexity, monotonicity, cotype of Banach lattices. With these results, we obtain the following results: a K$ö$the-Bochner function space E(X) is strictly (resp. uniformly) complex convex if and only if E is strictly (resp. uniformly) monotone and X is strictly (resp. uniformly) complex convex.