Embedding $\varLambda _{\widetilde{V }}^{0}$ ⊆ X ⊆ M_V*

Abstract

In this chapter, we prove the embedding theorem for classes of symmetric spaces having the same fundamental functions. The embedding theorem asserts that for every symmetric space X with a given fundamental function V = φ _X , there are continuous embeddings $\boldsymbol{\varLambda }_{\widetilde{V }}^{0} \subseteq \mathbf{X} \subseteq \mathbf{M}_{V _{{\ast}}}$. This means that the minimal part $\boldsymbol{\varLambda }_{\widetilde{V }}^{0}$ of the Lorentz space $\boldsymbol{\varLambda }_{\widetilde{V }}$ is the smallest symmetric space whose (concave) fundamental function $\widetilde{V }$ is equivalent to V, while the Marcinkiewicz space $\mathbf{M}_{V _{{\ast}}}$ is the largest symmetric space X with $\varphi _{\mathbf{X}} =\varphi _{\mathbf{M}_{V_{{\ast}}}} = V$. The renorming theorem and other consequences of the embedding theorem are considered.