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In this paper we prove that if      n &gt; 1 n &gt; 1       and      0 &gt; p &gt; 1 0 &gt; p &gt; 1       then the Lumer’s Hardy space      ( L H ) p ( B n ) {(LH)^p}({{\mathbf {B}}_n})       of the unit ball      B n {{\mathbf {B}}_n}       in      C n {{\mathbf {C}}^n}       does not have the Orlicz-Pettis property.

The Orlicz-Pettis theorem fails for Lumer’s Hardy spaces (𝐿𝐻)^{𝑝}(𝐵)