Simple Proofs for Bochner-Schoenberg-Eberlein and the Bochner-Schoenberg-Eberlein Module Properties on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:msup><a:mrow><a:mi>ℓ</a:mi></a:mrow><a:mrow><a:mi>p</a:mi></a:mrow></a:msup><a:mfenced open="(" close=")"><a:mrow><a:mi>X</a:mi><a:mo>,</a:mo><a:mi>A</a:mi></a:mrow></a:mfenced></a:math>

Let <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"><a:mi>X</a:mi></a:math> be a nonempty set, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"><c:mi>A</c:mi></c:math> be a commutative Banach algebra, and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M4"><e:mn>1</e:mn><e:mo>≤</e:mo><e:mi>p</e:mi><e:mo><</e:mo><e:mo>∞</e:mo></e:math> . In this paper, we present a concise proof for the result concerning the BSE (Banach space extension) property of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M5"><g:msup><g:mrow><g:mi>ℓ</g:mi></g:mrow><g:mrow><g:mi>p</g:mi></g:mrow></g:msup><g:mfenced open="(" close=")"><g:mrow><g:mi>X</g:mi><g:mo>,</g:mo><g:mi>A</g:mi></g:mrow></g:mfenced></g:math> . Specifically, we establish that <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M6"><k:msup><k:mrow><k:mi>ℓ</k:mi></k:mrow><k:mrow><k:mi>p</k:mi></k:mrow></k:msup><k:mfenced open="(" close=")"><k:mrow><k:mi>X</k:mi><k:mo>,</k:mo><k:mi>A</k:mi></k:mrow></k:mfenced></k:math> possesses the BSE property if and only if <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M7"><o:mi>X</o:mi></o:math> is finite and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M8"><q:mi>A</q:mi></q:math> is BSE. Additionally, we investigate the BSE module property on Banach <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M9"><s:msup><s:mrow><s:mi>ℓ</s:mi></s:mrow><s:mrow><s:mi>p</s:mi></s:mrow></s:msup><s:mfenced open="(" close=")"><s:mrow><s:mi>X</s:mi><s:mo>,</s:mo><s:mi>A</s:mi></s:mrow></s:mfenced></s:math> -modules and demonstrate that a Banach space <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M10"><w:msup><w:mrow><w:mi>ℓ</w:mi></w:mrow><w:mrow><w:mi>p</w:mi></w:mrow></w:msup><w:mfenced open="(" close=")"><w:mrow><w:mi>X</w:mi><w:mo>,</w:mo><w:mi>Y</w:mi></w:mrow></w:mfenced></w:math> serves as a BSE Banach <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M11"><ab:msup><ab:mrow><ab:mi>ℓ</ab:mi></ab:mrow><ab:mrow><ab:mi>p</ab:mi></ab:mrow></ab:msup><ab:mfenced open="(" close=")"><ab:mrow><ab:mi>X</ab:mi><ab:mo>,</ab:mo><ab:mi>A</ab:mi></ab:mrow></ab:mfenced></ab:math> -module if and only if <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" id="M12"><eb:mi>X</eb:mi></eb:math> is finite and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" id="M13"><gb:mi>Y</gb:mi></gb:math> represents a BSE Banach <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" id="M14"><ib:mi>A</ib:mi></ib:math> -module.