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We provide a representation of elements of the space lp(A,X) for a locally convex space X and 1≤p<∞ and determineits continuous dual for normed space X and 1<p<∞. In particular, we study the extension and characterization of isometries on lp(N,X) space, when X is a normed space with an unconditional basis and with asymmetric norm. In addition, we give a simple proof of the main resultof G. Ding (2002)

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Some Properties of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>l</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>Spaces

Some Properties oflp(A,X)Spaces