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I think that any (finite) vector space is a graded vector space, since any (finite) vector space admits a direct sum decomposable. I get stuck why to define graded vector space?

Thanks very much in advance!

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It only makes sense to call $V$ a graded vector space if you have a distinguished representation $V=\bigoplus_{k\geq0} V_k$ which is really meaningful in connection with the problem at hand. Important examples are of course spaces of polynomials, as explained in the quoted source. In this case you also have "natural" maps $D:\>V_k\to V_{k-1}$ of various sorts, the simplest being differentiation of polynomials of a single variable $x$.

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  • $\begingroup$ @ Christian Blatter Thank you for your attention. $\endgroup$ – bing Nov 16 '16 at 11:27

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