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Let $G$ be a group. A linear map $f:V\rightarrow W$ of $G$-graded vector spaces is said to be homogeneous of degree $g$ if $f(V_{h}) \subseteq W_{g\cdot h}$ for all $h\in G$. We denote the space all homogeneous maps of degree $g$ by $\operatorname{Hom}_g(V,W)$ and we set $$ \operatorname{Hom}^{\operatorname{gr}}(V,W) := \bigoplus_{g\in G}\operatorname{Hom}_{g}(V,W) $$ If $V$ is a finite-dimensional vector space, why does $\operatorname{Hom}^{\operatorname{gr}}(V,W)$ coincide with $\operatorname{Hom}(V,W)$? Thank you so much in advance.

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    $\begingroup$ Use a basis of $V$ consisting of homogeneous elements to decompose any linear map as a sum of homogeneous maps. $\endgroup$ – arkeet May 23 '18 at 2:47
  • $\begingroup$ Yeah, don’t overthink this, just use arkeet's hint. $\endgroup$ – Branimir Ćaćić May 23 '18 at 12:48
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The reason is that vector spaces of finite dimension are small. For details (in a more general setting) see Section 3 in this article.

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