# The Endomorphism algebra of graded vector space

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.

• Use a basis of $V$ consisting of homogeneous elements to decompose any linear map as a sum of homogeneous maps. – arkeet May 23 '18 at 2:47
• Yeah, don’t overthink this, just use arkeet's hint. – Branimir Ćaćić May 23 '18 at 12:48

## 1 Answer

The reason is that vector spaces of finite dimension are small. For details (in a more general setting) see Section 3 in this article.