# linear algebra – Proof of the Universal Property of the Tensor Produts

Suppose that $$h:Xtimes Yto mathbb{F}$$ is a bilinear map. Then prove that there exists a linear map $$h_otimes:Xotimes Ytomathbb{F}$$ such that $$h(x,y)=h_otimes(x,y)$$.

Logically, to every element $$(x,y)in Xtimes Y$$ there is an element $$xotimes yin Xotimes Y$$, therefore we there can be a one to one correspondance between both the maps.

But how do I prove that $$h_otimes$$ is a linear map ?

Let, $$h:Xtimes Ytomathbb{F}$$ such that $$h(x,y)=f(x)g(y)$$ where $$f:Xtomathbb{F}$$ and $$g:Ytomathbb{F}$$

Note : I only have understanding of the linear algebra basics.