# Confusion regarding definition of tensor product of two vector spaces

Consider two vector spaces $$E$$ and $$F$$ over the same field $$K$$. Now to form the tensor product $$E\otimes F$$ we typically take a particular vector space $$V_1$$ and quotient it. The vector space $$V_1$$ is from page 265 here, is defined as $$V_1=\oplus_{(e,f)\in E\times F}K(e,f)$$. My questions are:

(1) Is $$K(e,f)$$ the set of all formal expressions of the form $$\alpha\cdot(e,f)$$? If so, how is this a vector space? Alternatively is $$K(e,f)$$ the span of $$(e,f)$$ in the vector space $$E\times F$$?

(2) How do we visualize the direct sum? The way I see it, $$K(e,f)$$ the span of $$(e,f)$$ and the direct sum consists of all those elements of $$\prod K(e,f)$$ for which all but finitely many coordinates are zero. But then how are the basis elements of $$V_1$$ all the elements of $$E\times F$$?

Thanks

$$V_{1}$$ is the so-called ''free-vector space''. Using the defining property of the direct sum, this can be represented as:
$$F(V\times W):=\bigoplus_{(v,w)\in V\times W}\mathbb{K}\cdot (v,w)\cong$$
$$\bigg\{\sum_{(v,w)\in I}\lambda_{(v,w)}(v,w)\mid I\subset V\times W\text{ with } \vert I\vert<\infty\land \lambda_{(v,w)}\in\mathbb{K}\bigg\}$$
More generally, the free vector space of a given set $$X$$ is the vector space $$F(X)$$, which is ''generated by the set $$X$$'', which means that $$F(X)$$ is defined to be a vector space in such a way that $$X$$ is a basis of $$F(X)$$. In other words, $$F(X)$$ is the set of finite linear combinations of elements of $$X$$. So you see that $$V_{1}$$ is the collection of all finite linear combinations of $$(e,f)\in E\times F$$.
• No problem....But as I said, the direct sum is just a more compact and fancier form of the free vector space.....Depending on your exact definition on $\oplus$, you may also write a $=$ instead of $\cong$, because $\oplus$ is often defined as sum of all finite linear combinations. – Udalricus.S. May 17 '20 at 8:28