# Addition and scalar multiplication in vector space

In Gowers' explanation of tensor products, there is a passage where he writes, with vector spaces $$V$$ and $$W$$:

For every pair $$(v,w)$$ in $$V\times W$$, regard $$[[v,w]]$$ as a meaningless symbol. We can define a rather large vector space $$Z$$ by taking formal linear combinations of these symbols. By that I mean that $$Z$$ consists of all expressions of the form $$a_1[[v_1,w_1]]+ a_2[[v_2,w_2]]+...+ a_n[[v_n,w_n]]$$ with obvious definitions for addition and scalar multiplication.

I am a little confused by what he means at the end. When he talks about addition and scalar multiplication, I am wondering if he means the following: $$a_1[[v_1,w_1]]+a_2[[v_2,w_2]]=[[a_1v_1+a_2v_2,a_1w_1+a_2w_2]]$$ But this also seems incorrect because then there would seemingly be no difference between $$Z$$ and $$V\times W$$ other than the fact that instead of $$(\cdot,\cdot)$$ we write $$[[\cdot,\cdot]]$$.

No, $$Z$$ really is just the vector space whose basis is the set of all symbols $$[[v,w]]$$. $$a[[v,w]]$$ can be simplified no further, and there's no combination of elements when you take sums, in particular $$[[v+v',w]]$$ doesn't split into a sum of two elements. That comes later when we take the quotient to create the tensor product. Even $$[[0,0]]\neq 0$$, $$0$$ is the element $$0[[v,w]]$$ for any $$v,w$$, so for example $$[[0,0]]$$ and $$2[[0,0]]$$ are different elements.
• Just to make sure I'm understanding you correctly, in this case $[[v+v',w]]$ and $[[v,w]]+[[v',w]]$ are different elements, and they are only made to identify with each other when the quotient is taken, which explicitly sets all symbols of the form $[[v+v',w]]-[[v,w]]-[[v',w]]$ to zero. Commented Mar 27, 2020 at 19:33