The tensor product of two vector spaces U and V is defined as the dual of the vector space of all the bilinear forms on the direct sum of U and V. Is there a generalised form of this for the direct sums of more than two vector spaces? Is there a relation between the space of all multilinear forms on the direct sum of V 1 , V 2 , V 3 ,..., V k with their tensor product.Please explain without invoking other algebraic objects such as modules,rings etc and by using the concepts regarding vector spaces only (as the book assumes no such background either, it is unlikely that any reader of that book will benefit from such an exposition). Everywhere I searched, I found the explanation in terms of those concepts only and being unfamiliar to those I couldn't get them at all. Thanks in advance.

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The tensor product of two vector spaces U and V is defined as the dual of the vector space of all the bilinear forms on the direct sum of U and V. Is there a generalised form of this for the direct sums of more than two vector spaces? Is there a relation between the space of all multilinear forms on the direct sum of       V    1  ,      V    2  ,      V    3  ,...,      V    k   with their tensor product.Please explain without invoking other algebraic objects such as modules,rings etc and by using the concepts regarding vector spaces only (as the book assumes no such background either, it is unlikely that any reader of that book will benefit from such an exposition). Everywhere I searched, I found the explanation in terms of those concepts only and being unfamiliar to those I couldn&#039;t get them at all. Thanks in advance.

Bornejecbo · Accepted Answer

First of all: yes: just replace &quot;bilinear&quot; with &quot;multilinear&quot; and you have a definition that works for finite-dimensional vector spaces.Second: this is not the correct definition for vector spaces in general. The reason is that the correct definition is actually indirect: it says that the dual of the tensor product is isomorphic to the space of multilinear forms on the direct sum. For FDVS&#039;s, this is equivalent to what he wrote, because the double dual of an FDVS is itself, but for infinite-dimensional spaces, this is not so.Of course, for FDVS&#039;s you have the unsatisfying definition that &quot;the tensor product of a m-dimensional and an n-dimensional space is an mn-dimensional space&quot;, since they are all determined completely by their dimensions. But that&#039;s not what&#039;s important in tensor products: what matters is the relationships that exist between vector spaces. And the relevant relationship for tensor products is:For any vector spaces   V,  W and   U, there is a natural correspondence (in a sense that can be made precise using ideas you asked not to use) between linear maps   V  ⊗  W  →  U and bilinear maps on   V  ×  W valued in   U.If we used the 1-dimensional space for   U, we would recover the definition of   V  ⊗  W as the dual of the space of bilinear forms, but the above text is the real definition.

The tensor product of two vector spaces U and V is defined as the dual of the vector space of all th

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