The ONLY operation requiring closure for a vector space is ADDITION. The vector space is actually closed under cross product but that is NOT a requirement for a set to be a vector space. The reason is that cross product is defined under restrictive conditions: That the space is Euclidean and three dimensional.These are not essential for a set to be a vector space.
Of course the result of the tensor product is NOT the same as the arguments of the operation! Two vectors create a tensor. Consequently, there is no closure there.
The vector cross is not an operation. It is the definition of a particular tensor. The idea of closure in this circumstance is irrelevant.
In summary, when we are discussing closure for a vector space, we are only talking about addition – not any other operation.

well simplified

Good morning sir,

Is the vector space closed under cross product?

Is the vector space closed under tensor product?

Is the vector space closed under vector cross?

The ONLY operation requiring closure for a vector space is ADDITION. The vector space is actually closed under cross product but that is NOT a requirement for a set to be a vector space. The reason is that cross product is defined under restrictive conditions: That the space is Euclidean and three dimensional.These are not essential for a set to be a vector space.

Of course the result of the tensor product is NOT the same as the arguments of the operation! Two vectors create a tensor. Consequently, there is no closure there.

The vector cross is not an operation. It is the definition of a particular tensor. The idea of closure in this circumstance is irrelevant.

In summary, when we are discussing closure for a vector space, we are only talking about addition – not any other operation.

Sir, i noticed that you have not uploaded the updated version of the last slide (Tensor Algebra Part 2)