I thought w(f) where w=dual vector f=vector is Reals?

[precondition]

But lecturer said today that it's a function.
So for example he said, if we have w x f where x is supposed to be tensor product, then w(f) x f can be written w(f)f without tensor product sign because w(f) is just a function...

 

[HallsofIvy]

I'm not sure what you mean by "Reals". w(f) where w= dual vector f= vector is not "Reals", it is a single real number.

The definition of the "dual space" of a vector space, V, over a given field is "the vector space of all linear functions from V to the field with vector addition given by (w+q)(v)= w(v)+ q(v) (w and q functions and the sum on the right hand side is in V) and scalar multiplication given by (aw)(v)= a(w(v)) (again, the multiplication on ther right hand side is in V).

I doubt your instrutor meant that w(f) is a function. w itself is a function, w(f) is the real number value when w is applied to f. That's why we can talk about w(f)f- it's scalar multiplication.

Of course, that's a common "abuse of notation" just as when we refer to "f(x)" as a function. Actually, f is the function, f(x) is some value of the function.

 

[tacman]

It seems that there may be some confusion about the notation and terminology being used. In mathematics, the term "dual vector" typically refers to a vector in the dual space, which is the set of all linear functionals on a vector space. In this context, w(f) would indeed be a real number, as it is the value of the linear functional w at the vector f.

However, it seems that the lecturer in this case is using a different definition of "dual vector," where it is being used to refer to a function that maps vectors to real numbers. In this case, w(f) would indeed be a function, as it is taking a vector f as input and returning a real number as output.

As for the notation involving tensor products, it is important to clarify whether the lecturer is using the notation w x f to represent a tensor product or a cross product. If it is a tensor product, then w(f) x f would indeed be equivalent to w(f)f, as the tensor product of a number and a vector is simply the vector multiplied by that number. However, if it is a cross product, then w(f) x f would not be equivalent to w(f)f, as the cross product operation is only defined for vectors.

Overall, it is important to clarify the specific definitions and notations being used in order to fully understand the concepts being taught. It may also be helpful to ask the lecturer for clarification or further explanation if needed.