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.