Let x be a vector. How do you show that
By using this definition of
and Holder's inequality, I am able to prove that
But how do you show the other side of the inequality?
That shows that (which is a vector with the same components as x, but with the norm given as a functional on ) has norm at least (which is what you wanted).
It remains to deal with the fact that the components of x may not all be positive. If the scalars are real then you just have to ensure that, for each j, has the same sign as . (Then all the terms in the sum will be positive.) If the scalars are complex then you have to multiply each by an appropriate complex number of absolute value 1 to ensure the same result.
In other words, if , then you can only refer to the norm of x if you are thinking of x as an element of the space (in which case you must use the p-norm), and you can only refer to the norm of if you are thinking of x as defining an element of a dual space (a linear functional), in which case you must use the q-norm.