Is norm a functional (rather than a function)? Many authors of textbooks define the norm as a real-valued function that satisfies those well-known properties. For example, the definition of norm at Wiki here. In my opinion, however, the norm is indeed a functional rather than a function, because it maps sequences, functions, or some other "vectors" from a vector space to a real number: $X \to \mathbb{R}$. The norm is a function if and only if $X \subset \mathbb{R}^n$. Is my point of view wrong or the authors should improve the definition? 
PS. I think the idea of functional is kind of extended from the idea of function. I think a function can only be a mapping from numbers to a number, while a functional can be a mapping from any vectors in vector space to a number. Those vectors can themselves be functions, thus someone calls functional as function of function. In my opinion, therefore, functional is not a subset of function, but function is a subset of functional. If we talk about the generalized idea of function with respect to the mapping functionality, shall we say a mapping or an operator? 
 A: From yout point of view it can be either a function and a functional... But only pay attention to the following question: is the concept of function more or less restrictive than the one of functional?
Taking the definition of norm, we have that given a vector space $V$ over a field $K$ (actaully can also be a subfield, but for the sake of simplicity we can state also in this way), a norm on $V$ is a function $n\colon V \to \mathbb{R}$ that satisfies the usual properties of being absolutely homogeneous or absolutely scalable, being definite positive and making the triangle inequality hold.
If you now focus on what the vector space $V$ can be you can easily prove that, yes, continuous, discontinuous, $l^p$ functions and many others form a vector space. But they are only a very small part of what $V$ can be!
Greedily one could say: every function can be seen as a vector(at most, infinite dimensional) but not all vectors can be seen as function... 
So you are right when you say that a norm can be seen as a functional but the notion of function is more general, so the latter is preferable. (By the way, we are mathematicians, we like define objects in the most general case :) )
A: Removing psychological aspects of the question, if we define a functional to be a $\mathbb{R}$-valued or $\mathbb{C}$-valued or scalar-valued function, then a norm is a functional, and as such, a function.
