Can the norm of a vector be $\infty$? I am reading Pugh's Analysis and he defines a norm as a certain type of function from $V \to \mathbb{R}$. However, if we have two normed vector spaces, he later says that we can define the operator norm of a linear transformation by 
$$||T|| = \sup \left \{\dfrac {|T(v)|}{|v|}: v \not = 0 \right \}$$
However, from what he says later in the text, I infer that there are some linear transformations whose norm is $\infty$. Thus my question is: is a norm actually a function to the extended reals, or is the operator norm not actually a norm, but a different type of object?

Calling $\|\cdot\|$ a norm is a small abuse of terminology. Norms must be finite by definition. However, any function satisfying all the norm axioms except for finiteness becomes a norm when restricted to the domain for which it is finite. Even though $\|\cdot\|$ can be infinite, it is a norm on the space of continuous linear operators. 
For an example of a linear operator with infinite norm, consider the space of real sequences which are eventually zero with the sup norm, and the operator which takes such a sequence and multiplies its $n$th entry by $n.$
