§ Example of unbounded linear operator
- Simplest example is differentiation.
- Let be continuous functions on interval, and be differentiable functions on the interval.
- We equip both spaces with sup norm / infty norm.
- Consider the differentiation operator .
- Since every differentiable function is continuous, we have that
- Clearly differentiation is linear (well known).
- To see that the operator is not bounded, consider the sequence of functions .
- We have that for ann , while the , so clearly, there is no constant such that . Thus, the operator is unbounded.
- Note that in this definition, the space is not closed, as there are sequences of differentiable functions that coverge to non differentiable functions. Proof: polynomials which are differentiable functoins are dense in the full space of continuous functions.
- Thus, in the case of an unbounded operator, we consider where is some subspace of , not ncessarily closed!
- If we ask for an everywhere defined operator, then constructing such operators needs choice.
§ Nonconstructive example
- Regard as a normed vector space over . [Cannot call this a banach space, since a banach space needs base field ]
- Find an algebraic basis containing the numbers and and whatever else we need.
- define a function such that , and , and extend everywhere else by linearity.
- Now let be a sequence of rationals that converge to . Then , and thus , while . This shows that is not continuous, but is linear.