Verified answer

First of all,

ker L = {p(x): L(p(x)) = 0}, by definition of the kernel

........= {p(x): x p'(x) = 0}, by definition of L

........= {p(x): p'(x) = 0}

........= {a | a is any scalar}, since constants are the only functions with derivative equal to 0

........= span {1}.

---------------

Note: span{f(x), g(x)} signifies all linear combinations of f(x) and g(x).

That is, span {f(x), g(x)} = {a f(x) + b g(x) | a, b are scalars}.

[This readily generalizes to more than 2 functions.]

In the case of one function, span {f(x)} = {a f(x) | a is any scalar}.

So, span {1} = {a * 1 = a | a is any scalar}, as indicated above

----------------

I hope this helps!