I have that $K:C[0,1] \rightarrow C[0,1] $ and $K_N:C[0,1] \rightarrow C[0,1]$ where: $$K \phi (x) = \int_0^1 k(x,t) \phi (t) dt $$ $$K_N \phi (x) = \int_0^1 k_N(x,t) \phi (t) dt $$
Where $k(x,t):= xt +x^2t^2$
I want to prove that $K$ and $K_N$ are bounded and more specifically that $||K|| \leq 2$ and $||K_N|| \leq 2$ for every $N \in \mathbb{N}$.
I know that in general, if X is a normed space and $A:x \rightarrow X $ is a linear operator, then A is said to be bounded if there is a $C > 0 $ such that: $$||A \phi || \leq C|| \phi|| $$ for all $\phi \in X$
Therefore I should be able to use that in order to find how $K$ and $K_N$ are bounded.
I can then use the rule that, if A is bounded then the following $$ ||A\phi || / ||\phi|| : \phi \in X, \phi \neq 0$$
Is bounded above, by the constant C from earlier. The Supremum is the norm here written as $||A||$.
Therefore I would be use that in order to prove $||K|| \leq 2$ and $||K_N|| \leq 2$ for every $N \in \mathbb{N}$.
This is an example from mt lecture notes however the actual mathematics isn't really explained, just the theory I have written above. If anyone could explain how to do this mathematically that would be great!
Norm of the first operator: I will assume that you are providing $C[0,1]$ with the usual sup norm.
Let $\|\phi\| \leq 1$. Then $|K(\phi (x))|\leq \sup_x \int_0^{1} |k(x,t)|dt$ and a simple calculation of the integral shows that $\|K\| \leq \frac 12 +\frac 13$ which is a better bound than $2$.