TITLE: What is the general form of linear operators on continuous functions?
QUESTION [6 upvotes]: I was wondering if there was a representation for a set of operators dense
in the space of linear operators $B$ mapping $C(a,b) \to C(c,d)$. I
thought that maybe integral operators give a general representation; that is
given the right $E(u,t)$, $$\int f(t)E(u,t) dt$$ approximates an
arbitrary linear, bounded operator $T$ resulting in $$\int
f(t)E(u,t) dt = (Tf)(u)$$
I am honestly very new to operator theory and I am really clueless as to
whether or not a general representation exists. I would very much appreciate it
if someone could guide me in the right direction.

REPLY [5 votes]: Every linear functional $x^{\star}$ on $C[a,b]$ has the form
$$
               x^{\star}(f) = \int_{a}^{b}f(t)\,d\mu(t)
$$
for a unique finite complex Borel measure $\mu$ on $[a,b]$. Furthermore,
$$
                               \|x^{\star}\| = \|\mu\|,
$$
where $\|\mu\|$ is the total variation of $\mu$ on $[a,b]$.
If $K : C[a,b]\rightarrow C[a,b]$ is a bounded linear operator, then, for each $t \in [a,b]$, let $\delta_{t}^{\star}$ be the evaluation functional at $t$. That is, $\delta_{t}^{\star}(f)=f(t)$. The functional $\delta_{t}^{\star}$ is bounded on $C[a,b]$. So $\delta_{t}^{\star}\circ K=K^{\star}\delta_{t}^{\star}$ is a bounded linear functional on $C[a,b]$. Hence, there is unique finite complex Borel measure $\mu_{t}$ such that
$$
               (Kf)(t)= \delta_{t}^{\star}(Kf) = (K^{\star}\delta_{t}^{\star})(f)=\int_{a}^{b}f(s)d\mu_{t}(s),\\
                    \|\mu_{t}\|=\|K^{\star}x_{t}^{\star}\|.
$$
If there is a dominating positive finite meausre $\mu$ such that $\mu_{t} << \mu$ for all $t\in[a,b]$, then you can write $d\mu_{t}(s) = K_{t}(s)d\mu(s)$ where $K_{t}(s) \in L^{1}_{\mu}[a,b]$ for each $t$. That gets you close to such a representation, under the additional assumption:
$$
          Kf = \int_{a}^{b}f(s)K_{t}(s)\,d\mu(s) = \int_{a}^{b}f(s)K(t,s)\,d\mu(s).
$$
Without the additional assumption, the integrals of continuous functions with respect to finite complex Borel measures can be expressed in terms of functions of bounded variation, which gives a representation as a Riemann-Stieltjes integral
$$
          Kf = \int_{a}^{b}f(s)d_{s}L_{t}(s)=\int_{a}^{b}f(s)d_{s}L(t,s).
$$