Consider an integral equation $$u(x) - \int_a^x K(x,y) u(y) dy = f(x), \qquad x \in [a,b] \qquad (1) $$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $f$ and $K$ are known continuous functions.
Volterra's Theorem states that (1) has a unique solution $u \in C[a,b] $ (where $C[a,b]$ is the set of continuous functions on $[a,b]$).
My question is the following. Is the continuity of the solution a property or an assumption of the solution? That is, can we assume that $u \in X$, where $X$ is a normed space (instead of $u \in C[a,b] $) and show that there is a unique solution to (1)?
In the latter question I am thinking about the following corollary of the Riesz's theorem:
Let $A:X\to X$ be a compact linear operator on a normed space $X$. If the homogeneous equation $$u - Au = 0$$ only has the trivial solution $u=0$, then for each $f\in X$ the inhomogeneous equation $$ u - Au = f $$ has a unique solution $u \in X$ and this solution depends continuously on $f$.
(Reader may see the Volterra's theorem and the corollary, e.g., from Kress, R. (2014). In Linear Integral Equations. Springer, New York, NY, Theorem 3.10 and Corollary 3.5.)
The Volterra operator is quasinilpotent, hence $1$ is not an eigenvalue of $A$. This implies that the equation $Au=u$ has only the trivial solutio $u=0.$