We denote by $E$ the vector space of continuous functions defined on $[0,1]$ and taking values in $\mathbb{R}$ equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t)\,\mathrm{d}t$$ We consider a function $A:[0,1] \times [0,1] \rightarrow \mathbb{R}$ continuous. We are interested in the application $T: E \rightarrow E$ defined by $$T(f)(x) = \int_0^1 A(x,t) f(t)\,\mathrm{d}t$$ We suppose that $\ker T$ is finite-dimensional. Justify that $T$ induces an isomorphism from $(\ker T)^\perp$ onto $\operatorname{Im} T$.
We denote by $E$ the vector space of continuous functions defined on $[0,1]$ and taking values in $\mathbb{R}$ equipped with the inner product defined by
$$\langle f, g \rangle = \int_0^1 f(t) g(t)\,\mathrm{d}t$$
We consider a function $A:[0,1] \times [0,1] \rightarrow \mathbb{R}$ continuous. We are interested in the application $T: E \rightarrow E$ defined by
$$T(f)(x) = \int_0^1 A(x,t) f(t)\,\mathrm{d}t$$
We suppose that $\ker T$ is finite-dimensional.
Justify that $T$ induces an isomorphism from $(\ker T)^\perp$ onto $\operatorname{Im} T$.