= Solution
Take a minimizing sequence for the <variational regularization> functional
$$
F(u)=\frac12\|Au-f\|_Y^2+\alpha J(u).
$$
Its <coercivity> makes the sequence bounded. Since $X$ is a <reflexive Banach space>, a subsequence converges weakly to some $u\in X$. A convex norm-lower-semicontinuous functional has <weak lower semicontinuity>, so this applies to both $J$ and the convex continuous map $u\mapsto\|Au-f\|_Y^2$. Therefore
$$
F(u)\leq\liminf_nF(u_n)=\inf_XF,
$$
and $u$ is a minimizer. This is the <direct method in the calculus of variations>.
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