Equibounded pointwise approximation implies the uniform one

Key Points

• The study aims to establish the conditions under which a continuous function can be uniformly approximated by elements of a linear subspace through equibounded sequences.
• Presented a proof focusing on continuous functions in compact Hausdorff spaces.
• Analyzed the relationship between pointwise and uniform approximation.
• Explored the approximation of functions defined on compacta using rational functions and holomorphic functions.
• A continuous function can be uniformly approximated if and only if it can be pointwise approximated by equibounded sequences in its linear subspace.
• Every continuous function can be approximated by rational functions if pointwise approximation by holomorphic functions holds, without equibounded conditions.