Teorema de Banach-Steinhaus

The Banach-Steinhaus theorem, also known as the Uniform Boundedness Principle, states that for a family of linear operators acting on a Banach space (a complete mathematical space), if each operator doesn't "blow up" when applied to every vector in the space individually, then there is a uniform bound controlling their size. In simpler terms, if all these operators perform well individually, they collectively do not become arbitrarily large; their "sizes" are uniformly limited. This principle ensures that pointwise boundedness of operators implies a uniform bound, which is essential for analyzing stability and convergence in functional analysis.