The Uniform Boundedness Principle

As a consequence of the Baire Category Theorem, it was proven that a real-valued mapping that is the pointwise limit of a sequence of continuous mappings on a complete metric space must be continuous at all points of a dense subset of its domain. Adapt that proof so that it applies to mappings into any metric space. Use this to prove that the pointwise limit of a sequence of continuous linear operators on a Banach space has a limit that is continuous at some point and hence, by linearity, is continuous.

Let $\left\{ f_{n}\right\}$ be a sequence in $L^{\infty }[a, b]$. Suppose that for each $g \in L^{1}[a, b], \lim_{n \rightarrow \infty } \int_{a}^{b} g \cdot f_{n} \mathrm{dm}$ exists. Show that there is a function $f \in L^{\infty }[a, b]$ such that $\lim_{n \rightarrow \infty } \int_{a}^{b} g \cdot f_{n} \mathrm{dm} = \int_{a}^{b} g \cdot f \mathrm{dm}$ for all $g \in L^{1}[a, b]$.

Let $X$ be the linear space of all polynomials defined on $\mathbb {R}$. For $p \in X$, define $\left\| p\right\|$ to be the sum of the absolute values of the coefficients of $p$. Show that this is a norm on $X$. For each $n$, define $\psi_{n}: X \rightarrow \mathbb {R}$ by $\psi_{n}(p) = p^{(n)}(0)$. Use the properties of the sequence $\left\{ \psi_{n}\right\}$ in $\mathcal{L}(X, \mathbb {R})$ to show that $X$ is not a Banach space.