Bounded Linear Operators - Main Theorem

The Uniform Boundedness Principle, also known as the Banach-Steinhaus Theorem, is one of the three fundamental theorems of functional analysis (along with the Open Mapping Theorem and the Hahn-Banach Theorem).

TheoremUniform Boundedness Principle

Let $X$ be a Banach space, $Y$ a normed space, and $\{T_\alpha\}_{\alpha \in A}$ a family of bounded linear operators from $X$ to $Y$ . Suppose that for each $x \in X$ , the set $\{T_\alpha(x) : \alpha \in A\}$ is bounded in $Y$ .

Then the family $\{T_\alpha\}$ is uniformly bounded in operator norm: $\sup_{\alpha \in A} \|T_\alpha\| < \infty$

This theorem is remarkable: pointwise boundedness (for each fixed $x$ , the sequence $\|T_\alpha(x)\|$ is bounded) automatically implies uniform boundedness (there is a single constant $C$ such that $\|T_\alpha\| \leq C$ for all $\alpha$ ).

Proof

The proof uses the Baire Category Theorem. For each $n \in \mathbb{N}$ , define $E_n = \{x \in X : \|T_\alpha(x)\| \leq n \text{ for all } \alpha \in A\}$

By hypothesis, $X = \bigcup_{n=1}^\infty E_n$ . Each $E_n$ is closed (as intersection of closed sets $\{x : \|T_\alpha(x)\| \leq n\}$ ).

Since $X$ is a Banach space (complete metric space), the Baire Category Theorem implies that some $E_N$ has non-empty interior. Thus there exist $x_0 \in X$ and $r > 0$ such that $B(x_0, r) \subset E_N$ .

For any $x \in X$ with $\|x\| < r$ , we have $x_0 + x \in E_N$ and $x_0 - x \in E_N$ , so $\|T_\alpha(x)\| = \frac{1}{2}\|T_\alpha(x_0 + x) - T_\alpha(x_0 - x)\| \leq \frac{1}{2}(2N + 2N) = 2N$

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ExampleApplications

Fourier Series: If $f \in L^1[0,2\pi]$ and the Fourier series converges pointwise, the sequence of partial sum operators $S_n$ is uniformly bounded
Weak Convergence: If $x_n \rightharpoonup x$ weakly in a Banach space, then $\sup_n \|x_n\| < \infty$
Convergence of Operators: If $T_n(x) \to T(x)$ for all $x$ in a Banach space, then $T$ is bounded and $\sup_n \|T_n\| < \infty$

Remark

The completeness of $X$ is essential. The theorem fails for incomplete normed spaces. The requirement that $X$ be a Banach space cannot be relaxed to just a normed space.

This principle is one of the most frequently applied results in functional analysis, appearing in the theory of Fourier series, weak convergence, and the study of operator sequences.