Intereting Posts

Modern notational alternatives for the indefinite integral?
Union of $2$ set $A, B$ where $ A$ is a subset of $B$
Sort vectors according to their distance between them
Dimension of direct sum of vector spaces
Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$
Other Algebraically Independent Transcendentals
How prove this inequality $(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$
How to determine whether a number can be written as a sum of two squares?
Integral of rational function with higher degree in numerator
Travelling with a car
Are there infinitely many primes of the form $4n^{2}+3$?
Generalized Bernoulli's inequality
Essential ideals
Sequences of integers with lower density 0 and upper density 1.
Finite field, every element is a square implies char equal 2

There’s like three applications of the uniform boundedness principle in wikipedia:

1) If a sequence of bounded operators converges pointwise to an operator, then the limit operator is also bounded, and the convergence is uniform on compact sets.

2) Any weakly bounded subset of a normed space is bounded.

- For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.
- Dimension for a closed subspace of $C$.
- The weak topology on an infinite dimensional linear space is not first-countable
- Operator norm and tensor norms
- Banach spaces and their unit sphere
- How do I prove the completeness of $\ell^p$?

3) A result in pointwise convergence of Fourier series.

I am just asking if there’s more interesting applications of the uniform boundedness principle.

- Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals.
- Surprising identities / equations
- Pointwise approximation of a closed operator
- Proofs that every mathematician should know?
- A Riesz-type norm-preserving and bijective mapping between a Banach space and its dual
- Is it possible for a function to be in $L^p$ for only one $p$?
- Applications of complex numbers to solve non-complex problems
- List of interesting math podcasts?
- Other interesting consequences of $d=163$?
- Axiom of choice - to use or not to use

Let $f\in L^{p}(\mathbb{T})$, for some $1\leq p<\infty$, where $\mathbb{T}$ denotes the one-dimensional torus. Let $(a_{m})_{m\in\mathbb{Z}}$ be a bounded complex sequence. For $R\geq 0$, let $(a_{m}(R))_{m=1}^{\infty}$ be a compactly supported sequence such that $a_{m}(R)=a_{m}$ for all $\left|m\right|\leq R$. Define

$$S_{R}(f)(x):=\sum_{m\in\mathbb{Z}}a_{m}(R)\widehat{f}(m)e^{2\pi im\cdot x},\ \forall x\in\mathbb{T}$$

For each $R$, the above expressions defines an operator $L^{p}\rightarrow L^{p}$ that maps a function $f$ to the $\lfloor{R}\rfloor^{th}$ symmetric partial sum of its Fourier series.

Using, in part, a simple application of the Uniform Boundedness Principle, we can turn the question of $L^{p}$ convergence of $S_{R}(f)$ to $f$ as $R\rightarrow\infty$, for arbitrary $f\in L^{p}$, into a question of the uniform boundedness of the operators $S_{R}$, $R\geq 0$.

Here’s a couple more examples.

If $f : \Omega\subseteq\mathbb{C} \rightarrow \mathcal{L}(X)$ is a function from an open subset $\Omega$ of the complex plane into the bounded linear operators $\mathcal{L}(X)$ on a complex Banach space $X$, then $f$ is holomorphic iff $\lambda\mapsto x^{\star}(f(\lambda)x)$ is holomorphic on $\Omega$ for all $x \in X$, $x^{\star}\in X^{\star}$.

If $f$ is a holomorphic vector function on the punctured disk $0 < |\lambda| < \delta$ with values in a Banach space $X$, then $f$ has an essential singularity at $0$ iff there exists $x^{\star}\in X^{\star}$ such that $x^{\star}\circ f$ has an essential singularity at $0$.

- Is a polynomial also a covering map?
- Sum of two squares $n = a^2 + b^2$
- Show that $\operatorname{int}(A \cap B)= \operatorname{int}(A) \cap \operatorname{int}(B)$
- A question on a quotient of Alexandroff's double segment space
- Result due to Cohn, unique division ring whose unit group is a given group?
- Continuous Collatz Conjecture
- Wedge product and cross product – any difference?
- Calculating $a^n\pmod m$ in the general case
- Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
- Prove that $f(x) = x^3 -x $ is NOT Injective
- How is Cauchy's estimate derived?
- Easy proof for sum of squares $\approx n^3/3$
- Gamma Distribution out of sum of exponential random variables
- In the card game “Projective Set”: Compute the probability that $n$ cards contain a set
- Prove the product of a polynomial function of the roots of another polynomial is an integer.