Editing Uniform boundedness principle (section)

==Example: pointwise convergence of Fourier series==

Let <math>\mathbb{T}</math> be the [[Circle group|circle]], and let <math>C(\mathbb{T})</math> be the Banach space of continuous functions on <math>\mathbb{T},</math> with the [[uniform norm]]. Using the uniform boundedness principle, one can show that there exists an element in <math>C(\mathbb{T})</math> for which the Fourier series does not converge pointwise.

For <math>f \in C(\mathbb{T}),</math> its [[Fourier series]] is defined by
<math display=block>\sum_{k \in \Z} \hat{f}(k) e^{ikx} = \sum_{k \in \Z} \frac{1}{2\pi} \left (\int_0 ^{2 \pi} f(t) e^{-ikt} dt \right) e^{ikx},</math>
and the ''N''-th symmetric partial sum is
<math display=block>S_N(f)(x) = \sum_{k=-N}^N \hat{f}(k) e^{ikx} =  \frac{1}{2 \pi} \int_0^{2 \pi} f(t) D_N(x - t) \, dt,</math>
where <math>D_N</math> is the <math>N</math>-th [[Dirichlet kernel]]. Fix <math>x \in \mathbb{T}</math> and consider the convergence of <math>\left\{S_N(f)(x)\right\}.</math> 
The functional <math>\varphi_{N,x} : C(\mathbb{T}) \to \Complex</math> defined by
<math display=block>\varphi_{N, x}(f) =  S_N(f)(x), \qquad f \in C(\mathbb{T}),</math>
is bounded. 
The norm of <math>\varphi_{N,x},</math> in the dual of <math>C(\mathbb{T}),</math> is the norm of the signed measure <math>(2(2 \pi)^{-1} D_N(x - t) d t,</math> namely
<math display=block>\left\|\varphi_{N,x}\right\| = \frac{1}{2 \pi} \int_0^{2 \pi} \left|D_N(x-t)\right| \, dt = \frac{1}{2 \pi} \int_0^{2 \pi} \left|D_N(s)\right| \, ds = \left\|D_N\right\|_{L^1(\mathbb{T})}.</math>

It can be verified that
<math display=block>\frac{1}{2 \pi} \int_0 ^{2 \pi} |D_N(t)| \, dt \geq \frac{1}{2\pi}\int_0^{2\pi} \frac{\left|\sin\left( (N + \tfrac{1}{2})t \right)\right|}{t/2} \, dt \to \infty.</math>

So the collection <math>\left(\varphi_{N, x}\right)</math> is unbounded in <math>C(\mathbb{T})^{\ast},</math> the dual of <math>C(\mathbb{T}).</math> 
Therefore, by the uniform boundedness principle, for any <math>x \in \mathbb{T},</math> the set of continuous functions whose Fourier series diverges at <math>x</math> is dense in <math>C(\mathbb{T}).</math>

More can be concluded by applying the principle of condensation of singularities. 
Let <math>\left(x_m\right)</math> be a dense sequence in <math>\mathbb{T}.</math> 
Define <math>\varphi_{N, x_m}</math> in the similar way as above. The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each <math>x_m</math>  is dense in <math>C(\mathbb{T})</math> (however, the Fourier series of a continuous function <math>f</math> converges to <math>f(x)</math> for almost every <math>x \in \mathbb{T},</math> by [[Carleson's theorem]]).