Boundedness of functional

Question

In the setting of $2\pi$-periodic $C^1$ functions (whose Fourier series converge to themselves), and given a linear functional $D:C^1_{\text{per}}\to\mathbb R$ satisfying $\sup_{n}|D(e^{inx})|<\infty$ I would like to show that $D$ is continuous (or equivalently, bounded).

Attempt

The supremum condition seems it should imply boundedness, but I'm not managing to formalise that and keep running into a circular argument. For example, by contradiction, let's assume a series $\{f_n\}$ where $\|f_n\|=1$ such that $D(f_n) > n^2$, then taking $\frac {f_n}n$ which has norm $\frac 1n$ we have $D(\frac {f_n}n) > n$ which is a contradiction if $D$ is continuous, but that's what we're trying to prove...

Cheers.

Edit

I think in $L^2$ the hypothesis is wrong since we can construct a sequence \begin{align} f_1 &= e_1 \\ f_2 &= \frac 1 {\sqrt 2}e_1 + \frac 1{\sqrt 2}e_2 \\ &\ \vdots \\ f_n &= \sum_{k=1}^n \frac 1{\sqrt n}e_k \end{align}

So $\|f_n\|_2^2 = 1$ but if $D(e_k)=M$ (which satisfies our bound condition) we have $D(f_n) = \frac {nM}{\sqrt n}\to \infty$

In light of the failure of the $L^2$ norm, I suppose the $C^1$ norm $\|f\|_\infty+\|f'\|_\infty$ is the focus here.

Answer 1

This is not true. The functions $\{e^{inx}\}$ do not span $C^1_{\text{per}}$; indeed, no countable set can. So using the axiom of choice, one can show the existence of a linear functional that vanishes on all the $e^{inx}$ but is not continuous.

You can't tell whether a linear functional is continuous by looking at a proper subspace, even if that subspace is dense.

You can, however, show that there exists a unique continuous linear functional $D_1$ such that $D_1(e^{inx}) = D(e^{inx})$ for every $n$. Is that what you want?

Answer 2

I will use complex Fourier series for ease of writing; everything works the same for real series. Suppose $f\in C^1_{\rm per}$, $f(x)=\sum c_n e^{i n x}$. Since $f'$ is continuous, it belongs to $L^2$. The Fourier series of $f'$ is $f'(x)=\sum inc_n e^{i n x}$. By Parseval's identity, $$\sum n^2 |c_n|^2 = (2\pi)^{-1} \int_0^{2\pi} |f' |^2 \le \|f\|_{C^1}^2\tag{1}$$

Let $D$ be a functional from the problem. Define $b_n=D(e^{inx})$. The sequence $(b_n)$ is bounded. From the Cauchy-Schwarz, for every integer $N$, $$\sum_{0<|n|\le N} b_n c_n \le \sup|b_n| \sum_{n\ne 0} |c_n| \le \sup|b_n| \sqrt{\sum_{n\ne 0} n^{-2}}\sqrt{\sum_{n\ne 0} n^{ 2 }|c_n|^2} \le C \|f\|_{C^1} $$ Since $b_0c_0$ is also controlled by $\|f\|_{C^1}$, the operator $D$ is bounded on trigonometric polynomials. Since trigonometric polynomials are dense in $C^1_{\rm per}$, the restriction of $D$ to polynomials uniquely extends to a bounded operator on $C^1_{\rm per}$.