Finding the value of bounded linear functional

Question

In the $\ell^2$ space with:$$\|f\| = \sup\{|f(x)| : x\in \ell, \|x\| \le 1\}$$ Where $f(x)=\sum \frac{x_n} {3^n}$.

I need to find the value of $||f||$. I am a beginner and I haven't ever dealt with such problems. Please suggest how can I calculate its value?

Answer 1

The Riesz Representation Theorem establishes that fact that any bounded linear functional over a Hilbert space $H$ is of the form $$ f(x)=\langle x,y\rangle $$ for some $y\in H$. And in that case, $\|f\|=\|y\|$. This solves is automatically in your case.

A less sophisticated way of doing exactly the same is to apply Cauchy Schwarz and work toward finding which $x$ approximates equality. In your case that would be $$ |f(x)|\leq \|x\|\,\Big(\sum_n\frac1{3^{2n}}\Big)^{1/2} $$ In this case it is kind of obvious that the inequality will be an equality if $$ x=\sum_n\frac1{3^n}\,e_n. $$