Find the norm of functional

Question

Consider the functional from $l_2$. $$ x=(x_n)\mapsto \sum \frac{x_n+x_{n+1}}{2^n}. $$ What is the norm of the functional?

Answer 1

It should not be very difficult to rewrite your functional to the form $$x\mapsto \sum x_n y_n$$ where $y=y_n\in\ell_2$.

Then the norm of this functional is precisely $\|y\|_2$, i.e., it is the same as the $\ell_2$-norm of the sequence $y$.

To see this, just notice that you have the functional of the form $$f(x)=\langle x,y \rangle,$$ where $\langle \cdot,\cdot \rangle$ is the standard inner product on $\ell_2$.

From Cauchy-Schwarz inequality you have $$|f(x)| = |\langle x,y \rangle| \le \|y\|\cdot\|x\|,$$ i.e., $$\frac{|f(x)|}{\|x\|} \le \|y\|$$ which means that $\|f\|\le\|y\|$.

The opposite inequality follows from $$f(y)=\langle y,y \rangle =\|y\|^2$$ or $$\frac{f(y)}{\|y\|}=\|y\|.$$