Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 22G (Linear Analysis)

Let $m_N={\inf }_{n\ge N}{x}_n$ and $M_N={\sup }_{n\ge N}{x}_n$. Positivity and normalization giveShift invariance makes the middle term equal to $\varphi (x)$. Letting $N\to \infty$ yields

No. Shift invariance makes all $\varphi (e_n)$ equal. Positivity givesso $\varphi (e_n)=0$. If an ${\ell }^1$ sequence represented $\varphi$, then $y_n=\varphi (e_n)=0$ for every $n$, which would give $\varphi (1)=0$, a contradiction.

Take $x_n=(-1{)}^n$. Its shift is $-x$, so linearity and shift invariance giveThus every such functional assigns the common valuealthough the sequence does not converge.

Let $S$ be the left shift and defineShift invariance gives $\varphi (A_{p}x)=\varphi (x)$. The hypothesis says $A_{p}x$ converges uniformly in $n$, hence in the ${\ell }^{\infty }$ norm, to the constant sequence $y1$. By continuity,