The positivity condition ensures that the Banach limit of a real-valued sequence is real-valued, and that limits can be compared: if $x\leq y$, then $\phi(x)\leq\phi(y)$. In particular, given a real-valued sequence $x$, by comparison with the sequences $y(n)=\inf_{{k\geq n}}x(k)$ and $z(n)=\sup_{{k\geq n}}x(k)$, it follows that $\liminf_{n}x(n)\leq\phi(x)\leq\limsup_{n}x(n)$.

The shift invariance allows any finite number of terms of the sequence to be neglected when taking the Banach limit, as is possible with the classical limit.

On the other hand, $\phi$ can never be multiplicative, meaning that $\phi(xy)=\phi(x)\phi(y)$ fails. For a counter-example, set $x=(0,1,0,1,\ldots)$; then we would have $\phi(0)=\phi(x\cdot Sx)=\phi(x)\phi(Sx)=\phi(x)^{2}$, so $\phi(x)=0$, but $1=\phi(1)=\phi(x+Sx)=\phi(x)+\phi(Sx)=2\phi(x)=0$.

That $\phi$ is continuous means it is compatible with limits in $\ell^{\infty}$. For example, suppose that $\{x_{k}\}_{{k\in\mathbb{N}}}\subset\ell^{\infty}$, and that $\sum_{{k=0}}^{\infty}x_{k}$ is absolutely convergent in $\ell^{\infty}$. (In other words, $\sum_{{k=0}}^{\infty}\lVert x_{k}\rVert_{\infty}<\infty$.) Then $\phi(\sum_{{k=0}}^{\infty}x_{k})=\sum_{{k=0}}^{\infty}\phi(x_{k})$ by continuity. Observe that this is just the dominated convergence theorem, specialized to the case of the counting measure on $\mathbb{N}$, in disguise.

In some definitions of the Banach limit, condition (i) is replaced by the seemingly weaker condition that $\phi(1)=1$ — the Banach limit of a constant sequence is that constant. In fact, the latter condition together with shift invarance implies condition (i).

If we restrict to real-valued sequences, condition (ii) is clearly redundant, in view of the other conditions.