I'm trying to find a Banach space $V$ with closed unit ball $B$ and a sequence of isometries $(f_n:V\to V)$ such that $(f_n)$ converges pointwise in $B$ but not uniformly in $B$.
My first attempts were to make it kind of simple. If $V=c_0$ (the space with sequences $x_n\to 0$ with the sup norm) take for instance the linear functions $T_n(x)=(0,\ldots,0,x_{n+1},x_{n+2},\ldots)$. It is easy to see that $T_n$ converges pointwise to the zero function, but it doesn't so uniformly in $B$. However, $T_n$ is not an isometry although it satisfies $\left\|{T(x)}\right\|\le\left\|{x}\right\|$.
Can someone give me a hint?
Thanks.
Let $V=l^{2}, f_n(x_1,x_2,...))=(x_1,x_2,...,x_n,0,x_{n+1},...)$ (where a $0$ is inserted at the n-th position). Then $f_n(x) \to x$ for every $x \in B$ but $\|f_n(x)-x\|^{2}\geq |x_{n+1}|^{2}$ so the convergence is not uniform on $B$.