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Suppose I have a finite dimensional normed vector space $V$, and a sequence of $f_i$ in $V^*$ with norm all equal to 1. I want to know when does $\lim_{n->\infty} \frac{1}{n}\sum_{i=1}^{n} f_i$ converges. Now because $||f_i|| = 1$ for all $i$, I know if such limit exists then it is a bounded linear operator.

And if in addition I know for some {$x_1, x_2 ,...$} in $V$, $f_i(x_j)$ is eventually always non-negative (i.e. for each $x_j$, there exists some large enough $N$ such that for all $i>N$, $f_i(x_j) \geq 0$), does that help determine whether the limit exists?

Kerwin Yi
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1 Answers1

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In general, the limit may not exists. For example, take an increasing sequence of integers $n_{k}$ such that $\lim\frac{n_{k}}{n_{k+1}} = 0$, for $V$ take $R^{2}$ and lastly define $f_{n}(x_{1},x_{2}) = x_{1}$ if n is between $n_{k}$ and $n_{k+1}$ for event k and $f_{n}(x_{1},x_{2}) = -x_{1}$ if n is between $n_{k}$ and $n_{k+1}$ for odd k.
Now it is easy to see that the average for $n = n_{k}$ , where k is even will be close to $x_{1}$ but for odd $k$ it will be close to $-x_{1}$.
As far as the second case is concerned, since $V$ is finite dimensional it is isomorphic to Hilbert space (any two finite dimensional Banach spaces are isomorphic) so we can assume that any functional has form $<x,y>$. In words, $f_{n}(x) = <x,y_{n}>$ for some $y_{n}$. By assumption if $e_{k}$ is part of orthogonal basis on $V$ we have $f_{n}(e_{k})$ is $>=0$ for $n$ large enough. But this is also true for $-e_{k}$ so we must have $<e_{k},y_{n}> = 0$ for $n$ large enough. Since this is true for any $k$ and the space is finite dimensional, for $n$ large enough we must have $y_{n} = 0$. Then of course the average of $f_{n}$ converges to $0$.

Salcio
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