Let $(x_n)$ be a sequence in a normed vector space $X$. Is it possible to calculate the sequence of arithmetic means of $x_n$ as follows: $$a_n=\frac{x_1+x_2+...+x_n}{n}.$$ or, under which conditions the above formula is valid?
Arithmetic mean of a sequence in normed vector spaces
0
$\begingroup$
real-analysis
sequences-and-series
vector-spaces
normed-spaces
means
-
2Cesaro means I think would be. – 2017-02-10
-
0Is it meaningful for all normed vector spaces? – 2017-02-10
-
1you can add elements and you can multiply by constant, so why not? – 2017-02-10
-
1Are you sure of the formulation of your question ? If you are asking whether you can multiply the scalar $\frac 1n$ by the vector $x_1+\cdots+x_n$, then the answer is yes ! And the structure of normed vector space is not necessary for doing so : any (real or complex) vector space would do the job. – 2017-02-10
-
0Actually I wondered if $(a_n)$ preserves the limit when $(x_n)$ is convergent. I guess the answer is Yes. – 2017-02-10
-
1You have to assume that $x_n$ converges to $x$ in $X$. – 2017-02-10