Vector space endomorphisms of sequence space Suppose we are working in the sequence space $K^\mathbb{N}$, defined as follows:
Let $K$ be the field either of real or complex numbers. We denote $K^\mathbb{N}$ the set of scalars $(x_n)_{n\in\mathbb{N}}, \ x_n\in K$ and we define the vector addition $(x_n)_{n\in\mathbb{N}}+(y_n)_{n\in\mathbb{N}}=(x_n+y_n)_{n\in\mathbb{N}}$ and scalar multiplication $a(x_n)_{n\in\mathbb{N}}=(ax_n)_{n\in\mathbb{N}}$, so $K^\mathbb{N}$ is a vector space (of infinite dimension).
I am trying to understand the form of vector space endomorphisms of $K^\mathbb{N}$. Firstly, all vector space endomorphisms of $K$ are of the form $x\mapsto ax$, for some $a\in K$.
Two examples of endomorphisms of $K^\mathbb{N}$ are: 
$(x_n)_{n\in\mathbb{N}}\mapsto (x_{n+1})_{n\in\mathbb{N}}$, namely it "moves" the original sequence to the subsequence $(x_{n+1})_{n\in\mathbb{N}}$.
$(x_n)_{n\in\mathbb{N}}\mapsto (x_1,0,0,\ldots)$, it acts as the projection on the first term of the sequence. 
As it seems, these two examples have a very different form that is not as simple as in the case of endomorsphisms of $K$. 
Can we find the general form of all the endomorphisms of $K^\mathbb{N}$?
 A: if you look at a question that I asked in this pretty  web cite,
and with what I had a lot of - 1 perhaps by specialist, within
reaction leaving in the comment, I think we come to describe these
kinds of endomorphisms easily.
an endomorphism $f$  of $K^{\Bbb{N}}$ is an application that for
each $ (x_n)_{n\in \Bbb{N}}$ assign   $(y_n)_{n\in\Bbb{N}}$  from
$K^{\Bbb{N}}$ to $K^{\Bbb{N}}$ such that for all $m\in{\Bbb{N}}$
the restriction of $f$ to $K^m$  is a linear map from $K^m$ to
$K^{\Bbb{N}}$, so
the linearity of the restriction of $f$ to $ K^m$ requires that
coordinate $y_i$ are image of $(x_1, ..., x_m)$ by a polynomial in
$m$ variables, homogeneous of degree $1$.
So we must to define the value of a function  polynomial  in $n$
variable on $K^m$  :  denoted this  value by (this is a function
polynomial) $\overbrace{P}(x_1, ..., x_m) = P (x_1, ..., x_m,
0,...,0)$ if $n\geq m $ or $= P (x_1, ..., x_n, 0,...,0)$ if $
n\leq m$.
So $ f = (\overbrace{P_i})_{i\in\Bbb{N}}$ with $P_i$ a homogeneous
polynomial (on finite variable ) of degree $1$.
check  it is linear  is immediate.
