How to demonstrate a property that can hold for linear maps between an infinite vector spaces via sequences Hi I am trying to find a simple ( if possible) way to show that for some infinite dimensional vector space V there can exist linear transformations $f:V \to V$ such that they are injective but not surjective, and surjective but not injective.
My thought was possible something could be done using
$(a_{1},a_{2},a_{3},.......)  \to (b_{1},b_{2},b{3}......)$  entries in $\mathbb{N}$
in both directions, but I am not sure the best way to set it up. Any ideas?
Thanks 
 A: Take $\;V:=\left\{\{x_n\}_{n=1}^\infty\subset\Bbb R\right\}\;$ . We can see this set's elements as infinite vectors $\;\{x_1,x_2,...\}\;$ with real entries , and it becomes a real linear space if we define addition and scalar multiplication componentwise.
Now check the following transformations fulfill what you want:
$$\begin{align*}&T: V\to V\;,\;\;T(x_1,x_2,...):=(0,x_1,x_2,...)\\{}\\
&S:V\to V\;,\;\;S(x_1,x_2,...):=(x_2,x_3,...)\end{align*}$$
A: Let $B$ be an infinite Hamel (vector-space)basis for $V,$ where $V$ is a vector space over the field $F.$ That is, for each non-zero $v\in V$ there is a unique non-empty finite $B_v\subset B$ and a unique $\{c_{b,v}:b\in B_v\}\subset F\backslash \{0\}$ such that $v=\sum_{b\in B_v}c_{b,v}b.$ 
(1) Since $B$ is infinite there is an injection $J:B\to B$ which is not a surjection. Define $J^*:V\to V$ by letting $J^*(0)=0$ and by letting $$J^*(v)=\sum_{b\in B_v}c_{b,v}J(b)$$ when $v\ne 0.$ Then $J^*$ is linear and injective but not surjective: No member of $B\backslash \{J(b):b\in B\}$ belongs to the image of $J^*.$ 
(2) Since $B$ is infinite there is a surjection $S:B\to B$ which is not an injection. Define $S^*:V\to V$ by letting $S^*(0)=0$ and by letting$$S^*(v)=\sum_{b\in B_v}c_{b,v}S(b)$$ when $v\ne 0.$ Then $S^*$ is linear and surjective but not injective: For some $b,b'\in B$ with $b\ne b'$ we have $S(b)=S(b')$, so $S^*(b)=S^*(b')$. 
