Exercises 2.3 Exercises
1.
Show that the set \({\rm Seq(F)}\) of all sequences with values in a field \(F\) is a vector space under the componentwise addition
\begin{equation*}
(a_n)+(b_n)=(a_n+b_n)
\end{equation*}
and the scalar multiplication
\begin{equation*}
\alpha\cdot(x_n)=(\alpha x_n)\quad\text{where } \alpha,x_n\in F\text{ for every } n.
\end{equation*}
2.
Show that the set \({\rm Seq_0(\R)}\) of all real-valued sequences converging to \(0\) is a vector space. Use the same operations defined in an earlier exercise.3.
Show that the set \(\Q(\sqrt{2})=\{a+b\sqrt{2}:a,b\in\Q\}\) is a vector space over \(\Q\text{.}\) The addition is given by
\begin{equation*}
(a+b\sqrt{2})+(c+d\sqrt{2})=(a+c)+(b+d)\sqrt{2}
\end{equation*}
and the scalar multiplication by
\begin{equation*}
\alpha\cdot(a+b\sqrt{2})=\alpha a+\alpha b\sqrt{2} \quad\text{for }\alpha,a,b\in\Q.
\end{equation*}