Let $Q$ be the space of infinite sequences $$\mathbf { x } : = \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots \right)$$ of real numbers $x _ { n } \in [ 0,1 ]$, with the product topology coming from the identification $Q = [ 0,1 ] ^ { \mathbb { N } }$. ($[ 0,1 ]$ has the euclidean topology.) Let $S : Q \longrightarrow \mathbb { R }$ be the map $$S ( \mathbf { x } ) : = \sum _ { n } \frac { x _ { n } } { n ^ { 2 } } .$$ (A) Let $Q _ { 2 } : = \left\{ \left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \left\lvert \, 0 \leq y _ { n } \leq \frac { 1 } { n } \right. \right\}$. Show that $Q _ { 2 }$ is compact. (B) Let $D : Q _ { 2 } \longrightarrow Q$ be the map $$\left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \mapsto \left( y _ { 1 } , 2 y _ { 2 } , \ldots , n y _ { n } , \ldots \right)$$ Show that $D$ is a homeomorphism. (Hint: first show that $Q$ is Hausdorff.) (C) Show that $S \circ D : Q _ { 2 } \longrightarrow \mathbb { R }$ is continuous. (Hint: Show that there is a suitable inner-product space $( L , \langle - , - \rangle )$ and a vector $\mathbf { a } \in L$ such that $( S \circ D ) ( \mathbf { x } ) = \langle \mathbf { x } , \mathbf { a } \rangle$ for each $\mathbf { x } \in Q _ { 2 }$.) (D) Show that $S$ is continuous.
Let $Q$ be the space of infinite sequences
$$\mathbf { x } : = \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots \right)$$
of real numbers $x _ { n } \in [ 0,1 ]$, with the product topology coming from the identification $Q = [ 0,1 ] ^ { \mathbb { N } }$. ($[ 0,1 ]$ has the euclidean topology.) Let $S : Q \longrightarrow \mathbb { R }$ be the map
$$S ( \mathbf { x } ) : = \sum _ { n } \frac { x _ { n } } { n ^ { 2 } } .$$
(A) Let $Q _ { 2 } : = \left\{ \left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \left\lvert \, 0 \leq y _ { n } \leq \frac { 1 } { n } \right. \right\}$. Show that $Q _ { 2 }$ is compact.\\
(B) Let $D : Q _ { 2 } \longrightarrow Q$ be the map
$$\left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \mapsto \left( y _ { 1 } , 2 y _ { 2 } , \ldots , n y _ { n } , \ldots \right)$$
Show that $D$ is a homeomorphism. (Hint: first show that $Q$ is Hausdorff.)\\
(C) Show that $S \circ D : Q _ { 2 } \longrightarrow \mathbb { R }$ is continuous. (Hint: Show that there is a suitable inner-product space $( L , \langle - , - \rangle )$ and a vector $\mathbf { a } \in L$ such that $( S \circ D ) ( \mathbf { x } ) = \langle \mathbf { x } , \mathbf { a } \rangle$ for each $\mathbf { x } \in Q _ { 2 }$.)\\
(D) Show that $S$ is continuous.