Kreizhn
Jun21-08, 11:40 AM
1. The problem statement, all variables and given/known data
Define R^\infty_f = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \exists k_0 \text{ such that } t^{(k})=0 \; \forall k\geq k_0 \}
Define l^\infty = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \sup_{k\geq 1} | t^{(k})| < \infty \}
Observe that R^\infty_f is a linear subspace of l^\infty. Show that R^\infty_f is not closed in l^\infty, then show that the closure of R^\infty_f is the space c_0 ;
2. Relevant equations
The space c_0 is the set of all sequences converging to zero
3. The attempt at a solution
It's not too hard to show that this set is not closed. It suffices to show that there is a convergent sequence in l^\infty such that every term is in R^\infty_f , but whose limit is not in R^\infty_f . I constructed the following sequence
x_1 = (1, 0, \ldots, )
x_2 = (1, \frac{1}{2}, 0 , \ldots, )
\vdots
x_n = (1, \ldots, \frac{1}{n}, 0, \ldots} )
which converges to the point a = (1, \frac{1}{2}, \ldots, \frac{1}{n-1}, \frac{1}{n}, \frac{1}{n+1}, \ldots )
It's the closure part that I'm worried about. I'm not terribly sure how I would go about showing that...
Define R^\infty_f = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \exists k_0 \text{ such that } t^{(k})=0 \; \forall k\geq k_0 \}
Define l^\infty = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \sup_{k\geq 1} | t^{(k})| < \infty \}
Observe that R^\infty_f is a linear subspace of l^\infty. Show that R^\infty_f is not closed in l^\infty, then show that the closure of R^\infty_f is the space c_0 ;
2. Relevant equations
The space c_0 is the set of all sequences converging to zero
3. The attempt at a solution
It's not too hard to show that this set is not closed. It suffices to show that there is a convergent sequence in l^\infty such that every term is in R^\infty_f , but whose limit is not in R^\infty_f . I constructed the following sequence
x_1 = (1, 0, \ldots, )
x_2 = (1, \frac{1}{2}, 0 , \ldots, )
\vdots
x_n = (1, \ldots, \frac{1}{n}, 0, \ldots} )
which converges to the point a = (1, \frac{1}{2}, \ldots, \frac{1}{n-1}, \frac{1}{n}, \frac{1}{n+1}, \ldots )
It's the closure part that I'm worried about. I'm not terribly sure how I would go about showing that...