checking whether an operator is bounded or not let X be a real normed space with finitely many non zero terms,with supremum norm and let 
T:X$\to$X be a one-one and onto linear operator defined by $$T(x_1,x_2,x_3,......)=(x_1,\frac{x_2}{4},\frac{x_3}{9},.....)$$
then, which of the following is true:
(1) T is bounded but $T^{-1}$ is not bounded
(2) T is not bounded but $T^{-1}$ is bounded
(3) both T and $T^{-1}$ are bounded
(4) neither of the two is bounded
my thought:
i am trying to apply the result :-a linear operator T:X $\to$ Y is bounded iff it is continuous.now i can say just by looking at the operator that given T is continuous at  point ,say,(1,1,0,0,0,....) and being linear it will be cts on X and hence by above theorem it will be bounded.also,after constructing $T^{-1}$ and following the similar steps i am getting that $T^{-1}$ is bonded.so,(3) must be the correct option
is this approach correct??if not,please give other alternatives...
 A: An operator is bounded if there is a $C > 0$ st $$ \| T x \| \le C \|x\|$$  for all $x \in X$.
Now, let $e_1=(1,0,0,...), e_2=(0,1,0,0,...), $ etc. We see that $T e_n = (1/n^2) e_n$ for each $n \in \mathbb{N}$. This means $T^{-1} e_n = n^2 e_n$.
Notice that $\| T^{-1} e_n \| = n^2 \|e_n\|$. What does this say about boundedness?

Here is another approach. If $T^{-1}$ is bounded then, as you say, it is continuous. Consider the sequence $x_n = (1/n) e_n$. Notice that $\|x_n\| = 1/n \to 0$, and $x_n$ converges to zero. If $T^{-1}$ were continuous, then $T^{-1} x_n$ would converge.
What happens to the sequence $\{ T^{-1} x_n \}$?
A: Note that $X$ is infinite dimensional and so not all the linear operators are automatically continuous. The operator $T$ will be continuous if and only if
$$ ||T|| := \mathrm{sup}_{||x|| = 1} ||Tx|| < +\infty. $$
If $||x|| = 1$ then $|x_n| \leq 1$ for all $n \in \mathbb{N}$ and then $|(T(x_n))_m| = \left| \frac{x_n}{m^2} \right| \leq \frac{1}{m^2} \leq 1$ for all $m \in \mathbb{N}$ and thus $||T|| \leq 1$ so $T$ is continuous.
To see whether $T^{-1}$ is continuous, write $T^{-1}$ explicitly and try to estimate $||T^{-1}||$.
