let $T:L^2 \to L^2$ be a bounded linear map st for all $x=x_1,x_2,x_3...x_n$ which are elements of $L^2$,
$$Tx=(x_1,x_2/\sqrt 2,x_3/\sqrt 3,....x_n/\sqrt n)$$
show that this is not invertible
I dont understand why this isn't invertible , i assumed
$$T(x)=(x_1,\sqrt 2 x_2, \sqrt 3 x_3.. . etc)$$
would be the inverse.
I know that i need to show its not surjective to prove its not invertible, ie for some $T:u \to v$, there exists a $v \in V$ st $T(u)=v$ but how to I apply this when $L^2 \to L^2$?
Let $a_n=\frac 1 n$. Then $(a_n)\in \ell^{2}$. Suppose $(a_n)=T(x_n)$. Then $\frac 1 n =\frac {x_n} {\sqrt n}$ so $x_n=\frac 1 {\sqrt n}$. But then $\sum x_n^{2}=\infty$, a contradiction. So $T$ is not surjective.