MHB Convergence of bounded linear operators

[Fermat1]

Let $$(T_{n}) $$be a sequence in $${B(l_2}$$ given by
$$T_{n}(x)=(2^{-1}x_{1},...,2^{-n}x_{n},0,0,...). $$Show that $$T_{n}->T$$ given by
$$T(x)==(2^{-1}x_{1},2^{-2}x_{2},0,0,...). $$

I get a sequence of geometric series as my answer for the norm, but not sure whether that's correct.

 

[Opalg]

Let $$(T_{n}) $$be a sequence in $${B(l_2}$$ given by
$$T_{n}(x)=(2^{-1}x_{1},...,2^{-n}x_{n},0,0,...). $$Show that $$T_{n}->T$$ given by
$$T(x)==(2^{-1}x_{1},2^{-2}x_{2},0,0,...). $$

I get a sequence of geometric series as my answer for the norm, but not sure whether that's correct.

You have written the limit operator wrongly. Its definition should be $T(x)=(2^{-1}x_{1},2^{-2}x_{2},\color{red}{2^{-3}x_{3},2^{-4}x_{4},\ldots}).$ To show that $T_n\to T$, you need to show that $$\sup_{\|x\|\leqslant1}\|T(x) - T_n(x)\| = 0.$$

 

[Fermat1]

You have written the limit operator wrongly. Its definition should be $T(x)=(2^{-1}x_{1},2^{-2}x_{2},\color{red}{2^{-3}x_{3},2^{-4}x_{4},\ldots}).$ To show that $T_n\to T$, you need to show that $$\sup_{\|x\|=1}\|T(x) - T_n(x)\| = 0.$$

Do you mean $$\sup_{\|x\|=1}\|T(x) - T_n(x)\| -> 0.$$ ?

Well, $$\|T(x) - T_n(x)\|$$ is a sequence of geometric series, each of whose sum is zero, so the sequence tends to zero. Is this correct?

 

[Fermat1]

Actually I think that's wrong. Can you help me out?

 

[Opalg]

Do you mean $$\sup_{\|x\|=1}\|T(x) - T_n(x)\| \to 0$$ ?

Yes (careless typo on my part).

The coefficients in the definition of $Tx$ form a geometric sequence, but that does not help in the proof that $T_n\to T$. What you have to do is to look at the norm of $\|T(x) - T_n(x)\|$, where $x$ is a unit vector in $l_2$. If $x = (x_1,x_2,x_3,\ldots)$ then $T(x) - T_n(x) = (0,0,0,\ldots,0,2^{-(n+1)}x_{n+1},2^{-(n+2)}x_{n+2},\ldots)$ (where the first $n$ coordinates are all zero), and show that given $\varepsilon$ you can choose $n$ large enough to make the $l_2$-norm of that vector less than $\varepsilon$ for all $x$ in the unit ball of $l_2$.

 

[Fermat1]

I get that the sup of $$\|T(x) - T_n(x)\|$$ over norm vectors is equal to the square root of the series from k=1 to infinity of $$\frac{1}{4^{n+k}}$$. So I need to show this sequence of series converges to zero. Agree?

 

[Opalg]

I get that the sup of $$\|T(x) - T_n(x)\|$$ over norm vectors is equal to the square root of the series from k=1 to infinity of $$\frac{1}{4^{n+k}}$$. So I need to show this sequence of series converges to zero. Agree?

Yes. (Yes)