I have a question about one inequality. It is used to prove a Theorem.
This is the formulation of the Theorem.
Let $(X,∥⋅∥_X)$ be a normed linear space. If $X$ is a Banach space and if $T:X→X$ is a bounded linear operator and $∥T∥<1$ then $I−T$ is invertible and $(I−T)−I=\sum_{n=0}^{n}T^{n}$
How does one prove $$\|T^{n}\|\leqslant\|T\|^{n}.$$ Since this is used in the proof. Thank you so much.
This is a separate question, does $|\langle y,y\rangle|=\|y\|^{2}$ hold, where $\langle,\rangle$ is scalar product and the norm is given by $$\|y\|=(\langle y,y\rangle)^{\frac{1}{2}}$$
Please no hateful comments, I just want to learn. Thank you so much!
Since $T:X\to X$ is a bounded linear operator, the definition of $\|T\|$ is $$\|T\|=\sup_{x\ne 0}\frac{\|Tx\| }{\|x\|},$$ so $\|Tx\| \le \|T\| \|x\| $. And we get $$\|T^n\|=\sup_{\|x\|=1} \|T^n x\|\le \sup_{\|x\|=1} \|T\|^n\|x\| =\|T\|^n $$