Fixed point

Let $(X, ||\cdot||)$ be a Banach space and $T \in \mathbb{B} (X, X)$ with the property the series $\sum \limits_{n=1}^{\infty} ||T^n||$ converges. If $y \in X$ we define the map $S_y: X \rightarrow X$ such that:

$$S_y(x)= y+T(x)$$

Prove that $S_y$ has a fixed point.

Solution

Dense subspace

Let $\left(X,||\cdot||\right)$ be an $\mathbb{R}$ normed space and $Y$ be a subspace of $\left(X,+,\cdot\right)$ such that , if $f \in X^*= \mathbb{B} (X, \mathbb{R})$ with $f|_{Y}=\mathbb{O}$ then $f=\mathbb{O}$. Prove that $Y$ is dense on $\left(X,||\cdot||\right)$.

Solution