Simple Examples of Linear Bounded Operators

Example 1.13: Let $X,Y$ be Banach spaces and assume that $X$ is finite-dimensional. Let $T:X \rightarrow Y$ be a linear operator. Then $T$ is bounded.

Indeed, since all norms in $X$ are equivalent, it suffices to consider the case $X = \mathbb{C}^n$ with the Euclidean norm $\|\cdot\|_2$. Let $e_1,...,e_n$ be the standard basis in $\mathbb{C}^n$. For $x \in \mathbb{C}^n$, we have:

where $C^2 = \|Te_1\|^2_Y + \cdots + \|Te_n\|^2_Y$.

Example 1.14: Let $t_1,t_2,...$ be a bounded sequence of complex numbers. For $1 \leq p \leq \infty$, let $T : \ell^p \rightarrow \ell^p$ be defined by:

Then $T$ is bounded and $\|T\| = \sup_{n\geq 1} |t_n|.$

Example 1.15: Let $\Omega$ be any measurable space, and let $t \in L^{\infty}(\Omega)$. Then:

defines a bounded operator on $L^p(\Omega)$ (for any $p \geq 1$) and $\|T\| = \|t\|_{\infty}$.

Example 1.16: Let $S : \ell^2 \rightarrow \ell^2$ be the shift operator:

Then $S$ is bounded and $\|S\| = 1$. One can also consider the backwards shift operator $S^* : \ell^2 \rightarrow \ell^2$ (the reason for the notation will become clear in the next section),

Then we also have $\|S^*\| = 1$.

Example 1.17: Let $a \in \mathbb{R}$ and let $T_a : L^p(\mathbb{R}) \rightarrow L^p(\mathbb{R})$ be defined by:

Then $T_a$ is bounded and $\|T_a\| = 1$.