Linear Operators

Let $X$ and $Y$ be normed linear spaces and $T: X \rightarrow Y$ be linear.

For $X$ and $Y$ normed linear spaces and $T \in \mathcal{L}(X, Y)$, show that $\left\| T\right\|$ is the smallest Lipschitz constant for the mapping $T$, that is, the smallest number $c \geq 0$ for which

For $X$ and $Y$ normed linear spaces and $T \in \mathcal{L}(X, Y)$, show that

For $X$ and $Y$ normed linear spaces, let $\left\{ T_{n}\right\} \rightarrow T$ in $\mathcal{L}(X, Y)$ and $\left\{ u_{n}\right\} \rightarrow u$ in $X$. Show that $\left\{ T_{n}\left(u_{n}\right)\right\} \rightarrow T(u)$ in $Y$.

Let $X$ be a Banach space and $T \in \mathcal{L}(X, X)$ have $\left\| T\right\| < 1$.

Let $X$ be a Banach space and $T \in \mathcal{L}(X, X)$ have $\left\| T\right\| < 1$. Define $T^{0} = \mathrm{Id}$.

For $X$ and $Y$ normed linear spaces and $T \in \mathcal{L}(X, Y)$, show that $T$ is an isomorphism if and only if there is an operator $S \in \mathcal{L}(Y, X)$ such that for each $u \in X$ and $v \in Y$,

For $X$ and $Y$ normed linear spaces and $T \in \mathcal{L}(X, Y)$, show that $\operatorname {ker} T$ is a closed subspace of $X$ and that $T$ is one-to-one if and only if $\operatorname {ker} T = \left\{ 0\right\}$.

Let ($X, \rho$) be a metric space containing the point $x_{0}$. Define $\operatorname {Lip}_{0}(X)$ to be the set of realvalued Lipschitz functions $f$ on $X$ that vanish at $x_{0}$. Show that $\operatorname {Lip}_{0}(X)$ is a linear space that is normed by defining, for $f \in \operatorname {Lip}_{0}(X)$,

Show that $\operatorname {Lip}_{0}(X)$ is a Banach space. For each $x \in X$, define the linear functional $F_{x}$ on $\operatorname {Lip}_{0}(X)$ by setting $F_{x}(f) = f(x)$. Show that $F_{x}$ belongs to $\mathcal{L}\left(\operatorname {Lip}_{0}(X), \mathbb {R}\right)$ and that for $x, y \in X,\left\| F_{x}-F_{y}\right\| = \rho (x, y)$. Thus, $X$ is isometric to a subset of the Banach space $\mathcal{L}\left(\operatorname {Lip}_{0}(X), \mathbb {R}\right)$. Since any closed subset of a complete metric space is complete, this provides another proof of the existence of a completion for any metric space $X$. It also shows that any metric space is isometric to a subset of a normed linear space.

Use the preceding problem to show that every normed linear space is a dense subspace of a Banach space.

For $X$ a normed linear space and $T, S \in \mathcal{L}(X, X)$, show that the composition $S \circ T$ also belongs to $\mathcal{L}(X, X)$ and $\left\| S \circ T\right\| \leq \left\| S\right\| \cdot \left\| T\right\|$.

Let $X$ be a normed linear space and $Y$ a closed linear subspace of $X$. Show that $\left\| x\right\|_{1} = \inf_{y \in Y}\left\| x-y\right\|$ defines a pseudonorm on $X$. The normed linear space induced by the pseudonorm $\left\| \cdot \right\|_{1}$ (see Problem 8) is denoted by $X / Y$ and called the quotient space of $X$ modulo $Y$. Show that the natural map $\varphi$ of $X$ onto $X / Y$ takes open sets into open sets.

Show that if $X$ is a Banach space and $Y$ a closed linear proper subspace of $X$, then the quotient $X / Y$ also is a Banach space and the natural map $\varphi : X \rightarrow X / Y$ has norm 1.

Let $X$ and $Y$ be normed linear spaces, $T \in \mathcal{L}(X, Y)$ and ker $T = Z$. Show that there is a unique bounded linear operator $S$ from $X / Z$ into $Y$ such that $T = S \circ \varphi$ where $\varphi : X \rightarrow X / Z$ is the natural map. Moreover, show that $\left\| T\right\| = \left\| S\right\|$.