Linear Operators on Normed Spaces

Show that for $\mathcal{B}(\mathcal{X}, \mathbb {F}) \neq (0)$, $\mathcal{B}(\mathcal{X}, \mathcal{Y})$ is a Banach space if and only if $\mathcal{Y}$ is a Banach space.

Let $\mathcal{X}$ be a normed space, let $\mathcal{Y}$ be a Banach space, and let $\hat{\mathcal{X}}$ be the completion of $\mathcal{X}$. Show that if $\rho : \mathcal{B}(\hat{\mathcal{X}}, \mathcal{Y}) \to \mathcal{B}(\mathcal{X}, \mathcal{Y})$ is defined by $\rho (A) = A|\mathcal{X}$, then $\rho$ is an isometric isomorphism.

If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, $\phi : X \to \mathbb {F}$ is an $\Omega$-measurable function, $1 \leq p \leq \infty$, and $\phi f \in L^p(\mu )$ whenever $f \in L^p(\mu )$, then show that $\phi \in L^\infty (\mu )$.

Verify the statements in Example 2.2: if $(X, \Omega , \mu )$ is a $\sigma$-finite measure space and $\phi \in L^\infty (X, \Omega , \mu )$, define $M_\phi : L^p(X, \Omega , \mu ) \to L^p(X, \Omega , \mu )$, $1 \leq p \leq \infty$, by $M_\phi f = \phi f$ for all $f$ in $L^p(X, \Omega , \mu )$. Then $M_\phi \in \mathcal{B}(L^p(X, \Omega , \mu ))$ and $\left\| M_\phi \right\| = \left\| \phi \right\|_{\infty }$.

Verify the statements in Example 2.3: if $(X, \Omega , \mu )$, $k$, $c_1$, and $c_2$ are as in Example II.1.6 and $1 \leq p \leq \infty$, then $K : L^p(\mu ) \to L^p(\mu )$, defined by

for all $f$ in $L^p(\mu )$ and $x$ in $X$, is a bounded operator on $L^p(\mu )$ and $\left\| K\right\| \leq c_1^{1/q} c_2^{1/p}$, where $1/p + 1/q = 1$.

Verify the statements in Example 2.4: if $X$ and $Y$ are compact spaces and $\tau : Y \to X$ is a continuous map, define $A : C(X) \to C(Y)$ by $(Af)(y) = f(\tau (y))$. Then $A \in \mathcal{B}(C(X), C(Y))$ and $\left\| A\right\| = 1$.

Let $A$ and $\tau$ be as in Example 2.4: $X$ and $Y$ are compact spaces, $\tau : Y \to X$ is a continuous map, and $A : C(X) \to C(Y)$ is defined by $(Af)(y) = f(\tau (y))$.