Banach Spaces

Complete the proof of Proposition 1.3: if $\mathcal{X}$ is a normed space, then

Complete the proof of Proposition 1.5: if $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ are two norms on $\mathcal{X}$, then these norms are equivalent if and only if there are positive constants $c$ and $C$ such that

for all $x$ in $\mathcal{X}$. (The implication ``constants exist $\Rightarrow$ equivalent'' has been proved in the text; only the converse implication remains.)

For $1 \leq p < \infty$ and $x = (x_1, \ldots , x_d)$ in $\mathbb {F}^d$, define $\left\| x\right\|_{p} \equiv \left[\sum_{j=1}^{d} \left|x_j\right|^p\right]^{1/p}$; define $\left\| x\right\|_{\infty } \equiv \sup \left\{ \left|x_j\right| : 1 \leq j \leq d\right\}$. Show that all of these norms are equivalent. For $1 \leq p, q \leq \infty$, what are the best constants $c$ and $C$ such that $c \left\| x\right\|_{p} \leq \left\| x\right\|_{q} \leq C \left\| x\right\|_{p}$ for all $x$ in $\mathbb {F}^d$?

If $1 \leq p \leq \infty$ and $\left\| \cdot \right\|_{p}$ is defined on $\mathbb {R}^2$ as in Exercise 3, graph $\left\{ x \in \mathbb {R}^2 : \left\| x\right\|_{p} = 1\right\}$. Note that if $1 < p < \infty$, $\left\| x\right\|_{p} = \left\| y\right\|_{p} = 1$, and $x \neq y$, then for $0 < t < 1$, $\left\| tx + (1-t)y\right\|_{p} < 1$. The same cannot be said for $p = 1, \infty$.

Let $c =$ the set of all sequences $\left\{ \alpha_n\right\}_1^\infty$, $\alpha_n$ in $\mathbb {F}$, such that $\lim \alpha_n$ exists. Show that $c$ is a closed subspace of $l^\infty$ and hence is a Banach space.

Let $X = \left\{ n^{-1} : n \geq 1\right\} \cup \left\{ 0\right\}$. Show that $C(X)$ and the space of $c$ of Exercise 5 are isometrically isomorphic.

If $l^\infty (I)$ and $l^\infty (J)$ are isometrically isomorphic, do $I$ and $J$ have the same cardinality?

Show that $l^\infty$ is not separable.

Complete the proof of Proposition 1.7: if $X$ is a locally compact space and $C_0(X) =$ all continuous functions $f : X \to \mathbb {F}$ such that for all $\varepsilon > 0$, $\left\{ x \in X : \left|f(x)\right| \geq \varepsilon \right\}$ is compact, then $C_0(X)$ is a closed subspace of $C_b(X)$ and hence is a Banach space. (It has been shown in the text that $C_0(X)$ is closed in $C_b(X)$; only the fact that $C_0(X)$ is a linear manifold in $C_b(X)$ remains.)

Verify the statements in Example 1.10: let $n \geq 1$ and let $C^{(n)}[0,1] =$ the collection of functions $f : [0,1] \to \mathbb {F}$ such that $f$ has $n$ continuous derivatives. Define $\left\| f\right\| = \sup_{0 \leq k \leq n} \left\{ \sup \left\{ \left|f^{(k)}(x)\right| : 0 \leq x \leq 1\right\} \right\}$. Then $C^{(n)}[0,1]$ is a Banach space.

Verify the statements in Example 1.11: let $1 \leq p < \infty$ and $n \geq 1$ and let $W_p^n[0,1] =$ the functions $f : [0,1] \to \mathbb {F}$ such that $f$ has $n - 1$ continuous derivatives, $f^{(n-1)}$ is absolutely continuous, and $f^{(n)} \in L^p[0,1]$. For $f$ in $W_p^n[0,1]$, define

Then $W_p^n[0,1]$ is a Banach space.

Let $X$ be locally compact and let $X_\infty = X \cup \left\{ \infty \right\}$ be the one-point compactification of $X$. Show that $C_0(X)$ and $\left\{ f \in C(X_\infty ) : f(\infty ) = 0\right\}$, with the norm it inherits as a subspace of $C(X_\infty )$, are isometrically isomorphic Banach spaces.

Let $X$ be locally compact and define $C_c(X)$ to be the continuous functions $f : X \to \mathbb {F}$ such that $\mathrm{spt}\, f \equiv \mathrm{cl}\left\{ x \in X : f(x) \neq 0\right\}$ is compact ($\mathrm{spt}\, f$ is the support of $f$). Show that $C_c(X)$ is dense in $C_0(X)$.

If $W_p^n[0,1]$ is defined as in Example 1.11 and $f \in W_p^n[0,1]$, let $\vertiii {f} \equiv \left[\int \left|f(x)\right|^p \, dx\right]^{1/p} + \left[\int \left|f^{(n)}(x)\right|^p \, dx\right]^{1/p}$. Show that $\vertiii {\cdot }$ is equivalent to the norm defined on $W_p^n[0,1]$.

Let $\mathcal{X}$ be a normed space and let $\hat{\mathcal{X}}$ be its completion as a metric space. Show that $\hat{\mathcal{X}}$ is a Banach space.

Show that the norm on $C([0,1]) = C_b([0,1])$ does not come from an inner product by showing that it does not satisfy the parallelogram law.

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 $(Af)(y) = f(\tau (y))$.

Show that if $\mathcal{X}$ is a locally compact normed space, then $\mathcal{X}$ is finite dimensional. (This same result, due to F. Riesz, is valid in the more general topological vector spaces -- see IV.1.1 for the definition. For a nice proof of this look at Pitcairn [1966].)

Let $\mathcal{X}$ be a finite dimensional vector space over $\mathbb {F}$ with Hamel basis $\left\{ e_1, \ldots , e_d\right\}$, and for $x = \sum_{j=1}^{d} x_j e_j$ define $\left\| x\right\|_{\infty } \equiv \max \left\{ \left|x_j\right| : 1 \leq j \leq d\right\}$, as in the proof of Theorem 3.1. Show that $\left\| \cdot \right\|_{\infty }$ is a norm.

Show that if $\mathcal{M} \leq \mathcal{X}$, then (4.1) defines a norm on $\mathcal{X}/\mathcal{M}$, where (4.1) is $\left\| x + \mathcal{M}\right\| = \inf \left\{ \left\| x+y\right\| : y \in \mathcal{M}\right\}$.

Prove that $\mathcal{X}$ is a Banach space if and only if whenever $\left\{ x_n\right\}$ is a sequence in $\mathcal{X}$ such that $\sum \left\| x_n\right\| < \infty$, then $\sum_{n=1}^{\infty } x_n$ converges in $\mathcal{X}$.

Show that if $(X, d)$ is a metric space and $\left\{ x_n\right\}$ is a Cauchy sequence such that there is a subsequence $\left\{ x_{n_k}\right\}$ that converges to $x_0$, then $x_n \to x_0$.

Find a Banach space $\mathcal{X}$ and a closed subspace $\mathcal{M}$ such that the natural map $Q : \mathcal{X} \to \mathcal{X}/\mathcal{M}$ is not a closed map. Can the natural map ever be a closed map?

Prove the converse of (4.2b) -- Theorem 4.2 states that if $\mathcal{M} \leq \mathcal{X}$ and $\left\| x + \mathcal{M}\right\|$ is defined as in (4.1), then $\left\| \cdot \right\|$ is a norm on $\mathcal{X}/\mathcal{M}$, and part (b) states that if $\mathcal{X}$ is a Banach space, then so is $\mathcal{X}/\mathcal{M}$: if $\mathcal{X}$ is a normed space, $\mathcal{M} \leq \mathcal{X}$, and both $\mathcal{M}$ and $\mathcal{X}/\mathcal{M}$ are complete, then $\mathcal{X}$ is complete. (This is an example of what is called a ``two-out-of-three'' result. If any two of $\mathcal{X}, \mathcal{M}$, and $\mathcal{X}/\mathcal{M}$ are complete, so is the third.)

Let $\mathcal{M} = \left\{ x \in l^p : x(2n) = 0 \text{ for all } n\right\}$, $1 \leq p \leq \infty$. Show that $l^p/\mathcal{M}$ is isometrically isomorphic to $l^p$.

Let $X$ be a normal locally compact space and $F$ a closed subset of $X$. If $\mathcal{M} \equiv \left\{ f \in C_0(X) : f(x) = 0 \text{ for all } x \text{ in } F\right\}$, then $C_0(X)/\mathcal{M}$ is isometrically isomorphic to $C_0(F)$.

Prove Proposition 4.4: let $\left\{ \mathcal{X}_i : i \in I\right\}$ be a collection of normed spaces and let $\mathcal{X} = \oplus_p \mathcal{X}_i$, $1 \leq p \leq \infty$, where

and $\oplus_\infty \mathcal{X}_i \equiv \left\{ x \in \prod_i \mathcal{X}_i : \left\| x\right\| \equiv \sup_i \left\| x(i)\right\| < \infty \right\}$. Then:

Formulate and prove a version of Proposition 4.4 for $\oplus_0 \mathcal{X}_n$, where, if $\left\{ \mathcal{X}_1, \mathcal{X}_2, \ldots \right\}$ is a sequence of normed spaces,

given the norm it has as a subspace of $\oplus_\infty \mathcal{X}_n$. (Proposition 4.4 states: let $\left\{ \mathcal{X}_i : i \in I\right\}$ be a collection of normed spaces and let $\mathcal{X} = \oplus_p \mathcal{X}_i$, $1 \leq p \leq \infty$. Then (a) $\mathcal{X}$ is a normed space and the projection $P_i : \mathcal{X} \to \mathcal{X}_i$ is a continuous linear map with $\left\| P_i(x)\right\| \leq \left\| x\right\|$ for each $x$ in $\mathcal{X}$; (b) $\mathcal{X}$ is a Banach space if and only if each $\mathcal{X}_i$ is a Banach space; (c) each projection $P_i$ is an open map of $\mathcal{X}$ onto $\mathcal{X}_i$.)

If $\left\{ \mathcal{X}_1, \ldots , \mathcal{X}_n\right\}$ is a finite collection of normed spaces and $1 \leq p \leq \infty$, show that the norms on $\oplus_p \mathcal{X}_k$ are all equivalent.

Here is an abstraction of Proposition 4.4. Suppose $\left\{ \mathcal{X}_i : i \in I\right\}$ is a collection of normed spaces and $Y$ is a normed space contained in $\mathbb {F}^I$. Define $\mathcal{X} \equiv \left\{ x \in \prod_i \mathcal{X}_i : \text{there is a } y \text{ in } Y \text{ with } \left\| x(i)\right\| \leq y(i) \text{ for all } i\right\}$. If $x \in \mathcal{X}$, define $\left\| x\right\| \equiv \inf \left\{ \left\| y\right\| : \left\| x(i)\right\| \leq y(i) \text{ for all } i\right\}$. Then $(\mathcal{X}, \left\| \cdot \right\| )$ is a normed space. Give necessary and sufficient conditions on $Y$ that each of the parts of (4.4) be valid for $\mathcal{X}$.

Let $\mathcal{X}$ be a normed space and $\mathcal{M} \leq \mathcal{X}$.

Let $\left\{ \mathcal{X}_i : i \in I\right\}$ be a collection of non-zero normed spaces. For $1 \leq p < \infty$, put $\mathcal{X} = \oplus_p \mathcal{X}_i$. Show that $\mathcal{X}$ is separable if and only if $I$ is countable and each $\mathcal{X}_i$ is separable. Show that $\oplus_\infty \mathcal{X}_i$ is separable if and only if $I$ is finite and each $\mathcal{X}_i$ is separable.

Show that $\oplus_0 \mathcal{X}_n$ is separable if and only if each $\mathcal{X}_n$ is separable.

Let $J \subseteq I$, and $\mathcal{X} \equiv \oplus_p \left\{ \mathcal{X}_i : i \in I\right\}$, $\mathcal{M} \equiv \left\{ x \in \mathcal{X} : x(j) = 0 \text{ for } j \text{ in } J\right\}$. Show that $\mathcal{X}/\mathcal{M}$ is isometrically isomorphic to $\oplus_p \left\{ \mathcal{X}_j : j \in J\right\}$.

Let $\mathcal{H}$ be a Hilbert space and suppose $\mathcal{M} \leq \mathcal{H}$. Show that if $Q : \mathcal{H} \to \mathcal{H}/\mathcal{M}$ is the natural map, then $Q : \mathcal{M}^\perp \to \mathcal{H}/\mathcal{M}$ is an isometric isomorphism.

Complete the proof of Proposition 5.4: if $\mathcal{X}$ is a normed space, $\mathcal{X}^*$ is a Banach space. (It has been shown in the text that $\mathcal{X}^*$ is complete; only the fact that $\mathcal{X}^*$ is a normed space remains.)

Show that $\mathcal{X}^*$ is a normed space.

Give an example of a measure space $(X, \Omega , \mu )$ that is not $\sigma$-finite for which the conclusion of Theorem 5.6 is false, where Theorem 5.6 states: if $(X, \Omega , \mu )$ is a $\sigma$-finite measure space and $g \in L^\infty (X, \Omega , \mu )$, define $F_g : L^1(\mu ) \to \mathbb {F}$ by $F_g(f) = \int fg \, d\mu$. Then $F_g \in L^1(\mu )^*$ and the map $g \mapsto F_g$ defines an isometric isomorphism of $L^\infty (\mu )$ onto $L^1(\mu )^*$.

Let $\left\{ \mathcal{X}_i : i \in I\right\}$ be a collection of normed spaces. If $1 \leq p < \infty$, show that the dual space of $\oplus_p \mathcal{X}_i$ is isometrically isomorphic to $\oplus_q \mathcal{X}_i^*$, where $1/p + 1/q = 1$.

If $\mathcal{X}_1, \mathcal{X}_2, \ldots$ are normed spaces, show that $(\oplus_0 \mathcal{X}_n)^*$ is isometrically isomorphic to $\oplus_1 \mathcal{X}_n^*$.

Let $n \geq 1$ and let $C^{(n)}[0,1]$ be defined as in Example 1.10 (the collection of functions $f : [0,1] \to \mathbb {F}$ having $n$ continuous derivatives, with norm $\left\| f\right\| = \sup_{0 \leq k \leq n}\left\{ \sup \left\{ \left|f^{(k)}(x)\right| : 0 \leq x \leq 1\right\} \right\}$). Show that $\left\| f\right\| = \sum_{k=0}^{n-1} \left|f^{(k)}(0)\right| + \sup \left\{ \left|f^{(n)}(x)\right| : 0 \leq x \leq 1\right\}$ is an equivalent norm on $C^{(n)}[0,1]$. Show that $L \in (C^{(n)}[0,1])^*$ if and only if there are scalars $\alpha_0, \alpha_1, \ldots , \alpha_{n-1}$ and a measure $\mu$ on $[0,1]$ such that $L(f) = \sum_{k=0}^{n-1} \alpha_k f^{(k)}(0) + \int f^{(n)} \, d\mu$. If $C^{(n)}[0,1]$ is given this new norm, find a formula for $\left\| L\right\|$ in terms of $\left|\alpha_0\right|, \left|\alpha_1\right|, \ldots , \left|\alpha_{n-1}\right|$, and $\left\| \mu \right\|$?

Give $\mathcal{X} = C([0,1])$ the norm $\left\| f\right\| = \int \left|f(t)\right| \, dt$ and define $L : \mathcal{X} \to \mathbb {F}$ by $L(f) = f(\tfrac {1}{2})$. Show directly (without using Theorem 5.6) that $L$ is not bounded. Now prove this as a consequence of (5.6), where Theorem 5.6 states: if $(X, \Omega , \mu )$ is a $\sigma$-finite measure space and $g \in L^\infty (X, \Omega , \mu )$, define $F_g : L^1(\mu ) \to \mathbb {F}$ by $F_g(f) = \int fg \, d\mu$. Then $F_g \in L^1(\mu )^*$ and the map $g \mapsto F_g$ defines an isometric isomorphism of $L^\infty (\mu )$ onto $L^1(\mu )^*$.

Complete the proof of Lemma 6.3.

Give the details of the proof of Corollary 6.5.

Show that $c^*$ is isometrically isomorphic to $l^1$. Are $c$ and $c_0$ isometrically isomorphic?

If $\mu$ is a Borel measure on $[0,1]$ and $\int x^n \, d\mu (x) = 0$ for all $n \geq 0$, show that $\mu = 0$.

If $n \geq 1$, show that there is a measure $\mu$ on $[0,1]$ such that for every polynomial $p$ of degree at most $n$,

If $n \geq 1$, does there exist a measure $\mu$ on $[0,1]$ such that $p'(0) = \int p \, d\mu$ for every polynomial of degree at most $n$?

Does there exist a measure $\mu$ on $[0,1]$ such that $\int p \, d\mu = p'(0)$ for every polynomial $p$?

Let $K$ be a compact subset of $\mathbb {C}$ and define $A(K)$ to be $\left\{ f \in C(K) : f \text{ is analytic on int } K\right\}$. (Functions here are complex valued.) Show that if $a \in K$, then there is a probability measure $\mu$ supported on $\partial K$ such that $f(a) = \int f \, d\mu$ for every $f$ in $A(K)$. (A probability measure is a nonnegative measure $\mu$ such that $\left\| \mu \right\| = 1$.)

If $K = \mathrm{cl}\, \mathbb {D}$ ($\mathbb {D} = \left\{ \left|z\right| < 1\right\}$) and $a \in K$, find the measure $\mu$ whose existence was proved in Exercise 8.

Let $P = \left\{ p|\partial \mathbb {D} : p = \text{an analytic polynomial}\right\}$ and consider $P$ as a manifold in $C(\partial \mathbb {D})$. Show that if $\mu$ is a real-valued measure on $\partial \mathbb {D}$ such that $\int p \, d\mu = 0$ for every $p$ in $P$, then $\mu = 0$. Give an example of a complex-valued measure $\mu$ such that $\mu \neq 0$ but $\int p \, d\mu = 0$ for every $p$ in $P$.

If $L$ is a Banach limit, show that there are $x$ and $y$ in $l^\infty$ such that $L(xy) \neq L(x)L(y)$.

Let $X$ be a set and $\Omega$ a $\sigma$-algebra of subsets of $X$. Suppose $\mu$ is a complex-valued countably additive measure defined on $\Omega$ such that $\left\| \mu \right\| = \mu (X) < \infty$. Show that $\mu (\Delta ) \geq 0$ for every $\Delta$ in $\Omega$. (Though it is difficult to see at this moment, this fact is related to the proof of (c) in Theorem 7.1 for the complex case.)

Show that if $x \in l^\infty$, $\left\| x\right\|_{\infty } \leq 1$, then there is a sequence $\left\{ x_n\right\}$, $x_n$ in $l^\infty$ such that $\left\| x_n\right\|_{\infty } \leq 1$, $\left\| x_n - x\right\|_{\infty } \to 0$, and each $x_n$ takes on only a finite number of values.

Let $\mu$ be a compactly supported measure on $\mathbb {C}$ that is boundedly absolutely continuous with respect to area measure. Show that $\hat{\mu }$ is continuous on $\mathbb {C}_\infty$.

Let $m =$ Lebesgue measure on $[0,1]$. Show that $\hat{m}$ is not continuous at any point of $[0,1]$.

Prove Proposition 9.3:

Prove Proposition 9.4: if $(\mathcal{X}, \leq )$ is an ordered vector space and $P$ is the wedge of positive elements, $\leq$ is antisymmetric if and only if $P \cap (-P) = (0)$.

Show that $e$ is an order unit for $(\mathcal{X}, \leq )$ if and only if for every $x$ in $\mathcal{X}$ there is a $\delta > 0$ such that $e \pm tx \geq 0$ for $0 \leq t \leq \delta$.

Show that $C(\mathbb {R})$, the space of all continuous real-valued functions on $\mathbb {R}$, has no order unit.

Prove (9.11): with $(\mathcal{X}, \leq )$ an ordered vector space, $\mathcal{Y}$ a linear manifold in $\mathcal{X}$ that is cofinal, $f : \mathcal{Y} \to \mathbb {R}$ a positive linear functional, and, for $x$ in $\mathcal{X}$,

the function $q$ is a sublinear functional on $\mathcal{X}$.

Characterize the order units of $C_b(X)$. Does $C_b(X)$ always have an order unit?

Characterize the order units of $C_0(X)$ if $X$ is locally compact. Does $C_0(X)$ always have an order unit?

Let $\mathcal{X} = M_2(\mathbb {R})$, the $2 \times 2$ matrices over $\mathbb {R}$. Define $A$ in $M_2(\mathbb {R})$ to be positive if $A = A^*$ and $\left\langle Ax, x \right\rangle \geq 0$ for all $x$ in $\mathbb {R}^2$. Characterize the order units of $M_2(\mathbb {R})$.

If $1 \leq p < \infty$ and $\mathcal{X} = L^p(0,1)$, define $f \leq g$ to mean that $f(x) \leq g(x)$ a.e. Show that $\mathcal{X}$ is an ordered vector space that has no order unit.

Show that $(\mathcal{X}^*)^{**}$ and $(\mathcal{X}^{**})^*$ are equal.

Show that for a locally compact space $X$, $C_b(X)$ is reflexive if and only if $X$ is finite.

Let $\mathcal{M} \leq \mathcal{X}$ and let $\rho_{\mathcal{X}} : \mathcal{X} \to \mathcal{X}^{**}$ and $\rho_{\mathcal{M}} : \mathcal{M} \to \mathcal{M}^{**}$ be the natural maps. If $i : \mathcal{M} \to \mathcal{X}$ is the inclusion map, show that there is an isometry $\phi : \mathcal{M}^{**} \to \mathcal{X}^{**}$ such that the diagram

commutes. Prove that $\phi (\mathcal{M}^{**}) = (\mathcal{M}^\perp )^\perp \equiv \left\{ x^{**} \in \mathcal{X}^{**} : x^{**}(\mathcal{M}^\perp ) = 0\right\}$.

Use Exercise 3 to show that if $\mathcal{X}$ is reflexive, then any closed subspace of $\mathcal{X}$ is also reflexive. See Yang [1967].

Suppose $\mathcal{X}$ and $\mathcal{Y}$ are Banach spaces. If $A \in \mathcal{B}(\mathcal{X}, \mathcal{Y})$ and $\mathrm{ran}\, A$ is a second category space, show that $\mathrm{ran}\, A$ is closed.

Give both $C^{(1)}[0,1]$ and $C[0,1]$ the supremum norm. If $A : C^{(1)}[0,1] \to C[0,1]$ is defined by $Af = f'$, show that $A$ is not bounded.