Continuous Linear Operators Between Banach Spaces

Show that a non-empty subset $S$ of a linear space $X$ is a subspace if and only if $S+S = S$ and $\lambda \cdot S = S$ for each $\lambda \in \mathbb {R}, \lambda \neq 0$.

If $Y$ and $Z$ are subspaces of the linear space $X$, show that $Y+Z$ also is a subspace and $Y+Z = \operatorname {span}[Y \cup Z]$.

Let $S$ be a subset of a normed linear space $X$.

For a normed linear space $X$, show that the function $\left\| \cdot \right\| : X \rightarrow \mathbb {R}$ is continuous.

For two normed linear spaces ($X,\left\| \cdot \right\|_{1}$) and ($Y,\left\| \cdot \right\|_{2}$), define a linear structure on the Cartesian product $X \times Y$ by $\lambda \cdot (x, y) = (\lambda x, \lambda y)$ and $\left(x_{1}, y_{1}\right)+\left(x_{2}, y_{2}\right) = \left(x_{1}+x_{2}, y_{1}+y_{2}\right)$. Define the product norm $\left\| \cdot \right\|$ by $\left\| (x, y)\right\| = \left\| x\right\|_{1}+\left\| y\right\|_{2}$, for $x \in X$ and $y \in Y$. Show that this is a norm with respect to which a sequence converges if and only if each of the two component sequences converges. Furthermore, show that if $X$ and $Y$ are Banach spaces, then so is $X \times Y$.

Let $X$ be a normed linear space.

Show that the set $\mathcal{P}$ of all polynomials on $[a, b]$ is a linear space. For $\mathcal{P}$ considered as a subset of the normed linear space $C[a, b]$, show that $\mathcal{P}$ fails to be closed. For $\mathcal{P}$ considered as a subset of the normed linear space $L^{1}[a, b]$, show that $\mathcal{P}$ fails to be closed.

A non-negative real-valued function $\left\| \cdot \right\|$ defined on a vector space $X$ is called a pseudonorm if $\left\| x+y\right\| \leq \left\| x\right\| +\left\| y\right\|$ and $\left\| \alpha x\right\| = \left|\alpha \right|\left\| x\right\|$. Define $x \cong y$, provided that $\left\| x-y\right\| = 0$. Show that this is an equivalence relation. Define $X / \cong$ to be the set of equivalence classes of $X$ under $\cong$ and for $x \in X$ define $[x]$ to be the equivalence class of $x$. Show that $X / \cong$ is a normed vector space if we define $\alpha [x]+\beta [y]$ to be the equivalence class of $\alpha x+\beta y$ and define $\left\| [x]\right\| = \left\| x\right\|$. Illustrate this procedure with $X = L^{p}[a, b], 1 \leq p < \infty$.

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 $\left\| T\right\| = \left\| S\right\|$.

Show that a subset of a finite dimensional normed linear space $X$ is compact if and only if it is closed and bounded.

Complete the proof of Riesz's Lemma for $\epsilon \neq 1 / 2$.

Exhibit an open cover of the closed unit ball of $X = \ell^{2}$ that has no finite subcover. Then do the same for $X = C[0,1]$ and $X = L^{2}[0,1]$.

For normed linear spaces $X$ and $Y$, let $T: X \rightarrow Y$ be linear. If $X$ is finite dimensional, show that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\operatorname {ker} T$ is closed.

Let $X$ be an infinite dimensional normed linear space, $B$ the closed unit ball in $X$, and $B_{0}$ the unit open ball in $X$. Suppose $B$ is compact. Then the open cover $\left\{ x+(1 / 3) B_{0}\right\}_{x \in B}$ of $B$ has a finite subcover $\left\{ x_{i}+(1 / 3) B_{0}\right\}_{1 \leq i \leq n}$. Use Riesz's Lemma with $Y = \operatorname {span}\left[\left\{ x_{1}, \ldots , x_{n}\right\} \right]$ to derive a contradiction.

Let $X$ be a normed linear space. Show that $X$ is separable if and only if there is a compact subset $K$ of $X$ for which $\overline{\operatorname {span}}[K] = X$.

Let $X$ be a finite dimensional normed linear space and $Y$ a normed linear space. Show that every linear operator $T: X \rightarrow Y$ is continuous and open.

Let $X$ be a Banach space and $P \in \mathcal{L}(X, X)$ be a projection. Show that $P$ is open.

Let $T: X \rightarrow Y$ be a continuous linear operator between the Banach spaces $X$ and $Y$. Show that $T$ is open if the image under $T$ of the open unit ball in $X$ is dense in a neighborhood of the origin in $Y$.

Let $\left\{ u_{n}\right\}$ be a sequence in a Banach space $X$. Suppose that $\sum_{k = 1}^{\infty }\left\| u_{k}\right\| < \infty$. Show that there is an $x \in X$ for which

Let $T$ be a linear operator from a normed linear space $X$ to a finite-dimensional normed linear space $Y$. Show that $T$ is continuous if and only if $\operatorname {ker} T$ is a closed subspace of $X$.

Let $X$ be a Banach space, the operator $T \in \mathcal{L}(X, X)$ be open and $X_{0}$ be a closed subspace of $X$. The restriction $T_{0}$ of $T$ to $X_{0}$ is continuous. Is $T_{0}$ necessarily open?

Let $V$ be a linear subspace of a linear space $X$. Argue as follows to show that $V$ has a linear complement in $X$.

Verify (15) and (16).

Let $Y$ be a normed linear space. Show that $Y$ is a Banach space if and only if there is a Banach space $X$ and a continuous, linear, open mapping of $X$ onto $Y$.

As a consequence of the Baire Category Theorem, it was proven that a real-valued mapping that is the pointwise limit of a sequence of continuous mappings on a complete metric space must be continuous at all points of a dense subset of its domain. Adapt that proof so that it applies to mappings into any metric space. Use this to prove that the pointwise limit of a sequence of continuous linear operators on a Banach space has a limit that is continuous at some point and hence, by linearity, is continuous.

Let $\left\{ f_{n}\right\}$ be a sequence in $L^{\infty }[a, b]$. Suppose that for each $g \in L^{1}[a, b], \lim_{n \rightarrow \infty } \int_{a}^{b} g \cdot f_{n} \mathrm{dm}$ exists. Show that there is a function $f \in L^{\infty }[a, b]$ such that $\lim_{n \rightarrow \infty } \int_{a}^{b} g \cdot f_{n} \mathrm{dm} = \int_{a}^{b} g \cdot f \mathrm{dm}$ for all $g \in L^{1}[a, b]$.

Let $X$ be the linear space of all polynomials defined on $\mathbb {R}$. For $p \in X$, define $\left\| p\right\|$ to be the sum of the absolute values of the coefficients of $p$. Show that this is a norm on $X$. For each $n$, define $\psi_{n}: X \rightarrow \mathbb {R}$ by $\psi_{n}(p) = p^{(n)}(0)$. Use the properties of the sequence $\left\{ \psi_{n}\right\}$ in $\mathcal{L}(X, \mathbb {R})$ to show that $X$ is not a Banach space.