Math Problems Vault

Problem 18.42

Let $X$ be a normed linear space and $W$ be a subspace of $X^{*}$ . Show that the $W$ -weak topology on $X$ is Hausdorff if and only if $W$ separates points in $X$ .

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Problem 18.43

Let $X$ be a normed linear space and $\psi : X \rightarrow \mathbb {R}$ be linear. Show that $\psi$ is continuous with respect to the weak topology if and only if it is continuous with respect to the strong topology.

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Problem 18.44

Let $X$ be a locally convex topological vector space and $\psi : X \rightarrow \mathbb {R}$ be linear. Show that $\psi$ is continuous if and only if it is continuous at the origin.

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Problem 18.45

Let $X$ be a locally convex topological vector space and $\psi : X \rightarrow \mathbb {R}$ be linear. Show that $\psi$ is continuous if and only if there is a neighborhood $\mathcal{O}$ of the origin for which $f(\mathcal{O}) \neq \mathbb {R}$ .

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Problem 18.46

Let $X$ be a normed linear space and $W$ a subspace of $X^{*}$ that separates points. For any topological space $Z$ , show that a mapping $f: Z \rightarrow X$ is continuous, where $X$ has the $W$ -weak topology, if and only if $\psi \circ f: Z \rightarrow \mathbb {R}$ is continuous for all $\psi \in W$ .

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Problem 18.47

Show that the topology on a finite dimensional locally convex topological vector space is induced by a norm.

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Problem 18.48

Let $X$ be a locally convex topological space. Show that the linear space $X^{\prime }$ of all linear continuous functionals $\psi : X \rightarrow \mathbb {R}$ also has a topology with respect to which it is a locally convex topological space on which, for each $x \in X$ , the linear functional $\psi \mapsto \psi (x)$ is continuous.

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Problem 18.49

Let $X$ and $Y$ be locally convex topological vector spaces and $T: X \rightarrow Y$ be linear, one-toone, and onto. Show that $T$ is a topological homeomorphism if and only if it maps a base at the origin for the topology on $X$ to a base at the origin for the topology on $Y$ .

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Problem 18.50

Let $X$ be a linear space and the function $\sigma : X \rightarrow [0, \infty )$ have the following properties: for all $u, v \in X$ ,

$\sigma (u+v) \leq \sigma (u)+\sigma (v)$ ;
$\sigma (u) = 0$ if and only if $u = 0$ ;
$\sigma (u) = \sigma (-u)$ . Define $\rho (u, v) = \sigma (u-v)$ . Show that $\rho$ is a metric on $X$ .

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Problem 18.51

Let $X$ be the linear space of all Lebesgue measurable functions on $[0,1]$ . Define

\sigma (f) = \int _{[0,1]} \frac{\left|f\right|}{1+\left|f\right|} d m \text{ for all } f \in X .

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(i)

After identifing functions that are equal almost everywhere, use Problem 50 to show that $\rho (u, v) = \sigma (u-v)$ defines a metric on $X$ .

(ii)

Show that $\left\{ f_{n}\right\} \rightarrow f$ with respect to the metric $\rho$ if and only if $\left\{ f_{n}\right\} \rightarrow f$ in measure.

(iii)

Show that $(X, \rho )$ is a complete metric space.

(iv)

Show that the mapping $(f, g) \mapsto f+g$ is a continuous mapping of $X \times X$ into $X$ .

(v)

Show that the mapping $(\lambda , f) \mapsto \lambda \cdot f$ is a continuous mapping of $\mathbb {R} \times X$ into $X$ .

(vi)

Show that there are no non-zero continuous linear functionals $\psi$ on $X$ .

Problem 18.52

For $0 < p < 1$ , let $X$ be the linear space of all measurable (with respect to the Lebesgue measure $m$ ) real-valued functions on $[0,1]$ for which $\left|f\right|^{p}$ is integrable. Define

\sigma (f) = \int _{[0,1]}\left|f\right|^{p} d m \text{ for all } f \in X

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(i)

Use Problem 50 to show that $\rho (u, v) = \sigma (u-v)$ defines a metric on $X$ .

(ii)

Show that the linear space $X$ , with the topology determined by $\rho$ , is a topological vector space.

(iii)

For a non-negative function $f$ in $X$ and natural number $n$ , show that there is a partition $0 = x_{0} < x_{1} < \ldots x_{n} = 1$ of $[0,1]$ for which $\int_{x_{k-1}}^{x_{k}} f^{p} d m = 1 / n \cdot \int_{0}^{1} f^{p} d m$ , for all $1 \leq k \leq n$ .

(iv)

For a non-negative function $f$ in $X$ and natural number $n$ , show that there are functions $f_{1}, \ldots , f_{n}$ for which $\rho \left(f_{k}, 0\right) < 1 / n^{1-p} \int_{0}^{1} f^{p} d m$ for $1 \leq k \leq n$ and

f = \sum _{k = 1}^{n} 1 / n \cdot f_{k} .

(v)

Show that there are no continuous non-zero linear functionals on $X$ .

Problem 18.53

Let $\mathcal{S}$ be the space of all sequences of real numbers, and define

\sigma (x) = \sum \frac{\left|x_{n}\right|}{2^{n}\left[1+\left|x_{n}\right|\right]} \text{ for all } x = \left\{ x_{n}\right\} \in \mathcal{S} .

Show that $\sigma$ induces a metric on $\mathcal{S}$ for which $\mathcal{S}$ is a topological vector space with the metric topology. What is the most general continuous linear functional on $\mathcal{S}$ ?

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