Math Problems Vault

Problem 5.67

Prove that if $X$ is a reflexive space, then for any functional $f \in X^{*}$ there exists an element $x \in X$ such that $\left\| x\right\| = 1$ and $f(x) = \left\| f\right\|$ (necessity in James's theorem).

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Problem 5.68

Prove that the canonical embedding $\pi : X \rightarrow X^{* *}$ is an isometry.

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Problem 5.69

Prove the non-reflexivity of the spaces:

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(a)

$C[a, b]$ ;

(b)

$C^{n}[0,1], n = 1,2, \ldots$ ;

(c)

$l_{1}$ ;

(d)

$c_{0}$ ;

(e)

$c$ ;

(f)

$L_{1}[a, b]$ .

Problem 5.70

Prove that if the space $X$ is reflexive, then $X^{*}$ is also reflexive.

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Problem 5.71

(a)(b)

Prove the non-reflexivity of the spaces:

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(a)

$l_{\infty }$ ;

(b)

$L_{\infty }[0,1]$ .

Problem 5.72

Prove that any finite-dimensional normed space and any Hilbert space is reflexive.

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Problem 5.73

(a)(b)

Prove the reflexivity of the spaces:

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(a)

$l_{p}, p \in (1, \infty )$ ;

(b)

$L_{p}[0,1], p \in (1, \infty )$ .

Problem 5.74

Prove that if $X$ is a Banach space and $X^{*}$ is reflexive, then $X$ is also reflexive.

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Problem 5.75

Prove that there does not exist a Banach space $X$ such that $X^{*} = c_{0}$ .

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Problem 5.76

Prove that a Banach space $X$ is reflexive if and only if every closed subspace $Y \subset X$ of codimension 1 is a subspace of existence (compare with problem 14.127).

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Problem 5.77

Give an example of a Banach space $X$ and a closed subspace $Y \subset X$ that is not a subspace of existence (compare with problem 5.22).

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Problem 5.78

(a)(b)(c)(d)

Let $M$ be an arbitrary set in a normed space $X$ .

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(a)

Prove that $M^{\perp }$ is a closed subspace of $X^{*}$ .

(b)

Prove that the set $\left\{ x \in X: f(x) = 0 \text{ for every } f \in M^{\perp }\right\}$ coincides with $\overline{\operatorname {Lin}(M)}$ .

(c)

Prove that if $M$ is a closed linear subspace in a reflexive space $X$ , then $\left(M^{\perp }\right)^{\perp } = M$ .

(d)

Prove that if $M$ is a linear subspace in $X$ , then $\operatorname {dim}\left(M^{\perp }\right) = \operatorname {codim} M$ .

Problem 5.79

Find the set $M^{\perp }$ if:

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(a)

$M = B(x, r)$ in the normed space $X$ ;

(b)

$M = \left\{ x \in C[-1,1]: x(t) = 0\text{ for every }t < 0\right\}$ in the space С $[-1,1]$ ;

(c)

$M = \left\{ x \in C[0,1]: x(0) = 0\right\}$ in the space $C[0,1]$ ;

(d)

$M = \left\{ x \in L_{1}[0,1]: x \text{ is continuous at the point } a \text{ and } x(a) = 0\right\}$ in the space $L_{1}[0,1]$ ;

(e)

$M = \left\{ x \in L_{1}[-1,1]: x(t) = x(-t) \text{ for every } t \in [-1,1]\right\}$ in the space $L_{1}[-1,1]$ ;

(f)

the set $M$ is the set of all polynomials in the space $C[0,1]$ ;

(g)

the set $M$ in the space $C[0,1]$ is the set of all polynomials $p$ such that $p(0) = 0$ ;

(h)

$M = \left\{ x \in C[0,1]: x(t) \geqslant 0\text{ for every }t \in [0,1]\right\}$ in the space $C[0,1]$ over the field $\mathbb {R}$ ;

(i)

$M = \left\{ x \in C[0,1]: x(a) \geqslant x(b)\text{ for every }a > b\right\}$ in the space $C[0,1]$ over the field $\mathbb {R}$ ;

(j)

$M = \left\{ x \in L_{3}[-1,1]: \int_{0}^{1} x(t) d t = 0\right\}$ in the space $L_{3}[0,1]$ ;

(k)

$M = c_{00}$ in the space $c_{0}$ ;

(l)

$M = c_{0}$ in the space $c$ ;

(m)

$M = \left\{ x \in l_{1}: x_{1} > x_{2} > x_{3} > \ldots \right\}$ in the space $l_{1}$ over the field $\mathbb {R}$ ;

(n)

$M = \left\{ x = \left(\lambda , \lambda^{2}, \lambda^{3}, \ldots \right): \lambda \in \left\{ 1, \frac{1}{2}, \frac{1}{3}, \ldots \right\} \right\}$ in the space с.

Problem 5.80

Let $Y$ be a linear subspace of a normed space $X$ . Prove that for any element $x \in X$ the equality $\operatorname {dist}(x, Y) = \max \left\{ f(x): f \in Y^{\perp },\left\| f\right\| = 1\right\}$ holds.

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Problem 5.81

Give an example of a normed space $X$ , a closed subspace $Y$ of $X^{*}$ , and a bounded linear functional $f_{0} \notin Y$ such that for every vector $x \in X$ , from the equalities $f(x) = 0$ for every $f \in Y$ it follows that $f_{0}(x) = 0$ (the closed subspace $Y$ and the functional $f_{0}$ cannot be separated by means of a vector). Prove that in a reflexive space there necessarily exists a vector $x_{0} \in X$ such that $f\left(x_{0}\right) = 0$ for every $f \in Y$ , but $f_{0}\left(x_{0}\right) \neq 0$ (compare with problem 5.37).

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