Linear continuous functionals

Let $f$ be a nonzero linear functional on a linear space $X$. Prove that $\operatorname {Ker} f$ is a linear subspace of codimension 1 (see Definition 2.22).

Definition 2.22: Let $X$ be a linear space, $X_{1}$ a linear subspace of it, and $X_{2}$ its linear complement, i.e., $X = X_{1} \oplus X_{2}$. Then the dimension of the space $X_{2}$ (finite or infinite) is called the codimension of the subspace $X_{1}$ and is denoted $\operatorname {codim} X_{1}$.

Let $f$ and $g$ be two nonzero linear functionals on a linear space $X$. Prove that if $\operatorname {Ker} f = \operatorname {Ker} g$, then $f = \alpha g$ for some $\alpha \in \mathbb {C}$.

Let $f_{1}, f_{2}, \ldots , f_{n}$ be linear functionals on a linear space $X$. Denote by $X_{0}$ the intersection of the kernels of all these functionals. Prove that if $\operatorname {codim} X_{0} < n$, then there exist complex numbers $\alpha_{1}, \ldots , \alpha_{n},\left|\alpha_{1}\right|+\ldots +\left|\alpha_{n}\right| \neq 0$, such that $\sum_{k = 1}^{n} \alpha_{k} f_{k} = 0$ (in other words, the functionals $\left\{ f_{k}\right\}_{1}^{n}$ are linearly dependent).

Let $f_{1}, f_{2}, \ldots , f_{n}$ be a linearly independent system of linear functionals on a linear space $X$, and let $\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n}$ be complex numbers. Prove that there exists a vector $x \in X$ such that $f_{k}(x) = \lambda_{k}$ for all $1 \leqslant k \leqslant n$.

Prove Theorem 5.1.

Theorem 5.1: The set of all continuous linear functionals on a normed space $X$, equipped with the norm $\left\| f\right\| = \sup_{x \neq 0} \frac{\left|f(x)\right|}{\left\| x\right\| }$, is a Banach space. This Banach space is called the dual space of $X$ and is denoted by $X^{*}$.

Let $f$ be a linear functional on a normed space. Prove that $f$ is continuous if and only if for every sequence $x_{n} \rightarrow 0$ the numerical sequence $\left\{ f\left(x_{n}\right)\right\}_{1}^{\infty }$ is bounded.

Let $f$ be a linear functional on a normed space. Prove that $f$ is continuous if and only if the kernel $\operatorname {Ker} f$ is closed.

Give an example of a (nonlinear) functional $f: X \rightarrow \mathbb {R}$ on some Banach space $X$ that is continuous on $X$, but not bounded on the unit ball of this space.

Prove that in a finite-dimensional normed space every linear functional is continuous.

Let $f$ be a linear functional on a complex normed space $X$. Prove that $\operatorname {Re} f$ and $\operatorname {Im} f$ are real-linear functionals. Prove that if the functional $f$ is continuous, then the functionals $\operatorname {Re} f$ and $\operatorname {Im} f$ are also continuous and $\left\| \operatorname {Re} f\right\| = \left\| \operatorname {Im} f\right\| = \left\| f\right\|$ (the norms of the functionals $\operatorname {Re} f$ and $\operatorname {Im} f$ are defined analogously to equality (5.1)).

Let $\varphi$ be a real-linear functional on a complex linear space $X$. Prove that the unique complex linear functional $f$ with $\operatorname {Re} f = \varphi$ has the form $f(x) = \varphi (x)-i \varphi (i x)$.

Let $f$ be a nonzero linear continuous functional on a normed space $X$. Prove that for any $x \in X$ the equality $\operatorname {dist}(x, \operatorname {Ker} f) = \left|f(x)\right| /\left\| f\right\|$ holds.

Let $f$ be a nonzero linear continuous functional on a normed space $X$. Denote

Prove that $\operatorname {dist}(0, M) = \left|a\right| /\left\| f\right\|$. In finite-dimensional normed spaces any linear functional turns out to be bounded (see problem 5.9). In the infinite-dimensional case this is no longer so.

Let $X$ be the linear subspace of functions $x \in C[0,1]$ differentiable at the point $t_{0} \in (0,1)$. Prove that the subspace $X$ is not closed and is everywhere dense in $C[0,1]$. Prove that the linear functional $f(x) = x^{\prime }\left(t_{0}\right)$ on the normed space $X$ (the norm on $X$ is inherited from $C[0,1]$) is not bounded.

Let $\left\{ x_{\lambda }\right\}_{\lambda \in \Lambda }$ be a Hamel basis in a Banach space. Prove that there exists $n \in \mathbb {N}$ such that the set $\left\{ a_{1} x_{\lambda_{1}}+\ldots \ldots +a_{n} x_{\lambda_{n}}: a_{k} \in \mathbb {R}\right\}$ is everywhere dense in $X$.

Let $\left\{ x_{\lambda }\right\}_{\lambda \in \Lambda }$ be a Hamel basis in an infinite-dimensional normed space. Using this basis construct an example of an everywhere-defined unbounded linear functional.

Prove that in any infinite-dimensional normed space $X$ there exists a non-closed linear subspace of codimension 1.

Prove that if an unbounded functional in a normed space is defined on an everywhere dense set, then its kernel is everywhere dense.

Prove that the indicated functional is linear and continuous on the indicated space $X$, and find its norm:

Prove that the indicated functional is linear and continuous on the indicated space $X$, and find its norm:

Prove that in a finite-dimensional normed space and in a Hilbert space every functional attains its norm on the unit ball. In each of the spaces give examples of functionals that do not attain their norm on the unit ball:

Let $f$ be a nonzero linear continuous functional on a normed space $X$. Prove that if for some $x \notin \operatorname {Ker} f$ there exists an element of best approximation in $\operatorname {Ker} f$ (see Definition 2.16), then $f$ attains its norm on the unit ball. Prove that if $f$ attains its norm on the unit ball, then for any $x \in X$ there exists in $\operatorname {Ker} f$ an element closest to $x$.

Definition 2.16: Let $X$ be a normed space, $Y$ a subspace of it, and $x$ an arbitrary vector of $X$. A vector $y_{0} \in Y$ for which $\operatorname {dist}(x, Y) = \left\| x-y_{0}\right\|$ (if such a vector exists) is called an element of best approximation for the vector $x$ in the subspace $Y$.

Give an example of a Banach space $X$ and a closed subspace $X_{0}$ in it for which there is no $0$-perpendicular (see Definition 2.18).

Definition 2.18: Let $X$ be a normed space, $X_{0}$ a closed subspace of it, $\varepsilon \in [0,1]$. A vector $x \in X$ is called an $\varepsilon$-perpendicular (or almost perpendicular) to $X_{0}$ if $\left\| x\right\| = 1$ and $\operatorname {dist}\left(x, X_{0}\right) \geqslant 1-\varepsilon$.

Let $X$ be a normed space, $Y \subset X$ a subspace of finite codimension, and $f$ a linear continuous functional on $Y$. Prove that any extension of $f$ to a linear functional on $X$ is a continuous functional.

Let the Banach space $X$ be:

Let $X$ be a normed space. Prove that the Hahn-Banach extension of any linear continuous functional on $X$ is unique if and only if the dual space $X^{*}$ is strictly normed (see Definition 2.4).

Definition 2.4: A normed space $X$ is called strictly normed (or strictly convex) if for $\left\| x\right\| = \left\| y\right\| = 1$ the equality $\left\| x+y\right\| = \left\| x\right\| +\left\| y\right\|$ holds if and only if $x = y$ (in other words, the unit sphere of $X$ contains no line segments).

Prove that any linear continuous functional, defined on an everywhere dense linear subspace of a normed space $X$, has a unique Hahn-Banach extension.

Prove that for any nonzero $x \in X$ there exists a functional $f \in X^{*}$ such that $\left\| f\right\| = 1, f(x) = \left\| x\right\|$.

Prove that for any $x, y \in X, x \neq y$, there exists a functional $f \in X^{*}$ such that $f(x) \neq f(y)$.

Let $x_{1}, x_{2}, \ldots , x_{n}$ be a linearly independent system of vectors of a normed space $X$, and let $\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n}$ be complex numbers. Prove that there exists a linear continuous functional $f$ such that $f\left(x_{k}\right) = \lambda_{k}$ for all $1 \leqslant k \leqslant n$ (compare with problem 5.4).

Let $X$ be a real normed space, $\left\{ x_{n}\right\}$ a finite or countable system of vectors in $X$, and $\left\{ c_{n}\right\} \subset \mathbb {R}$. Prove that the existence of a functional $f \in X^{*}$ with the property $f\left(x_{n}\right) = c_{n}$ for every $n \in \mathbb {N}$ is equivalent to the condition that for an arbitrary finite collection $\left\{ \lambda_{n}\right\}_{n = 1}^{N} \subset \mathbb {R}$ we have $\left|\sum_{n = 1}^{N} \lambda_{n} c_{n}\right| \leqslant M\left\| \sum_{n = 1}^{N} \lambda_{n} x_{n}\right\|$ with some single constant $M > 0$.

Prove that there exists a linear continuous functional $f$ on the real space $l_{\infty }$ satisfying the following conditions:

1. $\inf_{n \in \mathbb {N}} x_{n} \leqslant f(x) \leqslant \sup_{n \in \mathbb {N}} x_{n}$;

2. if $\lim_{n \rightarrow \infty } x_{n} = a$ exists, then $f(x) = a$;

3. the functional is invariant under shifts, i.e., $f\left(\left(x_{1}, x_{2}, x_{3}, \ldots \right)\right) = f\left(\left(x_{2}, x_{3}, \ldots \right)\right)$ for every $x = \left(x_{1}, x_{2}, \ldots \right) \in l_{\infty }$.

Find:

Give examples of a smooth and a non-smooth space. Prove that in a smooth space the Hahn-Banach extension of any linear functional from any one-dimensional subspace is unique.

Let a linear functional $f$ be defined on a one-dimensional subspace $\operatorname {Lin}(x) \subset C[0,1]$, where:

Let a functional $f(\lambda x) = \lambda x\left(t_{0}\right)$ be defined on a one-dimensional subspace $\operatorname {Lin}(x) \subset C[0,1]$, generated by some nonzero function $x(t)$, where $t_{0} \in [0,1]$ is a fixed point. Is the Hahn-Banach extension of such a functional unique?

Prove that for a linear subspace $Y$ of a normed space $X$ with $\bar{Y} \neq X$ and an arbitrary point $x \in X$, there exists a functional $f \in X^{*}$ such that $\left\| f\right\| = 1, f(x) = \operatorname {dist}(x, Y)$, and $f(y) = 0$ for every $y \in Y$.

Let $U_{1}$ and $U_{2}$ be two convex nonempty disjoint sets of a real Banach space $X$, one of which has nonempty interior. Prove that there exist a linear continuous functional $f$ and a number $c \in \mathbb {R}$ such that the affine hyperplane $X_{f, c}$ separates $U_{1}$ and $U_{2}$.

Let $P$ be the subspace of the real space $C[0,1]$ consisting of all polynomials. Prove that $P_{+}$ and $P_{-}$, the sets consisting of polynomials with positive (negative) leading coefficient, are convex, disjoint, but not separated by any hyperplane.

Let $B_{1}$ and $B_{2}$ be two convex closed nonempty disjoint sets of a real Banach space $X$, one of which is compact. Prove that there exist a linear continuous functional $f$ and a number $c \in \mathbb {R}$ such that the affine hyperplane $X_{f, c}$ strictly separates $B_{1}$ and $B_{2}$.

Let $X$ be a normed space. Prove that the space $X^{*}$ is a Banach space.

Prove that if the dual space $X^{*}$ of a normed space $X$ is separable, then $X$ is also separable. Give an example showing that the converse statement is false.

Let a normed space $Z$ be the Cartesian product of normed spaces $X$ and $Y$ with $\left\| z = (x, y)\right\|_{Z} = \left\| x\right\|_{X}+\left\| y\right\|_{Y}$. Prove that the dual space $Z^{*}$ coincides with $X^{*} \times Y^{*}$, and find the norm on it.

Let $X$ be a finite-dimensional normed space. Prove that $X^{*}$ is isomorphic to $X$ (and hence also finite-dimensional).

Let the dual space $X^{*}$ of a normed space $X$ be finite-dimensional. Prove that $X$ is isomorphic to $X^{*}$ (and hence also finite-dimensional).

Let $X_{0}$ be an incomplete normed space, and $X$ its completion. Prove that the spaces $X^{*}$ and $X_{0}^{*}$ are isometrically isomorphic.

Let $X_{1}$ and $X_{2}$ be normed spaces, $X_{1} \subset X_{2}$, $\left\| \cdot \right\|_{X_{2}} \prec \left\| \cdot \right\|_{X_{1}}$ on the space $X_{1}$ (see Definition 2.3), and $X_{1}$ dense in $X_{2}$ with respect to the norm $\left\| \cdot \right\|_{X_{2}}$. Prove that then $X_{2}^{*} \subset X_{1}^{*}$ and $\left\| \cdot \right\|_{X_{1}^{*}} \prec \left\| \cdot \right\|_{X_{2}^{*}}$ on the space $X_{2}^{*}$.

Definition 2.3: Let $\left(X,\left\| \cdot \right\|_{1}\right)$ and $\left(X,\left\| \cdot \right\|_{2}\right)$ be two different normed spaces on a linear space $X$. The norm $\left\| \cdot \right\|_{1}$ is called subordinate to the norm $\left\| \cdot \right\|_{2}$ (denoted $\left\| \cdot \right\|_{1} \prec \left\| \cdot \right\|_{2}$) if there is a constant $C > 0$ such that $\left\| x\right\|_{1} \leqslant C\left\| x\right\|_{2}$ for every $x \in X$. If $\left\| \cdot \right\|_{1} \prec \left\| \cdot \right\|_{2}$ and $\left\| \cdot \right\|_{2} \prec \left\| \cdot \right\|_{1}$, such norms are called equivalent: $\left\| \cdot \right\|_{1} \sim \left\| \cdot \right\|_{2}$.

Construct an example of normed spaces $X_{1}$ and $X_{2}$ such that $X_{1} \subsetneq X_{2},\left\| \cdot \right\|_{X_{2}} \prec \left\| \cdot \right\|_{X_{1}}$, but $X_{2}^{*} \supsetneq X_{1}^{*}$ and $\left\| \cdot \right\|_{X_{1}^{*}} \succ \left\| \cdot \right\|_{X_{2}^{*}}$.

Prove statement item 2 in the list of dual spaces following the scheme described in the previous problem.

Find the elements of the space $l_{1}$ corresponding to the following functionals on the spaces с and с $c_{0}$:

Prove the following statements in the list of dual spaces:

Find the elements of the space $B V_{0}[-1,1]$ corresponding to the following functionals on the space $C[-1,1]$:

Find the elements of the space $B V_{0}[0,1]$ corresponding to the following functionals on the space $C[0,1]$:

Let $f$ be a positive linear functional on the real space $C[0,1]$, i.e., $f(x) \geqslant 0$ for every nonnegative function $x(t)$. Prove that $f$ is continuous and $\left\| f\right\| = f\left(x_{0}(t) \equiv 1\right)$.

Find the elements of the space $L_{\infty }[-1,1]$ corresponding to the following functionals on the space $L_{1}[-1,1]$:

Find the elements of the space $L_{2}[-1,1]$ corresponding to the following functionals on the space $L_{2}[-1,1]$:

Find the elements of the space $l_{\infty }$ corresponding to the following functionals on $l_{1}$:

Find the elements of the space $l_{2}$ corresponding to the following functionals on $l_{2}$:

Prove that the space $l_{1}$ is isometrically embedded in the space $l_{\infty }^{*}$, but is not isomorphic to it.

Prove that the following functionals are linear and continuous on the space $l_{\infty }$, and find their norms:

Prove that the Hahn-Banach extension of any linear continuous functional from $c_{0}$ to $l_{\infty }$ is unique. ¹

Prove that $L_{1}[0,1]$ is isometrically embedded in the space $(L_{\infty }[0,1])$*, but is not isomorphic to it.

Prove that the following functionals are linear and continuous on the space $L_{\infty }[-1,1]$, and find their norms:

Find the elements of the space $B V_{0}[-1,1] \times \mathbb {C}$ corresponding to the following functionals on the space $C^{1}[-1,1]$ with the norm $\left\| x\right\|_{C^{1}[-1,1]} = \left|x(-1)\right|+\max_{-1 \leqslant t \leqslant 1}\left|x^{\prime }(t)\right|$:

Find the elements of the space $W_{2}^{1}[0,1]$ corresponding to the following functionals on the space $W_{2}^{1}[0,1]$:

Solve items а, б of problem 5.20 using Theorem 5.4.

Theorem 5.4 (M. Fréchet, 1907; F. Riesz, 1907): let $H$ be a Hilbert space. Then every vector $y \in H$ generates a linear continuous functional $f_{y}$ on $H$, acting by the rule $f_{y}(x) = (x, y)$. Conversely, for every linear continuous functional $f$ on the space $H$ there exists a vector $y \in H$ such that $f = f_{y}$. The map $y \mapsto f_{y}$ is not only a bijection but an isometry and an antilinear map, i.e. $\alpha y_{1}+\beta y_{2} \mapsto \bar{\alpha } f_{y_{1}}+\bar{\beta } f_{y_{2}}$.

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).

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

Prove the non-reflexivity of the spaces: