Main properties. Calculation of norms

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 $\operatorname {dist}\left(x, X_{0}\right) \geqslant 1-\varepsilon$.