Complete and minimal systems of vectors

Let $X$ be a normed space. Prove that it is separable if and only if there exists in $X$ a countable complete system.

Prove that in every infinite-dimensional Banach space there exists a linearly independent system $\left\{ x_{k}\right\}_{1}^{\infty }$ (i.e., none of the vectors of the system is a finite linear combination of the others) which is not minimal.

Prove that a system $\left\{ x_{k}\right\}_{k = 1}^{\infty }$ of vectors of a normed space $X$ is complete if and only if it is total.

Let $\left\{ x_{k}\right\}_{k = 1}^{\infty }$ be a system of vectors of a normed space $X$. Prove that it is minimal if and only if there exists for it a biorthogonal system of functionals $\left\{ f_{k}\right\}_{k = 1}^{\infty }$. Prove that these functionals also form a minimal system in the space $X^{*}$.

Give an example of a Banach space $X$ and a minimal system $\left\{ f_{k}\right\}_{k = 1}^{\infty } \subset X^{*}$ for which there is no biorthogonal system $\left\{ x_{k}\right\}_{k = 1}^{\infty }$. Prove that such an example is impossible in a reflexive space.

Let $X$ be a normed space. Prove that for any finite linearly independent system of functionals $\left\{ f_{k}\right\}_{k = 1}^{n} \subset X^{*}$ there exists a biorthogonal system of vectors $\left\{ x_{k}\right\}_{k = 1}^{n} \subset X$ (cf. problem 5.4). Prove that for any finite linearly independent system of vectors $\left\{ x_{k}\right\}_{k = 1}^{n} \subset X$ there exists a biorthogonal system of functionals $\left\{ f_{k}\right\}_{k = 1}^{n} \subset X^{*}$.

Let $\left\{ x_{k}\right\}_{k = 1}^{\infty }$ be a complete minimal system of vectors of a normed space $X$. Prove that the biorthogonal system $\left\{ f_{k}\right\}_{k = 1}^{\infty }$ (it exists by problem 10.4) is unique. Prove that in the case of a reflexive space $X$ the system $\left\{ f_{k}\right\}_{k = 1}^{\infty }$ is complete in $X^{*}$ (by problem 10.4 it is minimal).

Let $X$ and $Y$ be normed spaces, the operator $T \in \mathcal{B}(X, Y)$ and $\overline{\operatorname {Im} T} = Y$. Prove that completeness of the system $\left\{ x_{k}\right\}_{1}^{\infty }$ in $X$ implies completeness of the system $\left\{ T x_{k}\right\}_{1}^{\infty }$ in $Y$.

Let the system $\left\{ x_{k}\right\}_{1}^{\infty }$ be complete in a normed space $X$ and let each vector $x_{k}$ be a finite linear combination of vectors of the system $\left\{ y_{k}\right\}_{1}^{\infty }$. Prove that the system $\left\{ y_{k}\right\}_{1}^{\infty }$ is also complete.

Let $X$ and $Y$ be normed spaces, $T \in \mathcal{B}(X, Y)$ and $\inf_{\left\| x\right\| = 1}\left\| T x\right\| = c > 0$. Prove that minimality of the system $\left\{ x_{k}\right\}_{1}^{\infty }$ in $X$ implies minimality of the system $\left\{ T x_{k}\right\}_{1}^{\infty }$ in $Y$.

Give an example of a Banach space $X$, an injective operator $T \in \mathcal{B}(X)$, and a minimal system $\left\{ x_{k}\right\}_{1}^{\infty } \subset X$ such that the system $\left\{ T x_{k}\right\}_{1}^{\infty }$ is not minimal.

Investigate the system $\left\{ e_{n} = \left(0,0, \ldots , 0, \frac{1}{n}, 0,0, \ldots \right)\right\}_{1}^{\infty }$ for completeness and minimality in the spaces

For every $\lambda$ with $\left|\lambda \right| < 1$, investigate the system of vectors

for completeness and minimality in the spaces

Investigate the system of vectors $\left\{ x_{n} = \left(1, \ldots , \frac{1}{n}, 0,0, \ldots \right)\right\}_{1}^{\infty }$ for completeness and minimality in the spaces

Investigate the following systems of vectors for completeness and minimality in the spaces $L_{p}[0,1], p \in [1, \infty )$, and $C[0,1]$:

Let $\left\{ \alpha_{n}\right\}_{1}^{\infty }$ be a strictly increasing sequence of positive numbers with the property $\alpha_{n} \rightarrow 1$. Prove that the system $\left\{ t^{\alpha_{n}}\right\}_{n = 1}^{\infty }$ is complete

(H. Müntz, 1914). The system of powers $\left\{ t^{n_{k}}\right\}_{k = 0}^{\infty }$, where $n_{0} < n_{1} < \ldots$, is complete in the space $C[0,1]$ if and only if $n_{0} = 0$ and $\sum_{k = 1}^{\infty } \frac{1}{n_{k}} = \infty$.

In the space $C[0,1]$, give an example of an incomplete system of vectors $\left\{ x_{n}\right\}_{1}^{\infty }$ such that for any continuous function $x \in C[0,1]$ the equalities $\int_{0}^{1} x(t) x_{n}(t) d t = 0, n = 1,2, \ldots$, imply $x = 0$.

Let $X$ be a Banach space, and let the system $\left\{ e_{n}\right\}_{1}^{\infty }$ be weakly complete in it, i.e., for every vector $x \in X$ there exists a sequence $\left\{ x_{k}\right\}_{1}^{\infty } \subset \operatorname {Lin}\left\{ e_{n}\right\}_{1}^{\infty }$, weakly convergent to $x$. Prove that then the system $\left\{ e_{n}\right\}_{1}^{\infty }$ is complete in $X$ in the usual sense.

Let $\left\{ x_{n}\right\}_{1}^{\infty }$ be a complete and minimal system in a Hilbert space $H$, and $\left\{ y_{n}\right\}_{1}^{\infty }$ a biorthogonal system for it. Prove that if $\left\| x_{n}\right\| = \left\| y_{n}\right\| = 1$ for all $n \in \mathbb {N}$, then $\left\{ x_{n}\right\}_{1}^{\infty }$ is an orthonormal basis in $H$.