Constructions of Banach spaces. Direct sums of subspaces

Let the linear space $X = X_{1} \times X_{2}$, where $(X_{1},\left\| \cdot \right\|_{1})$ and $(X_{2},\left\| \cdot \right\|_{2})$ are normed spaces. Verify the axioms of the norm $\left\| \binom {x_{1}}{x_{2}}\right\|_{X} = \left\| x_{1}\right\|_{X_{1}}+\left\| x_{2}\right\|_{X_{2}}$ in the space $X$. Prove that if $X_{1}$ and $X_{2}$ are Banach spaces, then the space $X$ is also a Banach space.

Prove that the Cartesian product $c_{0} \times c_{0}$ with the norm $\left\| \left(x_{1}, x_{2}\right)\right\| = \left\| x_{1}\right\|_{c_{0}}+\left\| x_{2}\right\|_{c_{0}}$ is not isometrically isomorphic to the space $c_{0}$.

Give an example of a norm $\left\| \binom {\cdot }{\cdot }\right\|$ on the linear space $\mathbb {R}^{2}$ and normed spaces $\left(X_{1},\left\| \cdot \right\|_{1}\right),\left(X_{2},\left\| \cdot \right\|_{2}\right)$ such that the function $p\left(x_{1}, x_{2}\right) = \left\| \binom {\left\| x_{1}\right\|_{1}}{\left\| x_{2}\right\|_{2}}\right\|$ on the linear space $X_{1} \times X_{2}$ is not a norm. Prove that if the norm $\left\| \binom {\cdot }{\cdot }\right\|$ is monotonic in each coordinate, then for any normed spaces $X_{1}, X_{2}$ the function $p\left(x_{1}, x_{2}\right)$ is a norm on the linear space $X_{1} \times X_{2}$.

Give an example of a norm on the Cartesian product $X_{1} \times X_{2}$ of normed spaces that is not given by the method indicated in problem 2.79.

Let $X_{0}$ be a closed subspace of a normed space $X$. Prove that $\left\| [x]\right\|_{0} = \inf_{x \in [x]}\left\| x\right\|$ defines a norm on the linear space $X / X_{0}$.

Let $X_{0}$ be a closed subspace of a Banach space $X$. Prove that the normed space $X / X_{0}$ is complete.

Let $X_{0}$ be a closed subspace of a normed space $(X,\left\| \cdot \right\| )$. Prove that if the spaces $(X_{0},\left\| \cdot \right\| )$ and $\left(X / X_{0},\left\| \cdot \right\|_{0}\right)$ are complete, then $X$ is complete as well.

Prove that:

Prove Theorem 2.6.

Theorem 2.6: If $X$ is an arbitrary normed space and $X_{1}$ and $X_{2}$ are two of its subspaces, one of which is closed and the other finite-dimensional, then their linear direct sum is topological.

Prove that the linear direct sum of the subspaces $c_{00}$ and $\operatorname {Lin}\left(x_{0}\right)$, where $x_{0} = \left(1, \frac{1}{2}, \frac{1}{3}, \ldots \right)$, in the space $c$ is not topological.

Let $X_{1}$ and $X_{2}$ be closed subspaces of a Banach space $X$. Prove that $X_{1} \oplus X_{2}$ is a closed subspace. Prove the converse: if $X_{1}$ and $X_{2}$ are linear subspaces of a Banach space $X$, and $X_{1} \oplus X_{2}$ is a closed subspace, then $X_{1}$ and $X_{2}$ are closed.

Give an example of an incomplete normed space $X$ and two closed subspaces in it $X_{1}$ and $X_{2}$ such that their topological direct sum $X_{1} \oplus X_{2}$ is a non-closed subspace.