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Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:

1. $X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary;
2. $X' \cong \ell_\infty \Rightarrow X'$ is complementary.

For the second one there is an option to try using Hahn-Banach theorem, as we do the same for proof of finite-dimensional subspaces.

More precisely, we can extend an identity operator $\ell_\infty \rightarrow \ell_\infty$ to the norm-one operator $X \rightarrow \ell_\infty$, but by the definition of complement subspace we should find closed subspace and moreover show that $\ell_\infty$ is closed.

Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:

1. $X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary;
2. $X' \cong \ell_\infty \Rightarrow X'$ is complementary.

For the second one there is an option to try using Hahn-Banach theorem, as we do the same for proof of finite-dimensional subspaces.

More precisely, we can extend an identity operator $\ell_\infty \rightarrow \ell_\infty$ to the norm-one operator $X \rightarrow \ell_\infty$, but by the definition of complement subspace we should find closed subspace (such subspace is a kernel of extension, as I know, but why does it form a closed subspace?) and moreover show that $\ell_\infty$ is closed.

Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:

1. $X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary;
2. $X' \cong \ell_\infty \Rightarrow X'$ is complementary.

The first proposition, as it seems, can be continued to the case of $\ell_p$, if the case of $\ell_1$ is at least true.

For the second one there is an option to try using Hahn-Banach theorem, as we do the same for proof of finite-dimensional subspaces.

Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:

1. $X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary;
2. $X' \cong \ell_\infty \Rightarrow X'$ is complementary.

For the second one there is an option to try using Hahn-Banach theorem, as we do the same for proof of finite-dimensional subspaces.

More precisely, we can extend an identity operator $\ell_\infty \rightarrow \ell_\infty$ to the norm-one operator $X \rightarrow \ell_\infty$, but by the definition of complement subspace we should find closed subspace and moreover show that $\ell_\infty$ is closed.

Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:

1. $X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary;
2. $X' \cong \ell_\infty \Rightarrow X'$ is complementary.

The first proposition seems being trivial, as thus $\ell_1$ is exactly the complement of $X'$, but I can miss something.

For the second one there is an option to try using Hahn-Banach theorem, as we do the same for proof of finite-dimensional subspaces.

Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:

1. $X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary;
2. $X' \cong \ell_\infty \Rightarrow X'$ is complementary.

The first proposition, as it seems, can be continued to the case of $\ell_p$, if the case of $\ell_1$ is at least true.

For the second one there is an option to try using Hahn-Banach theorem, as we do the same for proof of finite-dimensional subspaces.