In this post, all vector spaces are assumed to be real or complex.
Let $(X, ||\cdot||)$ be a Banach space, $Y \subset X$ a closed subspace. $Y$ is called $\underline{\mathrm{complemented}}$, if there is a closed subspace $Z \subset X$ such that $X =Y \oplus Z$ as topological vector spaces.
If $H$ is a Hilbert space every closed subspace $Y$ is complemented; the orthogonal complement $Y^{\bot}$ is a closed subspace of $H$ and we have $H=Y \oplus Y^{\bot}$. A famous theorem of Lindenstrauß and Tzafriri (which can be found in their article "On the complemented subspaces problem", Isreal Journal of Mathematics, Vol. 9, No.2, pp. 263-269) asserts that the converse is true as well. More precisely, if $(X, ||\cdot||)$ is a Banach space such that every closed subspace is complemented then $||\cdot||$ is induced by a scalarproduct, i.e. $(X,||\cdot||)$ is a Hilbert space.
Now to my question. Can you give me an example of a Banach space $(X,||\cdot||)$, which is not a Hilbert space, and of a closed subspace $Y \subset X$ which is not complemented? It is easily seen that $Y$ must be both infinite-dimensional and infinite-codimensional, for every finite-dimensional and every (closed) finite-codimensional subspace is complemented.
I thought about something like $c_{0} \subset (\ell^{\infty}, ||\cdot||_{\infty})$ the closed subspace of null sequences in the Banach space of bounded sequences but couldn't produce a proof that no closed complement exists in that case. Can you help me either proving that $c_{0}$ is not complemented (if that's true at all) or by giving me a different example?
Best Answer
Try the following article "A Survey of the Complemented Subspace Problem": https://arxiv.org/abs/math/0501048v1
Your suspicion about $c_0$ is correct. A couple of other examples: The disc algebra (those functions in $C(\mathbb{T})$ which are restrictions of functions analytic in the open unit disc) is closed in $C(\mathbb{T})$ but not complemented. Similarly, in $L^1(\mathbb{T})$, the subspace $H^1(\mathbb{T})$ consisting of functions whose negative Fourier coefficients vanish is closed but not complemented. See Rudin's Functional Analysis (the proof isn't very easy).