Existence of a non-expanding linear projection

Stream: maths

[Yury G. Kudryashov]

What are the conditions on a normed space E and its subspace s that guarantee existence of a linear projection π : E →L[Real] s such that ‖π x‖ ≤ ‖x‖ for all x?

[Yury G. Kudryashov]

E.g., it clearly holds for a closed subspace in a Hilbert space and for a 1-dimensional subspace in any space.

[Jireh Loreaux]

You don't require π to be continuous?

[Jireh Loreaux]

oops, sorry, I just saw the ₗ instead of L was confused

[Yury G. Kudryashov]

The inequality implies that π is continuous of norm ≤1.

[Yury G. Kudryashov]

(the norm will be zero, if all spaces are trivial)

[Jireh Loreaux]

right, of course, I just got tripped by the notation.

[Yury G. Kudryashov]

Edited

[Junyan Xu]

This question contains some relevant references.

[Yury G. Kudryashov]

Is there a simple counterexample in finite dimensions?

[Jireh Loreaux]

There's this paper about projections of norm 1.

[Jireh Loreaux]

But this is only about Banach spaces.

[Jireh Loreaux]

Theorem 5.8 [39]. Let X be a Banach space over ℂ or ℝ. Let M be a closed linear subspace of X. M is the range of a contractive projection if and only if there exists a weak∗-closed linear subspace L of X∗ with M ⊂ J⁻¹(L) and L ⊂ norm_closure(J(M)).

This appears to be the most general characterization of contractively complemented subspaces known.

[Yury G. Kudryashov]

Thanks! I think I don't need more details.

[Antoine Chambert-Loir]

Yury G. Kudryashov said:

E.g., it clearly holds for a closed subspace in a Hilbert space and for a 1-dimensional subspace in any space.

why, in the second case?

[Yury G. Kudryashov]

Due to Hahn-Banach extension theorem?

[Yury G. Kudryashov]

(it's past midnight here, so probability of me being wrong is larger than usual)

[Christopher Hoskin]

Yury G. Kudryashov said:

Thanks! I think I don't need more details.

Noted, but in case anyone else is interested, contractive projections on Banach spaces are of great interest in certain models of quantum mechanics because the class of JB star triples is stable under contractive projections (Kaup, Contractive Projections on Jordan C*-algebras and Generalizations, Math. Scand. 54 (1984), 95-100).

[Christopher Hoskin]

Coincidentally I opened #20330 yesterday which may be of interest?

[Yury G. Kudryashov]

We already have docs#Submodule.linearProjOfClosedCompl

[Jireh Loreaux]

Yury, Christopher knows about this (indeed, it is used in the construction). The point of #20330 is that the map is E →L[𝕜] E, not E →L[𝕜] ↥p.

[Yury G. Kudryashov]

It wasn't used in the previous version of the PR.