# A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces

Annales Polonici Mathematici (1992)

- Volume: 56, Issue: 3, page 303-309
- ISSN: 0066-2216

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topB. L. Chalmers, and F. T. Metcalf. "A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces." Annales Polonici Mathematici 56.3 (1992): 303-309. <http://eudml.org/doc/262300>.

@article{B1992,

abstract = {It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant $λ(v) = sup_X λ(v;X)$. The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in $L^∞(ν)$ and isometric to v and a projection $P_∞$ from C ⊕ V onto V such that $∥P_∞∥ = ∥P₁∥$, where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P₁ = ∑_\{i=1\}^2 U_i ⊗ v_i$, then $P_∞ = ∑_\{i=1\}^2 u_i ⊗ V_i$, where $dV_i = 2v_i dν$ and $dU_i = -2u_i dν$.},

author = {B. L. Chalmers, F. T. Metcalf},

journal = {Annales Polonici Mathematici},

keywords = {maximal overspace; two-dimensional spaces; relative projection constant},

language = {eng},

number = {3},

pages = {303-309},

title = {A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces},

url = {http://eudml.org/doc/262300},

volume = {56},

year = {1992},

}

TY - JOUR

AU - B. L. Chalmers

AU - F. T. Metcalf

TI - A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces

JO - Annales Polonici Mathematici

PY - 1992

VL - 56

IS - 3

SP - 303

EP - 309

AB - It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant $λ(v) = sup_X λ(v;X)$. The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in $L^∞(ν)$ and isometric to v and a projection $P_∞$ from C ⊕ V onto V such that $∥P_∞∥ = ∥P₁∥$, where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P₁ = ∑_{i=1}^2 U_i ⊗ v_i$, then $P_∞ = ∑_{i=1}^2 u_i ⊗ V_i$, where $dV_i = 2v_i dν$ and $dU_i = -2u_i dν$.

LA - eng

KW - maximal overspace; two-dimensional spaces; relative projection constant

UR - http://eudml.org/doc/262300

ER -

## References

top- [1] B. L. Chalmers, Absolute projection constant of the linear functions in a Lebesgue space, Constr. Approx. 4 (1988), 107-110. Zbl0647.41027
- [2] B. L. Chalmers and F. T. Metcalf, The determination of minimal projections and extensions in L¹, Trans. Amer. Math. Soc. 329 (1992), 289-305. Zbl0753.41018
- [3] B. L. Chalmers, F. T. Metcalf, B. Shekhtman and Y. Shekhtman, The projection constant of a two-dimensional real Banach space is no greater than 4/3, submitted. Zbl0970.46007
- [4] C. Franchetti and E. W. Cheney, Minimal projections in 𝓛₁-spaces, Duke Math. J. 43 (1976), 501-510. Zbl0335.41021
- [5] J. Lindenstrauss, On the extension of operators with a finite-dimensional range, Illinois J. Math. 8 (1964), 488-499. Zbl0132.09803
- [6] D. Yost, L₁ contains every two-dimensional normed space, Ann. Polon. Math. 49 (1988), 17-19. Zbl0679.46016

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