PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 12, Pages 3647–3656 S 0002-9939(01)05986-X Article electronically published on June 6, 2001 EXTENSION OF BILINEAR FORMS ON BANACH SPACES JES ´ US M. F. CASTILLO, RICARDO GARC ´ IA, AND JES ´ US A. JARAMILLO (Communicated by Dale Alspach) Abstract. We study the extension of bilinear and multilinear forms from a given subspace of a Banach space to the whole space. Precisely, an isomorphic embedding j : E → X is said to be (linearly) N-exact if N-linear forms on E can be (linear and continuously) extended to X through j . We present some necessary and sufficient conditions for j to be 2-exact, as well as several examples of 2-exact embeddings. We answer a problem of Zalduendo: in a cotype 2 space bilinear extendable and integral forms coincide. 1. Introduction and preliminaries Bilinear continuous forms on Banach spaces, contrary to what happens with linear continuous forms, do not extend from subspaces to the whole space. The typical example exhibited to prove that is the inner product on the real Hilbert space. This bilinear form cannot be extended to l ∞ since bilinear forms on l ∞ are weakly sequentially continuous (see [22]), something that the inner product is not. However, the preceding argument is misleading: it suggests that the property of making all polynomials weakly sequentially continuous matters, and suggests some unspecified essential property of either l 2 or l ∞ to be uncovered. However, the only thing that matters, as we shall see, is the inclusion map l 2 → l ∞ . A number of papers have considered the problem of the extension of bilinear and multilinear forms from a subspace E to a larger space X : Carando [6]; Carando and Zalduendo [7]; Galindo, Garc´ ıa, Maestre and Mujica [13]; Kirwan and Ryan [18] and Zalduendo [23]. However, all those papers dodge to consider the role of the embedding E → X which, in our opinion, should be the central object to be studied. We justify this assertion with a simple observation: Let j : E → X be an into isomorhism; a bilinear form b on E can be extended to a bilinear form B on X through j if and only if there exists an operator T ∈ L(X,X ∗ ) making commutative the diagram E j → X τ b ↓ ↓ T E ∗ j * ← X ∗ Received by the editors April 28, 2000. 2000 Mathematics Subject Classification. Primary 46B20, 46B28. The research of the first and second authors was supported in part by DGICYT project PB97- 0377. The research of the third author was supported by DGICYT project PB96-0607. c 2001 American Mathematical Society 3647 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use