# Characterizations of Inner Product Spaces (Operator Theory by Amir

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By Amir

Each mathematician operating in Banaeh spaee geometry or Approximation thought understands, from his personal experienee, that the majority "natural" geometrie houses may possibly faH to carry in a generalnormed spaee until the spaee is an internal produet spaee. To reeall the weIl identified definitions, this suggests IIx eleven = *, the place is an internal (or: scalar) product on E, Le. a functionality from ExE to the underlying (real or eomplex) box pleasant: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =

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Sample text

It is enough to assume (1) by symmetry (recall that we assume X reﬂexive). 1, item (5). 4, B ∗ D∗ is also bisectorial by general theory. This proves the equivalence. Checking details, one sees that the angles are the same. Bisectoriality of DB and D∗ B ∗ are already used in the proofs. 11]). 3. First-order constant coeﬃcients diﬀerential systems Assume now that D is a ﬁrst-order diﬀerential operator on Rn acting on functions valued in CN whose symbol satisﬁes the conditions (D0), (D1) and (D2) in [9].

Suppose x ∈ A. Then dist(x, Ac ) > 0 since A is open, and so dist(x, Ac ) > αt for some t > 0. Hence (x, t) ∈ T α (A) ⊂ T α (B), so that dist(x, B c ) > αt > 0. Therefore x ∈ B. 2. Let C ⊂ X + be cylindrical, and suppose α > 0. Then S α (C) is bounded. Proof. Write C ⊂ B(x, r) × (a, b) for some x ∈ X and r, a, b > 0. Then S α (C) ⊂ S α (B(x, r) × (a, b)), and one can easily show that S α (B(x, r) × (a, b)) ⊂ B(x, r + αb), showing the boundedness of S α (C). 3. Let C ⊂ X + , and suppose α > 0. Then T α (S α (C)) is the minimal α-tent containing C, in the sense that T α (S) ⊃ C for some S ⊂ X implies that T α (S α (C)) ⊂ T α (S).

Funct. Anal. 62 (1985), 304–335. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogenes, Lecture Notes in Mathematics, vol. 242, Springer-Verlag, Berlin, 1971. [9] J. J. , Vector measures, Mathematical Surveys, vol. 15, American Mathematical Society, Providence, 1977. [10] L. Forzani, R. Scotto, P. Sj¨ ogren, and W. Urbina, On the Lp -boundedness of the non-centred Gaussian Hardy–Littlewood maximal function, Proc. Amer. Math. Soc. 130 (2002), no. 1, 73–79. Tent Spaces over Metric Measure Spaces 29 [11] K.