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Geometry of spaces of quadratic forms. (English) Zbl 0918.46022

Rassias, Themistocles M. (ed.), Inner product spaces and applications. Harlow: Longman. Pitman Res. Notes Math. Ser. 376, 261-265 (1997).
Let \((E,\|\;\|)\) be the Euclidean space of dimension \(n\geq 2\) over \(\mathbb{R}\), with norm \(\|\;\|\) determined by the inner product. Let \(P_2(E)\) be the space of 2-homogeneous forms on \(E\) equipped with the norm \[ |||P|||= \sup_{\|x\|= 1}|P(x)|\quad\text{for }P\in P_2(E). \] Let \(U\) denote the closed unit ball in \((P_2(E), |||\;||||)\). Let \(\{e_1,e_2,\dots, e_n\}\) be a fixed ordered orthonormal basis in \(E\). For each \(P\), define the symmetric matrix \(\sigma(P)\equiv (a_{ij})\) given by \[ a_{ij}= \textstyle{{1\over 2}}[P(e_i+ e_j)- P(e_i)- P(e_j)]. \] Then the main result proved by the author is the following:
Theorem: (a) A 2-homogeneous form \(P\in P_2(E)\) is an extremal point of \(U\) iff the matrix \(\sigma(P)\) has unimodular eigenvalues.
(b) \(P\) is a smooth point of \(U\) iff the set \(\{\lambda_i\}^n_{i=1}\) of eigenvalues of \(\sigma(P)\) satisfies the condition: \[ \sup_{1\leq i\leq n}|\lambda_i|= 1,\quad\text{and }|\lambda_i|= 1\quad\text{for exactly one }i. \]
For the entire collection see [Zbl 0882.00018].

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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