## Characterizations of inner product spaces |

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### Contents

Notation | 4 |

2dimensional Characterization 1 The parallelogram equality and derived equalities | 8 |

Norm derivatives characterizations | 16 |

Copyright | |

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### Common terms and phrases

2-dimensional space 2-dimensional subspace assume dim Banach Banach space bounded characterizations chord closed maximal subspace codissipative condition conv dimE dimensional ellipsoid embeds in Eg equivalent Euclidean figure finite g e Jv geometric Hahn-Banach theorem hence Hilbert space immediate implies strict convexity inequality inner product spaces intersection isoceles triangle isometry Lemma linear isometries linear projection linear selection linearly independent Metric projections midpoint minimal nonexpansive norm-1 projection normed space numbered orthogonality is symmetric parallel to uv parallelogram equality plane properties proximinal Q Proof rE(A satisfies segment sgnx sgny smooth strictly convex subspace F sup inf supporting hyperplane tangent twice differentiable u,v e unique unit sphere Vu,v e Vu,v e Sg Vu.v e Vx,y e xlpy