## Characterizations of Inner Product Spaces |

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