Functional Analysis

a₁ arbitrary axiom B₁ B₂ Banach space basis called cardinal number Cauchy sequence complete complex numbers contains convergent countable defined definition denote denumerable disjoint eigenvalue element equipotent equivalence relation Example exists a positive finite dimensional follows function f Hence Hilbert space implies inequality infinite isomorphism Let f limit point linear combination linear subspace linear transformation linearly independent M₁ M₂ mapping ƒ matrix maximal element Meerut metric space non-empty subset non-zero vector nonvoid normed linear space null space one-one mapping open set open sphere operator on H ordinal orthonormal set partially ordered set positive integer Proof proper subset prove real numbers reflexive S₁ scalar multiplication Solution space and let space H subspace of H Suppose T₁ theorem unique x₁ y₁ zero vector