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absolutely convex arbitrary Banach space bounded linear operator bounded linear transformation bounded sequence bounded set called Cauchy sequence closed linear subspace closed unit ball compact operator complex numbers continuous linear functional convergent subsequence countable defined Definition Let denote dense eigen value element Example extreme point finite follows Hausdorff space Hence Hilbert space homeomorphism implies Inequality inner-product space invertible isometry isomorphic Lemma Let x e linear subspace linear transformation locally convex space maximal ideal non-empty non-zero normal operator normed linear space normed space open neighbourhood operator on H orthogonal orthonormal basis orthonormal set perpendicular projection positive integer Problem Proof Let Proof Step Prove real numbers reflexive Schauder basis self-adjoint operator sequence xn Solution Let space and let space H Step 2 Suppose subset surjective Theorem topological vector space totally bounded whence x e H