Hilbert and Banach spaces

Abstract

The inner product spaces ℂ n and l 2 share a further convenient property: informally speaking, any sequence in either of these spaces which looks convergent is convergent. To see what this means, recall the inner product space l 0 of Example 2.7. We saw there a sequence ( x k ) converging in l 2 but not in l 0 . If we tried to carry out all our analysis in l 0 this phenomenon would definitely complicate matters: it would be like trying to do real analysis in ℚ instead of ℝ. Let us formulate the requirement of the existence of limits. Definition Let ( M,d ) be a metric space. A sequence ( x k ) in M is a Cauchy sequence if, for every e > 0, there exists an integer k 0 such that k,l ≥ k 0 implies that d ( x k ,x l ) ( M,d ) is a complete metric space if every Cauchy sequence in M converges to a limit in M . Thus ℝ is a complete metric space with respect to its natural metric. So also is ℝ, for if ( z k ) is a Cauchy sequence in ℂ, then (Re z k ) and (Im z k ) are Cauchy sequences in ℝ. They thus have limits x,y ∈ℝ, and we have z k → x + i y in ℂ. Theorem ℂ n (for n ∈ℕ) and l 2 are complete metric spaces. We recall our convention that metric terminology refers to the metric determined by the norm according to Theorem 2.3 for an inner product space.