From Euclidean to Hilbert Spaces

4Banach Spaces and Hilbert Spaces

In this chapter, we shall consider normed or inner product spaces of infinite dimensions. Particular attention will be paid to “complete” spaces, for which several crucial theorems – which do not hold for non-complete spaces of infinite dimensions – can be formulated.

Before we can begin our analysis, it is important to note that all of the properties described previously for inner product spaces of finite dimension which rely solely on the algebraic nature of the inner product remain valid for infinite-dimensional vector spaces. For example:

• – a family of orthogonal vectors is free;
• – if 〈x, z〉 = 〈y, z〉 ∀z, then vectors x and y necessarily coincide;
• – the null vector is the only vector which is orthogonal to all other vectors;
• – the Gram-Schmidt orthonormalization procedure can be iterated, guaranteeing that an infinite system of mutually orthogonal vectors with a unitary norm will be obtained from any given infinite set of vectors.

The proofs for the first three properties are identical to those used for finite-dimensional vector spaces. The proof of the final property relies on the Zorn lemma.

Results for finite sums are harder to generalize; in this case, we need to take account of topological arguments in addition to algebraic considerations.

As we shall see, the definition and analysis of Banach and Hilbert spaces rely primarily on the analysis of the compatibility between the linear and topological structures of a normed or inner product ...