By Le Bellac M.

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Expanding this inequality proves the basic arithmetic– geometric mean inequality Using this result and the Cauchy–Schwarz inequality proves the following lemma. 2) A sequence ⊂ X is said to be convergent to a function υ ∈ X if Sometimes we will emphasize this convergence in norm by speaking of strong convergence. This is to distinguish from weak convergence. In particular, a sequence is said to converge weakly to a function υ ∈ X if for each φ ∈ X we have (υn, φ)X → (υ, φ)X as n → ∞. Unfortunately, bounded sets in a Hilbert space do not necessarily contain a convergent subsequence.

In particular, we note that these spaces have rather delicate and surprising properties concerning the density of smooth functions. We shall also discuss various decompositions of vector ﬁelds (in particular, the Helmholtz decomposition of a vector function into a curl-free and a divergence-free part) and prove a critical regularity result for solutions of Maxwell's equations. Related to this, we discuss scalar and vector potentials. The basic reference for this material is the excellent book of Girault and Raviart [143].

Let us suppose that μ is an eigenvalue of K of multiplicity m. e. 52Suppose ε > 0 is such that the disk of radius ε about μ contains no other eigenvalues of K. Then for h small enough the disk of radius ε centered at μ contains precisely m eigenvalues of the discrete problem denoted by μh, j, j = 1, …, m. The dimension of the eigenspace corresponding to μ, denoted by E(μ), is equal to that of . 23) Here is an X orthonormal basis for E(μ) and (K - Kh)|E(μ)is the restriction of (K - Kh) to E(μ). 53Osborn also provides an estimate for the distance of E(μ) to .