Duality and distance formulas in spaces defined by means of oscillation

For the classical space of functions with bounded mean oscillation, it is well known that VMO∗∗=BMOVMO∗∗=BMO and there are many characterizations of the distance from a function f in BMOBMO to VMOVMO. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as QK-spaces, weighted spaces, Lipschitz–Hölder spaces and rectangular BMOBMO of several variables

Multivitamin supplementation improves nutritional status and bone quality in aged care residents

ObjectivesTo assess the effectiveness of a multivitamin (MV) tablet on nutritional status, quantitative heel ultrasound (QUS), mobility, muscle strength and falls. The design comprised two groups matched on mobility levels, randomized to receive a daily MV or placebo (P) tablet for 6 months. The setting was an Australian residential care facility.SubjectsA total of 92 aged care residents. Serum micronutrients, body weight, QUS, rate of falls, hand grip strength, and the timed up and go test were assessed at baseline and 6 months.ResultsA total of 49 participants consumed a MV and 43, a matched P for 6 months. There was a greater increase in the MV vs P group for serum 25(OH)D (mean difference+/-standard error, 33.4+/-2.6 nmol l(-1)), folate (13.4+/-2.8 nmol l(-1)), and vitamin B12 (178.0+/-40.3 pmol l(-1)) (all P or =50 nmol l(-1)) were found among 77% of participants in the MV group vs 10% taking P (PConclusionsMV supplementation raised serum vitamin B12 and folate concentrations and increased serum 25(OH)D, which was accompanied by an apparent positive effect on bone density. We also found a trend towards a reduction in falls and this could contribute to a reduction in fractures.JA Grieger, CA Nowson, HF Jarman, R Malon and LM Acklan