On realcompact topological vector spaces

[EN] This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vector spaces appearing in Functional Analysis, including Frechet spaces, (L F)-spaces, and their duals, (D F)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (L F)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis.The research for the first named author was (partially) supported by Ministry of Science and Higher Education, Poland, Grant no. 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High-resolution palaeoclimatic records for the last millennium: Interpretation, integration and comparison with General Circulation Model control-run temperatures

Palaeoclimatology provides our only means of assessing climatic variations before the beginning of instrumental records. The various proxy variables used, however, have a number of limitations which must be adequately addressed and understood. Besides their obvious spatial and seasonal limitations, different proxies are also potentially limited in their ability to represent climatic variations over a range of different timescales. Simple correlations with instrumental data over the period since AD 1881 give some guide to which are the better proxies, indicating that coral- and ice-core-based reconstructions are poorer than tree-ring and historical ones. However, the quality of many proxy time series can deteriorate during earlier times. Suggestions are made for assessing proxy quality over longer periods than the last century by intercomparing neighbouring proxies and, by comparisons with less temporally resolved proxies such as borehole temperatures. We have averaged 17 temperature reconstructions (representing various seasons of the year), all extending back at least to the mid-seventeenth century, to form two annually resolved hemispheric series (NH10 and SH7). Over the 1901-91 period, NH10 has 36% variance in common with average NH summer (June to August) temperatures and 70% on decadal timescales. SH7 has 16% variance in common with average SH summer (December to February) temperatures and 49% on decadal timescales, markedly poorer than the reconstructed NH series. The coldest year of the millennium over the NH is AD 1601, the coldest decade 1691-1700 and the seventeenth is the coldest century. A Principal Components Analysis (PCA) is performed on yearly values for the 17 reconstructions over the period AD 1660-1970. The correlation between PC1 and NH10 is 0.92, even though PC1 explains only 13.6% of the total variance of all 17 series. Similar PCA is performed on thousand-year-long General Circulation Model (GCM) data from the Geophysical Fluid Dynamics Laboratory (GFDL) and the Hadley Centre (HADCM2), sampling these for the same locations and seasons as the proxy data. For GFDL, the correlation between its PC1 and its NH10 is 0,89, while for HADCM2 the PCs group markedly differently. Cross-spectral analyses are performed on the proxy data and the GFDL model data at two different frequency bands (0.02 and 0.03 cycles per year). Both analyses suggest that there is no large-scale coherency in the series on these timescales. This implies that if the proxy data are meaningful, it should be relatively straightforward to detect a coherent near-global anthropogenic signal in surface temperature data