Academic literature on the topic 'Banach spaces'

We study the weighted Banach spaces of vector-valued holomorphic functions defined on an open and connected subset of a Banach space. We use linearization results on these spaces to get conditions which ensure that a functionfdefined in a subsetAof an open and connected subsetUof a Banach spaceX, with values in another Banach spaceE, and admitting certain weak extensions in a Banach space of holomorphic functions can be holomorphically extended in the corresponding Banach space of vector-valued functions.

We characterize thek-smooth points in some Banach spaces. We will deal with injective tensor product, the Bochner spaceL∞(μ,X)of (equivalence classes of)μ-essentially bounded measurableX-valued functions, and direct sums of Banach spaces.

New Leray-Schauder results are presented for multivalued contractions defined on subsets of a Fréchet spaceE. The proof relies on fixed point results in Banach spaces and on viewingEas the projective limit of a sequence of Banach spaces.

This paper aims at describing Tim Gowers' contributions to Banach space theory that earned him the Fields medal in 1998. In particular, the construction of the Gowers-Maurey space, a Banach space not containing an unconditional basic sequence, is sketched as is the Gowers dichotomy theorem that led to the solution of the homogeneous Banach space problem. Moreover, Gowers' counterexamples to the hyperplane problem and the Schröder-Bernstein problem are discussed. The paper is an extended version of a talk given at Freie Universität Berlin in December 1999; hence the reference to the next millennium at the very end actually appeals to the present millennium. It should be accessible to anyone with a basic knowledge of functional analysis and of German.

In recent publications the concepts of fast completeness and local barreledness have been shown to be related to the property of all weak-*bounded subsets of the dual (of a locally convex space) being strongly bounded. In this paper we clarify those relationships, as well as giving several different characterizations of this property.

We define and study a new family of Banach spaces, the J ames-Schreier spaces, cre- ated by combining key properties in the definitions of two important classical Banach spaces, namely James' quasi-reflexive space and Schreier's space. We explore both the Banach space and Banach algebra theory of these spaces. The new spaces inherit aspects of both parent spaces: our main results are that the J ames-Schreier spaces each have a shrinking basis, do not embed in a Banach space with an unconditional basis, and each of their closed, infinite-dimensional subspaces contains a copy of Co. As Banach sequence algebras each James-Schreier space has a bounded approx- imate identity and is weakly amenable but not amenable, and the bidual and multiplier - algebra are isometrically isomorphic. We approach our study of Banach sequence algebras from the point of view of Schauder basis theory, in particular looking at those Banach sequence algebras for which the unit vectors form an unconditional or shrinking basis. We finally show that for each Banach space X with an unconditional basis we may construct a James-like Banach sequence algebra j(X) with a bounded approximate identity, and give a condition on the shift operators acting on X which implies that j(X) will contain a copy of X as a complemented ideal and hence not be amenable.

Capítulo 1. Después de estudiar algunos preliminares sobre familias adecuadas de conjuntos, formulamos y probamos algunas equivalencias, cada una de ellas son una condición suficiente para que la familia defina un conjunto compacto de Gul'ko. Damos una caracterización de conjunto compacto de Gul'ko en términos de emparejamiento con un conjunto $\mathcal{K}$-analítico. Capítulo 2. Estudiamos propiedades de los espacios de Banach débilmente Lindelöf determinados no-separables. Damos una caracterización por medio de la existencia de un generador proyeccional full sobre él. Estudiamos algunos aspectos sobre sistemas biortogonales en espacios de Banach. Usando técnicas de resoluciones proyeccionales de la identidad, probamos una extensión de un resultado de Argyros y Mercourakis. Capítulo 3. En el espacio $(c_0(\Gamma),\|\cdot\|_\infty)$, con $\Gamma\in\mathbb{R}$, damos una norma equivalente estrictamente convexa. Capítulo 4. Consideramos una caracterización de los subespacios de espacios de Banach débilmente compactamente generados, en términos de una propiedad de cubrimiento de la bola unidad por medio de conjuntos $\epsilon$-débilmente compactos. Reemplazamos este concepto por otro más preciso que llamamos $\epsilon$-débilmente auto-compactos, este concepto permite una mejor descripción. Capítulo 5. Damos condiciones intrínsecas, necesarias y suficientes para que un espacio de Banach sea generado por $c_0(\Gamma)$ o $\ell_p(\Gamma)$ para $p\in(1,+\infty)$. Ofrecemos una nueva demostración de un resultado de Rosenthal, sobre operadores de $c_0(\Gamma)$ en un espacio de Banach.
González Correa, AL. (2008). Compacta in Banach spaces [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8312
Palancia

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

We study the geometric classification of Banach spaces via Lipschitz, uniformly continuous, and coarse mappings. We prove that a Banach space which is uniformly homeomorphic to a linear quotient of lp is itself a linear quotient of lp when p<2. We show that a Banach space which is Lipschitz universal for all separable metric spaces cannot be asymptotically uniformly convex. Next we consider coarse embedding maps as defined by Gromov, and show that lp cannot coarsely embed into a Hilbert space when p> 2. We then build upon the method of this proof to show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a subspace of L0(??) for some probability space (Ω,B,??).

The notion of asymptotic structure of an infinite dimensional Banach space was introduced by Maurey, Milman and Tomczak-Jaegermann. The asymptotic structure consists of those finite dimensional spaces which can be found everywhere `at infinity'. These are defined as the spaces for which there is a winning strategy in a certain vector game. The above authors introduced the class of asymptotic $\ell_p$ spaces, which are the spaces having simplest possible asymptotic structure. Key examples of such spaces are Tsirelson's space and James' space. We prove some new properties of general asymptotic $\ell_p$ spaces and also compare the notion of asymptotic $\ell_2$ with other notions of asymptotic Hilbert space behaviour such as weak Hilbert and asymptotically Hilbertian. We study some properties of smooth functions defined on subsets of asymptotic $\ell_\infty$ spaces. Using these results we show that that an asymptotic $\ell_\infty$ space which has a suitably smooth norm is isomorphically polyhedral, and therefore admits an equivalent analytic norm. We give a sufficient condition for a generalized Orlicz space to be a stabilized asymptotic $\ell_\infty$ space, and hence obtain some new examples of asymptotic $\ell_\infty$ spaces. We also show that every generalized Orlicz space which is stabilized asymptotic $\ell_\infty$ is isomorphically polyhedral. In 1991 Gowers and Maurey constructed the first example of a space which did not contain an unconditional basic sequence. In fact their example had a stronger property, namely that it was hereditarily indecomposable. The space they constructed was `$\ell_1$-like' in the sense that for any $n$ successive vectors $x_1 < \ldots < x_n$, $\frac{1}{f(n)} \sum_{i=1}^n \| x_i \| \leq \| \sum_{i=1}^n x_i \| \leq \sum_{i=1}^n \| x_i \|,$ where $ f(n) = \log_2 (n+1) $. We present an adaptation of this construction to obtain, for each $ p \in (1, \infty)$, an hereditarily indecomposable Banach space, which is `$\ell_p$-like' in the sense described above. We give some sufficient conditions on the set of types, $\mathscr{T}(X)$, for a Banach space $X$ to contain almost isometric copies of $\ell_p$ (for some $p \in [1, \infty)$) or of $c_0$. These conditions involve compactness of certain subsets of $\mathscr{T}(X)$ in the strong topology. The proof of these results relies heavily on spreading model techniques. We give two examples of classes of spaces which satisfy these conditions. The first class of examples were introduced by Kalton, and have a structural property known as Property (M). The second class of examples are certain generalized Tsirelson spaces. We introduce the class of stopping time Banach spaces which generalize a space introduced by Rosenthal and first studied by Bang and Odell. We look at subspaces of these spaces which are generated by sequences of independent random variables and we show that they are isomorphic to (generalized) Orlicz spaces. We deduce also that every Orlicz space, $h_\phi$, embeds isomorphically in the stopping time Banach space of Rosenthal. We show also, by using a suitable independence condition, that stopping time Banach spaces also contain subspaces isomorphic to mixtures of Orlicz spaces.

Macroeconomic summary Annual inflation continued to rise in the first quarter (8.5%) and again outpaced both market expectations and the technical staff’s projections. Inflation in major consumer price index (CPI) baskets has accelerated year-to-date, rising in March at an annual rate above 3%. Food prices (25.4%) continued to contribute most to rising inflation, mainly affected by a deterioration in external supply and rising costs of agricultural inputs. Increases in transportation prices and in some utility rates (energy and gas) can explain the acceleration in regulated items prices (8.3%). For its part, the increase in inflation excluding food and regulated items (4.5%) would be the result of shocks in supply and external costs that have been more persistent than expected, the effects of indexation, accumulated inflationary pressures from the exchange rate, and a faster-than-anticipated tightening of excess productive capacity. Within the basket excluding food and regulated items, external inflationary pressures have meaningfully impacted on goods prices (6.4%), which have been accelerating since the last quarter of 2021. Annual growth in services prices (3.8%) above the target rate is due primarily to food away from home (14.1%), which was affected by significant increases in food and utilities prices and by a rise in the legal monthly minimum wage. Housing rentals and other services prices also increased, though at rates below 3%. Forecast and expected inflation have increased and remain above the target rate, partly due to external pressures (prices and costs) that have been more persistent than projected in the January report (Graphs 1.1 and 1.2). Russia’s invasion of Ukraine accentuated inflationary pressures, particularly on international prices for certain agricultural goods and inputs, energy, and oil. The current inflation projection assumes international food prices will increase through the middle of this year, then remain high and relatively stable for the remainder of 2022. Recovery in the perishable food supply is forecast to be less dynamic than previously anticipated due to high agricultural input prices. Oil prices should begin to recede starting in the second half of the year, but from higher levels than those presented in the previous report. Given the above, higher forecast inflation could accentuate indexation effects and increase inflation expectations. The reversion of a rebate on value-added tax (VAT) applied to cleaning and hygiene products, alongside the end of Colombia’s COVID-19 health emergency, could increase the prices of those goods. The elimination of excess productive capacity on the forecast horizon, with an output gap close to zero and somewhat higher than projected in January, is another factor to consider. As a consequence, annual inflation is expected to remain at high levels through June. Inflation should then decline, though at a slower pace than projected in the previous report. The adjustment process of the monetary policy rate wouldcontribute to pushing inflation and its expectations toward the target on the forecast horizon. Year-end inflation for 2022 is expected to be around 7.1%, declining to 4.8% in 2023. Economic activity again outperformed expectations. The technical staff’s growth forecast for 2022 has been revised upward from 4.3% to 5% (Graph 1.3). Output increased more than expected in annual terms in the fourth quarter of 2021 (10.7%), driven by domestic demand that came primarily because of private consumption above pre-pandemic levels. Investment also registered a significant recovery without returning to 2019 levels and with mixed performance by component. The trade deficit increased, with significant growth in imports similar to that for exports. The economic tracking indicator (ISE) for January and February suggested that firstquarter output would be higher than previously expected and that the positive demand shock observed at the end of 2021 could be fading slower than anticipated. Imports in consumer goods, retail sales figures, real restaurant and hotel income, and credit card purchases suggest that household spending continues to be dynamic, with levels similar to those registered at the end of 2021. Project launch and housing starts figures and capital goods import data suggest that investment also continues to recover but would remain below pre-pandemic levels. Consumption growth is expected to decelerate over the year from high levels reached over the last two quarters. This would come amid tighter domestic and external financial conditions, the exhaustion of suppressed demand, and a deterioration of available household income due to increased inflation. Investment is expected to continue to recover, while the trade deficit should tighten alongside high oil and other export commodity prices. Given all of the above, first-quarter economic growth is now expected to be 7.2% (previously 5.2%) and 5.0% for 2022 as a whole (previously 4.3%). Output growth would continue to moderate in 2023 (2.9%, previously 3.1%), converging similar to long-term rates. The technical staff’s revised projections suggest that the output gap would remain at levels close to zero on the forecast horizon but be tighter than forecast in January (Graph 1.4). These estimates continue to be affected by significant uncertainty associated with geopolitical tensions, external financial conditions, Colombia’s electoral cycle, and the COVID-19 pandemic. External demand is now projected to grow at a slower pace than previously expected amid increased global inflationary pressures, high oil prices, and tighter international financial conditions than forecast in January. The Russian invasion of Ukraine and its inflationary effects on prices for oil and certain agricultural goods and inputs accentuated existing global inflationary pressures originating in supply restrictions and increased international costs. A decline in the supply of Russian oil, low inventory levels, and continued production limits on behalf of the Organization of Petroleum Exporting Countries and its allies (OPEC+) can explain increased projected oil prices for 2022 (USD 100.8/barrel, previously USD 75.3) and 2023 (USD 86.8/barrel, previously USD 71.2). The forecast trajectory for the U.S. Federal Reserve (Fed) interest rate has increased for this and next year to reflect higher real and expected inflation and positive performance in the labormarket and economic activity. The normalization of monetary policy in various developed and emerging market economies, more persistent supply and cost shocks, and outbreaks of COVID-19 in some Asian countries contributed to a reduction in the average growth outlook for Colombia’s trade partners for 2022 (2.8%, previously 3.3%) and 2023 (2.4%, previously 2.6%). In this context, the projected path for Colombia’s risk premium increased, partly due to increased geopolitical global tensions, less expansionary monetary policy in the United States, an increase in perceived risk for emerging markets, and domestic factors such as accumulated macroeconomic imbalances and political uncertainty. Given all the above, external financial conditions are tighter than projected in January report. External forecasts and their impact on Colombia’s macroeconomic scenario continue to be affected by considerable uncertainty, given the unpredictability of both the conflict between Russia and Ukraine and the pandemic. The current macroeconomic scenario, characterized by high real inflation levels, forecast and expected inflation above 3%, and an output gap close to zero, suggests an increased risk of inflation expectations becoming unanchored. This scenario offers very limited space for expansionary monetary policy. Domestic demand has been more dynamic than projected in the January report and excess productive capacity would have tightened more quickly than anticipated. Headline and core inflation rose above expectations, reflecting more persistent and important external shocks on supply and costs. The Russian invasion of Ukraine accentuated supply restrictions and pressures on international costs. This partly explains the increase in the inflation forecast trajectory to levels above the target in the next two years. Inflation expectations increased again and are above 3%. All of this increased the risk of inflation expectations becoming unanchored and could generate indexation effects that move inflation still further from the target rate. This macroeconomic context also implies reduced space for expansionary monetary policy. 1.2 Monetary policy decision Banco de la República’s board of directors (BDBR) continues to adjust its monetary policy. In its meetings both in March and April of 2022, it decided by majority to increase the monetary policy rate by 100 basis points, bringing it to 6.0% (Graph 1.5).

Macroeconomic summary Several factors contributed to an increase in projected inflation on the forecast horizon, keeping it above the target rate. These included inflation in December that surpassed expectations (5.62%), indexation to higher inflation rates for various baskets in the consumer price index (CPI), a significant real increase in the legal minimum wage, persistent external and domestic inflationary supply shocks, and heightened exchange rate pressures. The CPI for foods was affected by the persistence of external and domestic supply shocks and was the most significant contributor to unexpectedly high inflation in the fourth quarter. Price adjustments for fuels and certain utilities can explain the acceleration in inflation for regulated items, which was more significant than anticipated. Prices in the CPI for goods excluding food and regulated items also rose more than expected. This was partly due to a smaller effect on prices from the national government’s VAT-free day than anticipated by the technical staff and more persistent external pressures, including via peso depreciation. By contrast, the CPI for services excluding food and regulated items accelerated less than expected, partly reflecting strong competition in the communications sector. This was the only major CPI basket for which prices increased below the target inflation rate. The technical staff revised its inflation forecast upward in response to certain external shocks (prices, costs, and depreciation) and domestic shocks (e.g., on meat products) that were stronger and more persistent than anticipated in the previous report. Observed inflation and a real increase in the legal minimum wage also exceeded expectations, which would boost inflation by affecting price indexation, labor costs, and inflation expectations. The technical staff now expects year-end headline inflation of 4.3% in 2022 and 3.4% in 2023; core inflation is projected to be 4.5% and 3.6%, respectively. These forecasts consider the lapse of certain price relief measures associated with the COVID-19 health emergency, which would contribute to temporarily keeping inflation above the target on the forecast horizon. It is important to note that these estimates continue to contain a significant degree of uncertainty, mainly related to the development of external and domestic supply shocks and their ultimate effects on prices. Other contributing factors include high price volatility and measurement uncertainty related to the extension of Colombia’s health emergency and tax relief measures (such as the VAT-free days) associated with the Social Investment Law (Ley de Inversión Social). The as-yet uncertain magnitude of the effects of a recent real increase in the legal minimum wage (that was high by historical standards) and high observed and expected inflation, are additional factors weighing on the overall uncertainty of the estimates in this report. The size of excess productive capacity remaining in the economy and the degree to which it is closing are also uncertain, as the evolution of the pandemic continues to represent a significant forecast risk. margin, could be less dynamic than expected. And the normalization of monetary policy in the United States could come more quickly than projected in this report, which could negatively affect international financing costs. Finally, there remains a significant degree of uncertainty related to the duration of supply chocks and the degree to which macroeconomic and political conditions could negatively affect the recovery in investment. The technical staff revised its GDP growth projection for 2022 from 4.7% to 4.3% (Graph 1.3). This revision accounts for the likelihood that a larger portion of the recent positive dynamic in private consumption would be transitory than previously expected. This estimate also contemplates less dynamic investment behavior than forecast in the previous report amid less favorable financial conditions and a highly uncertain investment environment. Third-quarter GDP growth (12.9%), which was similar to projections from the October report, and the fourth-quarter growth forecast (8.7%) reflect a positive consumption trend, which has been revised upward. This dynamic has been driven by both public and private spending. Investment growth, meanwhile, has been weaker than forecast. Available fourth-quarter data suggest that consumption spending for the period would have exceeded estimates from October, thanks to three consecutive months that included VAT-free days, a relatively low COVID-19 caseload, and mobility indicators similar to their pre-pandemic levels. By contrast, the most recently available figures on new housing developments and machinery and equipment imports suggest that investment, while continuing to rise, is growing at a slower rate than anticipated in the previous report. The trade deficit is expected to have widened, as imports would have grown at a high level and outpaced exports. Given the above, the technical staff now expects fourth-quarter economic growth of 8.7%, with overall growth for 2021 of 9.9%. Several factors should continue to contribute to output recovery in 2022, though some of these may be less significant than previously forecast. International financial conditions are expected to be less favorable, though external demand should continue to recover and terms of trade continue to increase amid higher projected oil prices. Lower unemployment rates and subsequent positive effects on household income, despite increased inflation, would also boost output recovery, as would progress in the national vaccination campaign. The technical staff expects that the conditions that have favored recent high levels of consumption would be, in large part, transitory. Consumption spending is expected to grow at a slower rate in 2022. Gross fixed capital formation (GFCF) would continue to recover, approaching its pre-pandemic level, though at a slower rate than anticipated in the previous report. This would be due to lower observed GFCF levels and the potential impact of political and fiscal uncertainty. Meanwhile, the policy interest rate would be less expansionary as the process of monetary policy normalization continues. Given the above, growth in 2022 is forecast to decelerate to 4.3% (previously 4.7%). In 2023, that figure (3.1%) is projected to converge to levels closer to the potential growth rate. In this case, excess productive capacity would be expected to tighten at a similar rate as projected in the previous report. The trade deficit would tighten more than previously projected on the forecast horizon, due to expectations of an improved export dynamic and moderation in imports. The growth forecast for 2022 considers a low basis of comparison from the first half of 2021. However, there remain significant downside risks to this forecast. The current projection does not, for example, account for any additional effects on economic activity resulting from further waves of COVID-19. High private consumption levels, which have already surpassed pre-pandemic levels by a large margin, could be less dynamic than expected. And the normalization of monetary policy in the United States could come more quickly than projected in this report, which could negatively affect international financing costs. Finally, there remains a significant degree of uncertainty related to the duration of supply chocks and the degree to which macroeconomic and political conditions could negatively affect the recovery in investment. External demand for Colombian goods and services should continue to recover amid significant global inflation pressures, high oil prices, and less favorable international financial conditions than those estimated in October. Economic activity among Colombia’s major trade partners recovered in 2021 amid countries reopening and ample international liquidity. However, that growth has been somewhat restricted by global supply chain disruptions and new outbreaks of COVID-19. The technical staff has revised its growth forecast for Colombia’s main trade partners from 6.3% to 6.9% for 2021, and from 3.4% to 3.3% for 2022; trade partner economies are expected to grow 2.6% in 2023. Colombia’s annual terms of trade increased in 2021, largely on higher oil, coffee, and coal prices. This improvement came despite increased prices for goods and services imports. The expected oil price trajectory has been revised upward, partly to supply restrictions and lagging investment in the sector that would offset reduced growth forecasts in some major economies. Elevated freight and raw materials costs and supply chain disruptions continue to affect global goods production, and have led to increases in global prices. Coupled with the recovery in global demand, this has put upward pressure on external inflation. Several emerging market economies have continued to normalize monetary policy in this context. Meanwhile, in the United States, the Federal Reserve has anticipated an end to its asset buying program. U.S. inflation in December (7.0%) was again surprisingly high and market average inflation forecasts for 2022 have increased. The Fed is expected to increase its policy rate during the first quarter of 2022, with quarterly increases anticipated over the rest of the year. For its part, Colombia’s sovereign risk premium has increased and is forecast to remain on a higher path, to levels above the 15-year-average, on the forecast horizon. This would be partly due to the effects of a less expansionary monetary policy in the United States and the accumulation of macroeconomic imbalances in Colombia. Given the above, international financial conditions are projected to be less favorable than anticipated in the October report. The increase in Colombia’s external financing costs could be more significant if upward pressures on inflation in the United States persist and monetary policy is normalized more quickly than contemplated in this report. As detailed in Section 2.3, uncertainty surrounding international financial conditions continues to be unusually high. Along with other considerations, recent concerns over the potential effects of new COVID-19 variants, the persistence of global supply chain disruptions, energy crises in certain countries, growing geopolitical tensions, and a more significant deceleration in China are all factors underlying this uncertainty. The changing macroeconomic environment toward greater inflation and unanchoring risks on inflation expectations imply a reduction in the space available for monetary policy stimulus. Recovery in domestic demand and a reduction in excess productive capacity have come in line with the technical staff’s expectations from the October report. Some upside risks to inflation have materialized, while medium-term inflation expectations have increased and are above the 3% target. Monetary policy remains expansionary. Significant global inflationary pressures and the unexpected increase in the CPI in December point to more persistent effects from recent supply shocks. Core inflation is trending upward, but remains below the 3% target. Headline and core inflation projections have increased on the forecast horizon and are above the target rate through the end of 2023. Meanwhile, the expected dynamism of domestic demand would be in line with low levels of excess productive capacity. An accumulation of macroeconomic imbalances in Colombia and the increased likelihood of a faster normalization of monetary policy in the United States would put upward pressure on sovereign risk perceptions in a more persistent manner, with implications for the exchange rate and the natural rate of interest. Persistent disruptions to international supply chains, a high real increase in the legal minimum wage, and the indexation of various baskets in the CPI to higher inflation rates could affect price expectations and push inflation above the target more persistently. These factors suggest that the space to maintain monetary stimulus has continued to diminish, though monetary policy remains expansionary. 1.2 Monetary policy decision Banco de la República’s board of directors (BDBR) in its meetings in December 2021 and January 2022 voted to continue normalizing monetary policy. The BDBR voted by a majority in these two meetings to increase the benchmark interest rate by 50 and 100 basis points, respectively, bringing the policy rate to 4.0%.