We formulate a general approximation problem involving reflexive and smooth Banach spaces, and give its explicit solution. Two applications are presented— the first is to the Bounded Completion Problem involving approximation of Hardy class functions, while the second involves the construction of minimal vec- tors and hyperinvariant subspaces of linear operators, generalizing the Hilbert space technique of Ansari and Enflo

Chalendar, I.

Partington, J.R.

Smith, M.

English

View looking up in the Chancel; The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

Montreuil, Pierre de

Chelles, Jean de

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Notre-Dame Cathedral

View from the south transept into the nave, looking northwest; The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

View looking up just inside the front central portal (Portal of the Last Judgment); The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

General view of the sacristy; The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

East façade (apse with flying buttresses), distant view from east; The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

General view of the High Altar; The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

Close view of head of gargoyle; The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

View of the south transept, depicting upper southwest corner; The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

View along the choir seats or stalls (by Robert de Cotte); The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

View looking up edge of roof, depicting gargoyles; The plan as conceived by the first architect was highly original. It included a five-aisled nave of five double bays, a wide non-projecting transept and a choir of two double bays flanked by double aisles, which continued into a double ambulatory. The chapels were contained within the ambulatory wall. The north transept wall was built ca. 1250 by Jean de Chelles with a sculptured portal and a glazed triforium surmounted by an enormous rose window. He designed an identical arrangement for the south arm, but the project was continued by Pierre de Montreuil from 1258. In the 14th century Jean Ravy, continuing the work of Pierre de Chelles, enlarged the gallery windows of the chevet, decorating them with gables, and put up fantastic flying buttresses with a single span of 15 m. Source: Grove Art Online; http://www.groveart.com/ (accessed 2/9/2008

https://eprints.whiterose.ac.uk/746/1/partingtonjr8_ijm.pdf

Approximation in reflexive Banach spaces and applications to the invariant subspace problem

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