* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

If E / F is a Galois extension, then Gal ( E / F ) can be given a topology, called the Krull topology, that makes it into a profinite group.

If F is equipped with the order topology arising from the total order ≤, then the axioms guarantee that the operations + and * are continuous, so that F is a topological field.

For compact space | compact 2-dimensional surfaces without boundary ( topology ) | boundary, if every loop can be continuously tightened to a point, then the surface is topologically Homeomorphism | homeomorphic to a 2-sphere ( usually just called a sphere ).

At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology — what would later be called combinatorial topology and then algebraic topology.

If N is a closed normal subgroup of a profinite group G, then the factor group G / N is profinite ; the topology arising from the profiniteness agrees with the quotient topology.

Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points.

The function N is called a neighbourhood topology if the axioms below are satisfied ; and then X with N is called a topological space.

is a continuous map from a closed subset A of X into the real numbers carrying the standard topology, then there exists a continuous map

More generally, if F is a subset of the algebraic dual space, then the initial topology of X with respect to F, denoted by σ ( X, F ), is the weak topology with respect to F.

If one takes F to be the whole continuous dual space of X, then the weak topology with respect to F coincides with the weak topology defined above.

If the field K has an absolute value, then the weak topology σ ( X, F ) is induced by the family of

Specifically, if F is a vector space of linear functionals on X which separates points of X, then the continuous dual of X with respect to the topology σ ( X, F ) is precisely equal to F.

If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space.

An important fact about the weak * topology is the Banach – Alaoglu theorem: if X is normed, then the closed unit ball in X * is weak *- compact ( more generally, the polar in X * of a neighborhood of 0 in X is weak *- compact ).

If a normed space X is separable, then the weak -* topology is metrizable.

For instance, if Y is a normed space, then this topology is defined by the seminorms indexed by x ∈ X:

More generally, if a family of seminorms Q defines the topology on Y, then the seminorms p < sub > q, x </ sub > on L ( X, Y ) defining the strong topology are given by

The finished -- and drastically cut -- product would begin with a hazy longshot of Joyce entering the suds, then bursting above the pool's surface clad in layers of lavender lather, and I had a hunch this item was going to sell tons and tons of soap ; ;

Without the good magazines, without their book reviews, their hospitality to European writers, without above all their awareness of literary standards, we might very well have had a generation of Krim's heroes -- Wolfes, Farrells, Dreisers, and I might add, Sandburgs and Frosts and MacLeishes in verse -- and then where would we be??

( For each State, make all computations set forth in items 1 to 8 above, and then add the results obtained for each State in item 8.

If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

Purified inactive chlorine was then added from one of the tubes described above and the mixture frozen out and sealed off in a flask equipped with a break seal.

It then follows that Af is large enough, but this is not obvious from the above expression.

There were three houses in Salyer's Canyon just at the foot of a low bluff, the road winding along the top, entering above, and then passing down in front of the houses, thence to the Vermejo.

At the very first, then, Brumidi was required, by the classically pyramidal shape of his central group, to fill in the triangular space above the seated girl on Liberty's right, before starting on the allegorical figures themselves.

Thus, in the example cited above Fromm rests his whole case on the premise that the workers are being deprived unconsciously, unknowingly, of fulfillment, and then supports this with survey data reporting conscious, experienced frustrations.

The 1899 act mentioned above was replaced in 1997 — until then the government had stated there were no ethnic minority groups.

The Seljuq Turks then arrived in 1071 and changed its name to Kara Hissar (" black castle ") after the ancient fortress situated upon a volcanic rock 201 meters above the town.

The Doya trace their descent matrilineally, marry their cross cousins, and embalm the deceased who are then placed in a foetal position in a circular sarcophagus above the ground.

Disraeli wrote a personal letter to Gladstone, asking him to place the good of the party above personal animosity: " Every man performs his office, and there is a Power, greater than ourselves, that disposes of all this ..." In responding to Disraeli Gladstone denied that personal feelings played any role in his decision then and previously to accept office, while acknowledging that there were differences between him and Derby " broader than you may have supposed.

The Bastille Day Military Parade opens with cadets from the École Polytechnique, Saint-Cyr, École Navale, and so forth, then other infantry troops, then motorized troops ; aircraft of the Patrouille de France aerobatics team fly above.

Analogous to the above reactions, the glucose produced can then undergo glycolysis in tissues that need energy, be stored as glycogen ( or starch in plants ), or be converted to other monosaccharides or joined into di-or oligosaccharides.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

In an " extended model " which includes hot dark matter in the form of neutrinos, then if the " physical baryon density " Ω < sub > b </ sub > h < sup > 2 </ sup > is estimated at about 0. 023 ( this is different from the ' baryon density ' Ω < sub > b </ sub > expressed as a fraction of the total matter / energy density, which as noted above is about 0. 046 ), and the corresponding cold dark matter density Ω < sub > c </ sub > h < sup > 2 </ sup > is about 0. 11, the corresponding neutrino density Ω < sub > v </ sub > h < sup > 2 </ sup > is estimated to be less than 0. 0062.

If the first theory is correct ( see above ), then the absence of cavalry removed the main Athenian tactical disadvantage, and the threat of being outflanked made it imperative to attack.

If the character encoding for a web page is chosen appropriately then HTML character references are usually only required for a markup delimiting characters mentioned above, and for a few special characters ( or not at all if a native Unicode encoding like UTF-8 is used ).

Calcium hydroxide ( slaked lime ) is used in many chemical refinery processes and is made by heating limestone at high temperature ( above 825 ° C ) and then carefully adding water to it.

The acid then dissolves the limestone from below, rather than from above, by acidic water percolating from the surface.

Non-avian dinosaur fossils are found only below the K – T boundary, indicating that they became extinct during the boundary event .< ref > A very small number of dinosaur fossils have been found above the K – T boundary, but they have been explained as reworked fossils, that is, fossils that have been eroded from their original locations then preserved in later sedimentary layers.

An example of an editing error can be seen in the film It's a Mad, Mad, Mad, Mad World ( 1963 ), where a scene of people climbing a slope at the start is seen from below and then replayed from above.

If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1 / g ′( f ( 0 )).