When the Soviets recaptured Lwów, Banach reestablished the University.

When World War I broke out, Banach was excused from military service due to his left-handedness and poor vision.

When the Russian Army opened its offensive toward Lwów, Banach left for Kraków, where he spent rest of the war.

When the theory of distributions was still not widely known nor used, property above was formulated by saying that the convolution of the function ' with a given function belonging to a proper Hilbert or Banach space converges as ε → 0 to this last one: this is exactly what Friedrichs did .< ref > See, properties PI, PII, PIII and their consequence PIII < sub > 0 </ sub >.</ ref > This also clarifies why mollifiers are related to approximate identities.

When A and B are Banach spaces a cross norm p on the algebraic tensor product

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

When it was reported that the Lambert W function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges — a fundamental problem in physics — Corless and developers of the Maple Computer algebra system made a library search to find that this function was in fact ubiquitous to nature.

When the algebra can be endowed with a metric, then the structure coefficients are written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations.

When treated in terms of homogeneous coordinates, projective geometry looks like an extension or technical improvement of the use of coordinates to reduce geometric problems to algebra, an extension reducing the number of special cases.

For example, the p < sub > i </ sub > may be the factors of the square-free factorization of g. When K is the field of the rational numbers, as it is typically the case in computer algebra, this allows to replace factorization by greatest common divisor to compute the partial fraction decomposition.

When one wants to work over a base ring R ( commutative ), there is the group scheme concept: that is, a group object in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra.

When abstract algebra was first being developed, the definition of a field usually did not include commutativity of multiplication, and what we today call a field would have been called either a commutative field or a rational domain.

When Σ is a 2-torus the states correspond to the integrable representations of the affine Lie algebra corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern – Simons theory.

When generalized to a Grassmann algebra, they also appear in quantum field theory.

When the algorithm succeeds, it has effectively solved the word problem for the specified algebra.

When n = 2, this is also sometimes called the Clifford algebra of an infinite separable Hilbert space.

When the Lie algebra is semisimple, its Killing form is nondegenerate, and hence can be used as a metric tensor to raise and lower indexes.

When negation is also present in T, then A is a Boolean algebra, provided the logic is classical.

When we have generated enough of these relations ( it's generally sufficient that the number of relations be a few more than the size of P ), we can use the methods of linear algebra ( for example, Gaussian elimination ) to multiply together these various relations in such a way that the exponents of the primes on the right-hand side are all even:

When n is even the algebra Cℓ < sub > n </ sub >( C ) is central simple and so by the Artin-Wedderburn theorem is isomorphic to a matrix algebra over C. When n is odd, the center includes not only the scalars but the pseudoscalars ( degree n elements ) as well.

When the order of G is divisible by the characteristic of K, the group algebra is not semisimple, hence has non-zero Jacobson radical.

** When talking about indefinite forms in multilinear algebra, see definite bilinear form.

When A is an operator algebra, the ranges of these projections are the initial and final subspaces of W respectively.

When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Dynkin diagrams and these correspond to twisted affine Lie algebras.

When Heidegger and Sartre speak of a contrast between being and existence, they may be right, I don't know, but their language is too philosophical for me.

When they express themselves it is incandescent hatred that shines forth, the rage of repudiation, the ecstasy of negation.

When I try to work out my reasons for feeling that this passage is of critical significance, I come up with the following ideas, which I shall express very briefly here and revert to in a later essay.

When the proprietor dies, the establishment should become a corporation until it is either acquired by another proprietor or the government decides to drop it.

When the reactionary response is thus bolstered by an intellectual defense, the characteristics of that defense are explicable only in terms of the basic attitudes of unanalyzed reaction.

When I take over Taliesin, the first thing I'll do is fire you ''.

When someone in the audience rose and asked how does it feel to be a celebrity, Carl said, `` A celebrity is a fellow who eats celery with celerity ''.

When decision makers act within this frame they determine whether a claim put forward in the name of religion is to be accepted by the larger community as appropriate to religion.

When we consider the disorganized state of the world community, and the legacy of predispositions adversely directed against all who are identified as Jews, it is obvious that the struggle for the minds and muscles of men needs to be prosecuted with increasing vigor and skill.

When we turn to Aristotle's ideas on the moral measure of literature, it is at once apparent that he is at times equally concerned about the influence of the art.

When their faith in civil liberties is tested against strong pressures of social expediency in specific issues, e.g., suppression of `` dangerous ideas '', many waver and give in.

There is, of course, nothing new about dystopias, for they belong to a literary tradition which, including also the closely related satiric utopias, stretches from at least as far back as the eighteenth century and Swift's Gulliver's Travels to the twentieth century and Zamiatin's We, Capek's War With The Newts, Huxley's Brave New World, E. M. Forster's `` The Machine Stops '', C. S. Lewis's That Hideous Strength, and Orwell's Nineteen Eighty-Four, and which in science fiction is represented before the present deluge as early as Wells's trilogy, The Time Machine, `` A Story Of The Days To Come '', and When The Sleeper Wakes, and as recently as Jack Williamson's `` With Folded Hands '' ( 1947 ), the classic story of men replaced by their own robots.

When I first came across Samuel Johnson's pronouncement, `` the remedy for the ills of life is palliative rather than radical '', it seemed to me to sum up the profoundest of political and social truths.

When all else fails, it is argued that open sessions slow down governmental operations.

When a person has thoughtlessly or deliberately caused us pain or hardship it is not always easy to say, `` Just forget it ''.

When a person meets God's requirements for the experience of forgiveness he is forgiven.

When I hold my son he stiffens his whole body in my arms until he is as straight and stiff as a board.

When different colors are used, she is just as likely to color trees purple, hair green, etc..

When we become firm enough to stand for those ideals which we know to be right, when we become hard enough to refuse to aid nations which do not permit self-determination, when we become strong enough to resist any more drifts towards socialism in our own Nation, when we recognize that our enemy is Communism not war, and when we realize that concessions to Communists do not insure peace or freedom, then, and only then will we no longer be `` soft ''.

When one powerful nation strives to emulate the success of another, it is only natural.

When a man is laid to rest, he is entitled to stay put.

When I talked to Ching about it, he said, Everyone can learn, if he is not a Reactionary or lazy.