* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If the boundary of S is rectifiable, then the Minkowski content is the ( n-1 )- dimensional Hausdorff measure.

If X is any topological space and Y is Hausdorff, then it is straightforward to show that the graph of T is closed whenever T is continuous.

* If Y is T < sub > 0 </ sub >, T < sub > 1 </ sub >, Hausdorff, regular, or Tychonoff, then the compact-open topology has the corresponding separation axiom.

* If X is Hausdorff and S is a subbase for Y, then the collection

If X is an uncountable set, any two open sets intersect, hence the space is not Hausdorff.

If dμ = ƒ dH is the product of a continuous function on S with the ( d − 1 )- dimensional Hausdorff measure, then at a point y of S, the normal derivative undergoes a jump discontinuity ƒ ( y ) when crossing the layer.

If K is a compact Hausdorff space, then C ( K ) is a projectionless algebra if and only if K is connected.

* If X is a compact Hausdorff topological space which satisfies the ccc then X is not the union of k or fewer nowhere dense subsets.

If X is Hausdorff, then A must be closed.

If Y is not locally compact Hausdorff, the exponential object may not exist ( the space Z < sup > Y </ sup > still exists, but it may fail to be an exponential object since the evaluation function need not be continuous ).

If is a second-countable Hausdorff space and contains the Borel sigma-algebra, then the set of recurrent points of has full measure.

If one has defined a homology theory over compact pairs, it is possible to extend it into a compactly supported homology theory in the wider category of Hausdorff pairs ( X, A ) with A closed in X, by defining that the homology of a Hausdorff pair ( X, A ) is the direct limit over pairs ( Y, B ), where Y, B are compact, Y is a subset of X, and B is a subset of A.

If the content of faith is to be presented today in a form that can be `` understanded of the people '' -- and this, it must not be forgotten, is one of the goals of the perennial theological task -- there is no other choice but to abandon completely a mythological manner of representation.

`` If you substitute ' atom ' for ' angel ', the problem is not insoluble, given the metallic content of the pin in question ''.

Immanuel Kant, writing in 1790, observes of a man " If he says that canary wine is agreeable he is quite content if someone else corrects his terms and reminds him to say instead: It is agreeable to me ," because " Everyone has his own ( sense of ) taste ".

If one knows the actual URL of a website that features sexual explicit or 18 + content, they have the ability to access it without using a search engine.

If the content or presentation of a course is inconsistent with preconceived information, the student will learn if he or she is open to varying concepts.

In It's Not Funny If I Have to Explain It, Adams recounts having been attacked for the alleged political content of his work, although in the case of one such strip ( where oil drilling kills a unicorn ) he excuses himself by saying " I just thought the image was funny.

If battle was refused by the defender, they would generally retreat to their city, in which case the attackers generally had to content themselves with ravaging the surrounding countryside, since siegecraft was not efficient, at least until the 5th century BC.

If we should say that no one could possibly know whether water exists a priori, it seems either we cannot know content externalism to be true on the basis of thought experiments or we cannot know what we are thinking without first looking into the world to see what it is like.

If all of the properties of a course can be precisely defined in a common format, the content can be serialized into a standard format such as XML and loaded into other systems.

# If not, it provides a descriptive label for the format of content, with respect to the presence of 8 bit or binary content.

: If any pilgrim monk come from distant parts, if with wish as a guest to dwell in the monastery, and will be content with the customs which he finds in the place, and do not perchance by his lavishness disturb the monastery, but is simply content with what he finds: he shall be received, for as long a time as he desires.

If the organization that is serving the content has control over the network between server and recipients ( i. e., educational, government, and corporate intranets ), then routing protocols such as Protocol Independent Multicast can be used to deliver stream content to multiple Local Area Network segments.

If there exist four ways to sort images, three choices of thumbnail size, two file formats, and an option to disable user-provided content, then the same set of content can be accessed with 48 different URLs, all of which may be linked on the site.

If such content is in a film, particularly violence in a plot development context ( i. e. separate from kids violence ), it is often edited out prior to submission to the OFLC to avoid being " Refused Classification " ( effectively banning the film ).

If not, production houses such as Warner Bros. and Sony Pictures can distribute their content to the different networks, and it is common that a certain production house may have programs on two or more rival networks.

If one uses a flour with a lower protein content ( 9 – 11 %) to produce bread, a longer mixing time will be required to develop gluten strength properly.

If, for example, you begin with 100 kilograms ( 220 pounds ) of natural uranium, it takes about 60 SWU to produce 10 kilograms ( 22 pounds ) of uranium enriched in U-235 content to 4. 5 %

If " good " coins have a face value below that of their metallic content, individuals may be motivated to melt them down and sell the metal for its higher intrinsic value, even if such destruction is illegal.

A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says " If A then B " when in fact A is inherently false.

If the sedimentation rate is low ( about 1 cm / kyr ), the organic carbon content is low ( about 1 % ), and oxygen is abundant, aerobic bacteria can use up all the organic matter in the sediments faster than oxygen is depleted, so lower-energy electron acceptors are not used.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.