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In , the support of a
is the set of points where the function is not zero-valued or, in the case of functions defined on a topological space, the
of that set. This concept is used very widely in . In the form of functions with support that is bounded, it also plays a major part in various types of
Suppose that f : X → R is a real-valued function whose
is an arbitrary set X. The set-theoretic support of f, written supp(f), is the set of points in X where f is non-zero
The support of f is the smallest subset of X with the property that f is zero on its complement, meaning that the non-zero values of f "live" on supp(f). If f(x) = 0 for all but a finite number of points x in X, then f is said to have finite support.
If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. The notion of support also extends in a natural way to functions taking values in more general sets than R and to other objects, such as measures or distributions.
The most common situation occurs when X is a
(such as the
or n-dimensional Euclidean space) and f : X → R is a continuous real (or complex)-valued function. In this case, the support of f is defined topologically as the
of the subset of X where f is non-zero i.e.,
Since the intersection of closed sets is closed, supp(f) is the intersection of all closed sets that contain the set-theoretic support of f.
For example, if f : R → R is the function defined by
then the support of f is the closed interval [-1,1], since f is non-zero on the open interval (-1,1) and the closure of this set is [-1,1].
The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on a topological space, and some authors do not require that f : X → R (or C) be continuous.
Functions with compact support on a topological space X are those whose support is a
subset of X. If X is the real line, or n-dimensional Euclidean space, then a function has compact support if and only if has bounded support, since the support is closed by definition and a subset of Rn is compact if and only if it is closed and bounded.
For example, the function f : R → R defined above is a continuous function with compact support [-1,1].
The condition of compact support is stronger than the condition of . For example, the function f : R → R defined by
vanishes at infinity, since f(x) →  0 as |x| → ∞, but its support R is not compact.
Real-valued compactly supported
are called .
are an important special case of bump functions as they can be used in
of smooth functions approximating nonsmooth (generalized) functions, via .
In , functions with compact support are
in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. As an intuition for more complex examples, and in the language of , for any ε & 0, any function f on the real line R that vanishes at infinity can be approximated by choosing an appropriate compact subset C of R such that
for all x ∈ X, where
of C. Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.
If X is a topological measure space with a Borel measure μ (such as Rn, or a Lebesgue measurable subset of Rn, equipped with Lebesgue measure), then one typically identifies functions that are equal μ-almost everywhere. In that case, the essential support of a measurable function f : X → R, written ess supp(f), is defined to be the smallest closed subset F of X such that f = 0 μ-almost everywhere outside F. Equivalently, ess supp(f) is the complement of the largest open set on which f = 0 μ-almost everywhere
The essential support of a function f depends on the measure μ as well as on f, and it may be strictly smaller than the closed support. For example, if f : [0,1] → R is the
that is 0 on irrational numbers and 1 on rational numbers, and [0,1] is equipped with Lebesgue measure, then the support of f is the entire interval [0,1], but the essential support of f is empty, since f is equal almost everywhere to the zero function.
In analysis one nearly always wants to use the essential support of a function, rather than its closed support, when the two sets are different, so ess supp(f) is often written simply as supp(f) and referred to as the support.
If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f : X→M. M may also be any
(such as a , , or ), in which the identity element assumes the role of zero. For instance, the family ZN of functions from the
set of integer sequences. The subfamily { f  in ZN :f  has finite support } is the countable set of all integer sequences that have only finitely many nonzero entries.
For more details on this topic, see .
In , the support of a
can be loosely thought of as the closure of the set of possible values of a random variable having that distribution. There are, however, some subtleties to consider when dealing with general distributions defined on a , rather than on a topological space.
Note that the word support can refer to the
It is possible also to talk about the support of a , such as the
δ(x) on the real line. In that example, we can consider test functions F, which are
with support not including the point 0. Since δ(F) (the distribution δ applied as
to F) is 0 for such functions, we can say that the support of δ is {0} only. Since
(including ) on the real line are special cases of distributions, we can also speak of the support of a measure in the same way.
Suppose that f is a distribution, and that U is an open set in Euclidean space such that, for all test functions
such that the support of
is contained in U, . Then f is said to vanish on U. Now, if f vanishes on an arbitrary family
of open sets, then for any test function
supported in , a simple argument based on the compactness of the support of
and a partition of unity shows that
as well. Hence we can define the support of f as the complement of the largest open set on which f vanishes. For example, the support of the Dirac delta is .
in particular, it is interesting to study the singular support of a distribution. This has the intuitive interpretation as the set of points at which a distribution fails to be a smooth function.
For example, the
can, up to constant factors, be considered to be 1/x (a function) except at x = 0. While x = 0 is clearly a special point, it is more precise to say that the transform of the distribution has singular support {0}: it cannot accurately be expressed as a function in relation to test functions with support including 0. It can be expressed as an application of a
improper integral.
For distributions in several variables, singular supports allow one to define
and understand
in terms of . Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails – essentially because the singular supports of the distributions to be multiplied should be disjoint).
An abstract notion of family of supports on a
X, suitable for , was defined by . In extending
that are not compact, the 'compact support' idea enters naturally on one see for example .
Bredon, Sheaf Theory (2nd edition, 1997) gives these definitions. A family Φ of closed subsets of X is a family of supports, if it is
and closed under . Its extent is the union over Φ. A paracompactifying family of supports that satisfies further than any Y in Φ is, with the , and has some Z in Φ which is a . If X is a , assumed
the family of all
satisfies the further conditions, making it paracompactifying.
Folland, Gerald B. (1999). Real Analysis, 2nd ed. New York: John Wiley. p. 132.
H?rmander, Lars (1990). Linear Partial Differential Equations I, 2nd ed. Berlin: Springer-Verlag. p. 14.
Pascucci, Andrea (2011). PDE and Martingale Methods in Option Pricing. Berlin: Springer-Verlag. p. 678. :.  .
Rudin, Walter (1987). Real and Complex Analysis, 3rd ed. New York: McGraw-Hill. p. 38.
(2001). Analysis. Graduate Studies in Mathematics 14 (2nd ed.). . p. 13.  .
In a similar way, one uses the
of a measurable function instead of its supremum.
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Code is Poetry.support的具体用法。。。。后面加名词，。动词的都要。。。人和物的也要_百度作业帮
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support的具体用法。。。。后面加名词，。动词的都要。。。人和物的也要
support的具体用法。。。。后面加名词，。动词的都要。。。人和物的也要
常用的用法：support / verb1. 支撑；支持The dome was supported by a hundred white columns. 这个圆屋顶由100根白柱支撑。2. 供应（足够的食品和水）；维持The land had lost its capacity to support life. 这块土地已经失去了维持人们生活的...Support | Ubuntu
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