In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological spaces (that is, topological vector spaces), then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on infinite-dimensional normed spaces, the answer is generally no, there exist discontinuous linear maps. If the domain of definition is complete, such maps can be proven to exist but not explicitly constructed; their existence relies on the axiom of choice. Discontinuous linear maps can exist on spaces more general than normed spaces. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... Partial plot of a function f. ... The fundamental concept in linear algebra is that of a vector space or linear space. ... Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

In finite dimensions all linear maps are continuous

Let X and Y be two normed spaces and f a linear map from X to Y. If X is finite-dimensional, choose a base (e₁, e₂, …, e_n) in X which may be taken to be unit vectors. Then, In mathematics, the dimension of a vector space V is the cardinality (i. ...

and so by the triangle inequality, In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...

Letting

and using the fact that

for some C>0 which follows from the fact that any two norms on a finite-dimensional space are equivalent, one finds

Thus, f is a bounded linear operator and so is continuous. In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...

If X is infinite-dimensional, this proof will fail as there is no guarantee that the supremum M exists. If Y is the zero space {0}, the only map between X and Y is the zero map which is trivially continuous. In all other cases, when X is infinite dimensional and Y is not the zero space, one can find a discontinuous map from X to Y. In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...

A concrete example

Examples of discontinuous linear maps are easy to construct in spaces that are not complete; on any Cauchy sequence of independent vectors which does not have a limit, a linear operator may grow without bound. In a sense, the linear operators are not continuous because the space has "holes".

For example, consider the space X of real-valued smooth functions on the interval [0, 1] with the uniform norm, that is, In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ...

The derivative at a point map, given by In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

defined on X and with real values, is linear, but not continuous. Indeed, consider the sequence

for n≥1. This sequence converges uniformly to the constantly zero function, but

as n→∞ instead of which would hold for a continuous map. Note that T is real-valued, and so is actually a linear functional on X (an element of the algebraic dual space X^*). The linear map X → X which assigns to each function its derivative is similarly discontinuous. Note that although the derivative operator is not continuous, it is closed. In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. ...

The fact that the domain is complete here is important. The derivative operator cannot be everywhere-defined on a complete domain. Discontinuous operators on complete spaces require a little more work.

A nonconstructive example

An algebraic basis for the real numbers as a vector space over the rationals is known as a Hamel basis (note that some authors use this term in a broader sense to mean an algebraic basis of any vector space). Note that any two noncommensurable numbers, say 1 and π, are linearly independent. One may find a Hamel basis containing them, and define a map f from R to R so that f(π) = 0, f acts as the identity on the rest of the Hamel basis, and extend to all of R by linearity. Let {r_n}_n be any sequence of rationals which converges to π. Then lim_n f(r_n) = π, but f(π) = 0. Thus f is linear over Q (by construction), but not continuous. Note that f is also not measurable; an additive real function is linear if and only if it is measurable, so for every such function there is a Vitali set. The construction of f relies on the axiom of choice. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... Commensurability in general Generally, two quantities are commensurable if both can be measured in the same units. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... Look up Additive in Wiktionary, the free dictionary When used as a noun, additive refers to something that is introduced to a larger quantity of something else, usually to alter characteristics of the larger quantity. ... In mathematics, the Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable. ...

This example can be extended into a general theorem about the existence of discontinuous linear maps on any infinite-dimensional normed space (as long as the codomain is not trivial).

General existence theorem

Discontinuous linear maps can be proven to exist more generally even if the space is complete. Let X and Y be normed spaces over the field K where K = R or K = C. Assume that X is infinite-dimensional and Y is not the zero space. We will find a discontinuous linear map f from X to K, which will imply the existence of a discontinuous linear map g from X to Y given by the formula g(x) = f(x)y₀ where y₀ is an arbitrary nonzero vector in Y. In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...

If X is infinite-dimensional, to show the existence of a linear functional which is not continuous then amounts to constructing f which is not bounded. For that, consider a sequence (e_n)_n (n ≥ 1) of linearly independent vectors in X. Define In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

for each n = 1, 2, ... Complete this sequence of linearly independent vectors to a vector space basis of X, and define T at the other vectors in the basis to be zero. T so defined will extend uniquely to a linear map on X, and since it is clearly not bounded, it is not continuous.

Notice that by using the fact that any set of linearly independent vectors can be completed to a basis, we implicitly used the axiom of choice, which was not needed for the concrete example in the previous section.

Axiom of choice

As noted above, the axiom of choice (AC) is used in the general existence theorem of discontinuous linear maps. In fact, there are no nonconstructive examples of discontinuous linear maps with complete domain (for example, Banach spaces). In analysis as it is usually practiced by working mathematicians, the axiom of choice is always employed (it is an axiom of ZFC set theory); thus, to the analyst, all infinite dimensional topological vector spaces admit discontinuous linear maps. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

On the other hand, in 1970 Robert M. Solovay exhibited a model of set theory in which every set of reals is measurable. This implies that there are no discontinuous linear real functions. Clearly AC does not hold in the model. Robert M. Solovay is a set theorist who spent many years as a professor at UC Berkeley. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

Solovay's result shows that it is not necessary to assume that all infinite-dimensional vector spaces admit discontinuous linear maps, and there are schools of analysis which adopt a more constructivist viewpoint. For example H. G. Garnir, in searching for so-called "dream spaces" (topological vector spaces from which every linear map into a normed space is continuous), was led to adopt ZF + DC + BP (dependent choice is a weakened form and the Baire property is a weak negation of AC) as his axioms to prove the Garnir-Wright closed graph theorem which states, among other things, that any linear map from an F-space to a TVS is continuous. Going to the extreme of constructivism, there is Ceitin's theorem, which states that every map is continuous (where this is to be understood in an appropriate framework). Such stances are held by only a small minority of working mathematicians. In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ... In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × V → R so that Scalar multiplication in V is continuous with respect to d and the standard metric on R or C. Addition in V is continuous...

The upshot is that it is not possible to obviate the need for AC: all discontinuous linear maps on complete spaces are nonconstructible. A corollary is that constructible discontinuous operators such as the derivative cannot be everywhere-defined on a complete space.

Closed operators

Many naturally occurring linear discontinuous operators occur are closed, a class of operators which share some of the features of continuous operators. It makes sense to ask the analogous question about whether all linear operators on a given space are closed. The closed graph theorem asserts that all everywhere-defined closed operators on a complete domain are continuous, so in the context of discontinuous closed operators, one must allow for operators which are not defined everywhere. Among operators which are not everywhere-defined, one can consider densely-defined operators without loss of generality. In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. ... In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph. ...

Thus let T be a map with domain . The graph Γ(T) of an operator T which is not everywhere-defined will admit a distinct closure . If the closure of the graph is itself the graph of some operator , T is called closable, and is called the closure of T.

So the right question to ask about linear operators that are densely-defined is whether they are closable. The answer is no; one can prove that every infinite-dimensional normed space admits a nonclosable linear operator. The proof requires the axiom of choice and so is in general nonconstructive, though again, if X is not complete, there are constructible examples.

In fact, an example of a linear operator whose graph has closure all of X×Y can be given. Such an operator is not closable. Let X be the space of polynomial functions from [0,1] to R and Y the space of polynomial functions from [2,3] to R. They are subspaces of C([0,1]) and C([2,3]) respectively, and so normed spaces. Define an operator T which takes the polynomial function x ↦ p(x) to on [0,1] to the same function on [2,3]. As a consequence of the Stone-Weierstrass theorem, the graph of this operator is dense in X×Y, so this provides a sort of maximally discontinuous linear map (confer nowhere continuous function). Note that X is not complete here, as must be the case when there is such a constructible map. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. ...

Impact for dual spaces

The dual space of a topological vector space is the collection of continuous linear maps from the space into the underlying field. Thus the failure of some linear maps to be continuous for infinite-dimensional normed spaces implies that for these spaces, one needs to distinguish the algebraic dual space from the continuous dual space which is then a proper subset. It illustrates the fact that an extra dose of caution is needed in doing analysis on infinite-dimensional spaces as compared to finite-dimensional ones. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

Beyond normed spaces

The argument for the existence of discontinuous linear maps on normed spaces can be extended to more general classes of topological vector spaces. For example, replacing the norm with a seminorm and adjusting the action appropriately, one sees that every infinite-dimensional locally convex space has discontinuous linear functionals. On the other hand, the Hahn-Banach theorem, which applies to all locally convex spaces, guarantees the existence of many continuous linear functionals, and so a large dual space. In fact, to every convex set, the Minkowski gauge associates a continuous linear functional. The upshot is that spaces with fewer convex sets has fewer functionals, and in the worst case scenario, a space may have no functionals at all other than the zero functional. This is the case for the L^p(R,dx) spaces with 0<p<1, from which it follows that these spaces are nonconvex. Note that here is indicated the Lebesgue measure on the real line. There are other L^p spaces with 0<p<1 which do have nontrivial dual spaces. In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In mathematics, the Lp and lp spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...

Another such example is the space of real-valued measurable functions on the unit interval with quasinorm given by In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

This non-locally convex space has a trivial dual space.

One can consider even more general spaces. For example, the existence of a homomorphism between complete separable metric groups can also be shown nonconstructively. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

References

• Constantin Costara, Dumitru Popa, Exercises in Functional Analysis, Springer, 2003. ISBN 1402015607.
• Schechter, Eric, Handbook of Analysis and its Foundations, Academic Press, 1997. ISBN 0-12-622760-8.