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 (see topological vector space), then it makes sense to ask whether all linear maps are continuous. It turns out that for normed spaces (which are the simplest and most used topologocial vector spaces), the answer is intimately tied to whether the dimension of the domain of the maps is finite or not. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... 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... In mathematics, a function returns a unique output for a given input. ... 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, 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 mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, the domain of a function is the set of all input values to the function. ...

Problem statement

Let X and Y be two normed spaces, and consider linear linear maps f from X to Y. If X is finite-dimensional, choose a base (e₁, e₂, …, e_n) in X for some n ≥ 1. Then, for any

in X one can write

where f_i = f(e_i) for each i. Using the fact that all norms on X are equivalent, it follows that f is a continous map.

If the space X is not necessarily finite-dimensional, but Y is just the zero space {0}, any linear map from X to Y will be constant and trivially continuous.

The non-trivial case is when X is infinite-dimensional and Y is nonzero, and it turns out that in this case one can always find a discontinuous linear map from X to Y.

A concrete example

Before showing a fully general example of a discontinuous linear map, which will be rather abstract and somewhat artificially constructed, it might be instructive to give example of a concrete discontinuous linear map in specific spaces.

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 one of the two central concepts of calculus. ...

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. 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). ...

General example

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 existance 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. ...

Let thus f:X → K be a linear map. Using the linearity property, one can check that f is continuous if and only if f maps the ball The solid interior of a sphere or circle; in mathematics, latter terms refer specifically to the (n-1)-dimensional surface of an n-dimensional solid ball. ...

to a bounded set in K, that is, if and only if f is a bounded linear operator. This property is in turn equivalent to requiring that there exists a constant M ≥ 0 such that In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...

for all x in X.

If X is infinite-dimensional, to show the existence of a linear functional which is not continuous then amounts to constructing f such that equation (1) fails to hold. For that, consider a sequence (e_n)_n (n ≥ 1) of linearly independent vectors in X. Define This is a page about mathematics. ... 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 clearly fails to satisfy (1), 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. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

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. And it emphasizes 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 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. ...

is not a locally convex space, which implies that the only continuous linear map from this space to the underlying field (the reals) is the constantly zero function. In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ...

References

• Constantin Costara, Dumitru Popa, Exercises in Functional Analysis, Springer, 2003. ISBN 1402015607.