the same as vector space (seeVECTOR SPACE). Functional analysis is primarily concerned with infinite-dimensional spaces. The space of all polynomials (with real or complex coefficients), with the usual definitions of addition and multiplication by numbers, is an example of an infinite-dimensional vector space. The hilbert space and the space C[a, b] of continuous functions defined on an interval [a, b] were among the first examples of an infinite-dimensional vector space. These spaces are normed, that is, they are vector spaces in which each element x has a norm—a nonnegative number ǀǀxǀǀ that vanishes only when x = 0 and has the properties ǀǀλxǀǀ = ǀλǀ ǀǀxǀǀ and ǀǀx + yǀǀ < ǀǀxǀǀ + ǀǀyǀǀ (the triangle inequality). The number ǀǀx – yǀǀ is called the distance between the elements x and y. In a normed vector space the concepts of open sphere, the limit point of a set, and continuity of a functional are introduced in much the same way as in three-dimensional space.

In a finite-dimensional space all norms are topologically equivalent: a sequence of points that converges in one norm converges in any other. In an infinite-dimensional space norms may differ greatly. For example, in solving P. L. Chebyshev’s problem of finding the polynomial that deviates least from zero (the problem of best approximation), it is necessary to find a polynomial P_{k - 1}(t) of degree k – 1 such that

has minimum value. In terms of the norm

in the space C[0,1] we can formulate our problem as follows: find the polynomial P_{k - 1}(t) whose distance from the function t^k is minimal. In studying polynomials that are orthogonal relative to the weight function p(t), it is natural to consider the norm

and to solve the problem of best approximation with respect to this norm. The norms ǀǀxǀǀ₁ and ǀǀxǀǀ₂ are radically different. Thus, for example, the sequence of functions

diverges in the first norm, but in the second norm, with p(t) = 1, it converges to the function

It should be noted that although all the functions x_n(t) are continuous, the function x(t) is discontinuous. This is related to the fact that the space of continuous functions with the norm ǀǀxǀǀ₂ is not complete. A normed vector space is said to be complete if for any sequence {x_n} of its elements satisfying the condition

there exists an element x in the space such that {x_n} converges to x, that is,

If the vector space is incomplete, then new elements must be added to it in such a way that it becomes complete. For example, the Hubert space L_p² is obtained by completing the space of continuous functions with the norm ǀǀxǀǀ₂. Complete normed vector spaces are called Banach spaces, after S. Banach, who studied their main properties.

The concept of topological vector space is a generalization of the concept of Banach space. Specifically, a set E is called a topological vector space if (1) it is a vector space, (2) it is a topological space, and (3) the operations of addition and multiplication by numbers defined in E are continuous with respect to the topology in E. All normed spaces are topological vector spaces. A. N. Kolmogorov established in 1934 the necessary and sufficient conditions for the normability of topological vector spaces.