Functionals and Dual Space

A sequence x in a metric space (AT, d) is referred to as a Cauchy sequence or fundamental sequence if d xm, x ) 0 form, n - oo. A convergent sequence is 

A metric space (X,d) is complete if any Cauchy sequence in X,d) is convergent. The space of real numbers R is complete under d x, y) = x — y, while the space of rational numbers Q is not complete under the same metric, since, e.g., e = 1 +1 +1/2 +1/3 H— (i.e., the l.h.s. term is an irrational number, while the r.h.s. terms are rational numbers). 

An operation to add all the limits of Cauchy sequences to X, d) is referred to as a completion. We recall that the space of real numbers R is a completion of the space of rational numbers Q under the uniform norm .  

A complete normed space is referred to as a Banach space, and a complete inner product space is a Hilbert space. 

A set of all linear transformations LfJA, M) from a LVS U = AT, +, into the real number space R is referred to as a dual space, which is denoted as U. An element 

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