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By Ralph P.Jr. Boas, R.C. Buck

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**Example text**

The spaces S(R, R), S(R, Rn ), S(R, C n ) are obviously deﬁned, starting in the ﬁrst case with real trigonometric polynomials instead of T (to be more speciﬁc, with Re T ) and then proceeding by completion. There are many references in the literature to the spaces S p (R, C), 1 ≤ p, of almost periodic functions in Stepanov’s sense. They are deﬁned in a manner similar to that used above for the space S = S 1 (R, C). Namely, the norm in S p (R, C) is given by 1/p t+1 x Sp |x(s)|p ds = sup t∈R , t and the relationship with the norm of S (or M ) follows from the inequality mentioned above, namely t+1 1/p t+1 |x(s)|ds ≤ t |x(s)|p ds 1 ≤ p < ∞.

X| = |λ||x| for any λ ∈ R (or C) and x ∈ E; and 3. |x + y| ≤ |x| + |y| for any x, y ∈ E. 8. The only diﬀerence between a norm and a seminorm is in the fact that a seminorm can vanish for nonzero elements of E. 19, we see that the set x ∈ E for which |x| = 0 constitutes a linear manifold in E. 20. A sequence of seminorms {|x|k ; k ≥ 1} on the linear space E is called suﬃcient if |x|k = 0, k ≥ 1, implies x = θ. In other words, the sequence is suﬃcient if for each x ∈ E, x = θ, there exists a natural number m such that |x|m > 0.

19), then we shall say that E is a Hilbert space if it is a complete metric space, the distance being deﬁned by d(x, y) = x − y . Since it is obvious that a Hilbert space is a Banach space in which the norm is deﬁned by means of an inner product, it is legitimate to ask the question: When is a Banach space a Hilbert space? 11 gives a rather simple answer to this question. 11. Let (E, · ) be a Banach space over the ﬁeld of reals. Then, a necessary and suﬃcient condition for E to be a Hilbert space also is the validity of the parallelogram law in E, x+y 2 + x−y 2 = 2( x 2 + y 2 ), for any x, y ∈ E.