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11 THE SPACES L¹ AND L² 163

We remark that, given /e JS?i(X, /z), it is not always possible to find a
function/' equivalent to /and a sequence of functions/, e «?T_R(X) such that
the sequence (/„(*)) converges to/'O) for allxeX (Problem 3).

We have already defined (13.10) the vector space -S?£(X, /*) of complex-
valued integrable functions. Likewise, we denote by <£ £(X, ju) (or <£ cOu), or
JS?£) the set of c0ra/?/ex-valued /i-measurable functions such that \f\² is
ji-integrable, and we define N^/) and N₂(/) by the formula (13.11.1) for
any complex-valued function/ Since a complex-valued function is measurable
if and only if ^?/and .//are measurable, and since

it follows that

#S(X,Ai) = JS? J(X,M) 0 fcS?2

All the properties of J$?£(X, ju) proved above extend immediately to
JSf£(X,/i) Q? = l,2), and the Banach space JS? J(X, //) (also denoted by
^ c(j") ^or «^c) ^ defined exactly as in the real case.

Furthermore, we have

(13.11.7) Iff, g belong to -SfJ(X, /*) (resp. JS?£(X, ^)) then their product f g
belongs to 4fi(X, ji) (resp. Jif i(X, ju)). T/ze space Lj(X, ^) (resp. Lj(X, ji))
w « separable Hilbert space with respect to the Hermitian form

(13.11.7.1) (/, g) h+(/|0) = tfWtfddrtx),

and the corresponding norm is N₂(/).

The product/^ is measurable (13.9.8.1), hence the first assertion follows
from (13.9.13) and (13.11.2.2). The second follows from the first together
with the Fischer-Riesz theorem (13.11.4) and (13.11.6).

PROBLEMS

1. Let A be Lebesgue measure on I = [0,1 [. For each integer « = 2"+ &(()<;/:< 2"),
let/_n be the function which is equal to 1 on the interval [k • 2~^h, (k+ 1) -2~*[,
and zero elsewhere. Show that the sequence (/„) converges to 0 in mean and in square
mean, but that the sequence (/„(*)) does not converge for any x e I,



	11 THE SPACES L¹ AND L² 163 We remark that, given /e JS?i(X, /z), it is not always possible to find a function/' equivalent to /and a sequence of functions/, e «?T_R(X) such that the sequence (/„()) converges to/'O) for allxeX* (Problem 3). We have already defined (13.10) the vector space -S?£(X, /) of complex- valued integrable functions. Likewise, we denote by <£ £(X, ju) (or <£ cOu), or JS?£) the set of c0ra/?/ex-valued /i-measurable functions such that \f\²* is ji-integrable, and we define N^/) and N₂(/) by the formula (13.11.1) for any complex-valued function/ Since a complex-valued function is measurable if and only if ^?/and .//are measurable, and since

	it follows that #S(X,Ai) = JS? J(X,M) 0 fcS?2 All the properties of J$?£(X, ju) proved above extend immediately to JSf£(X,/i) Q? = l,2), and the Banach space JS? J(X, //) (also denoted by ^ c(j") ^or «^c) ^ defined exactly as in the real case.

	Furthermore, we have (13.11.7) Iff, g belong to -SfJ(X, /) (resp. JS?£(X, ^)) then their product f g belongs to 4fi(X, ji) (resp. Jif i(X, ju)). T/ze space* Lj(X, ^) (resp. Lj(X, ji)) w « separable Hilbert space with respect to the Hermitian form (13.11.7.1) (/, g) h+(/\|0) = tfWtfddrtx), and the corresponding norm is N₂(/). The product/^ is measurable (13.9.8.1), hence the first assertion follows from (13.9.13) and (13.11.2.2). The second follows from the first together with the Fischer-Riesz theorem (13.11.4) and (13.11.6).

	PROBLEMS 1. Let A be Lebesgue measure on I = [0,1 [. For each integer « = 2"+ &(()<;/:< 2"), let/_n be the function which is equal to 1 on the interval [k • 2~^h, (k+ 1) -2~[, and zero elsewhere. Show that the sequence (/„) converges to 0 in mean and in square mean, but that the sequence (/„()) does not converge for any x e I,