Lp spaces, Hilbert space. Schwartz�s, Holder�s, Minkowski�s inequalities. Convergence in the mean. Cauchy sequences. Riesz-Fischer theorem. Convergence in measure.

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L^p spaces, Hilbert space. Schwartz’s, Holder’s, Minkowski’s inequalities. Convergence in the mean. Cauchy sequences. Riesz-Fischer theorem. Convergence in measure.

L^p spaces

Def. Function of class L^p. A function f(x) is of class L^p on an interval (or measurable set) Ω if it is (Lebesgue) measurable and the Lebesgue integral of | f(x)|^p over Ω is finite.

Example. A function f(x) is of class L³ on the interval [a, b] if the Lebesgue integral

is finite.

Def. L^p space. The set of all functions of class L^p (for a particular interval).

Syn. Lebesgue space

The L^p space corresponding to the interval [a, b] is denoted by L^p[a, b] (or simply L^p, if the particular interval is not required). We then say that f(x) belongs to L^p [a, b] or f(x) ε L^p [a, b]. If p = 1, we denote L^p by L.

Example. L³[0, 8] consists of all functions f(x) for which the integral

is finite.

Def. Hilbert space. The space L² consisting of all functions f(x) on an interval [a, b] for which the (Lebesgue) integral

is finite. Functions belonging to Hilbert space are often said to be square integrable.

Observation. Note that most functions that are well-behaved over an interval [a, b] (i.e. they don’t become infinite at some point in the interval), would qualify for membership in all the L^pclasses. Thus most functions that are bounded and well-behaved on an interval can be expected to be members of Hilbert (L²) space, L³ space, L⁴ space, .... , L^p space. For example, the functions shown in Figs. 1a and 1b would be members of all of these spaces. It is some functions with behavior problems that would not qualify.

Important inequalities

1. Schwartz’s inequality. Let f(x) and g(x) be real functions such that f(x) ε L² and g(x) ε L². Then Schwartz’s inequality is

Equality holds if and only if f(x)/g(x) is constant almost everywhere.

2. Holder’s inequality.

valid for

and f(x) ε L^p and g(x) ε L^q. The functions f(x) and g(x) may be real or complex.

Equality holds if and only if |f(x)|^p/|g(x)|^q is constant almost everywhere.

If p = 2 and q = 2, Holder’s inequality reduces to Schwartz’s inequality.

3. Minkowski’s inequality.

where p 1 and f(x) ε L^p and g(x) ε L^p.

Equality holds if f(x)/g(x) is constant almost everywhere.

L^p spaces as metric spaces and linear spaces. If, on L^p, addition and multiplication by scalars are taken as ordinary addition and multiplication, the space L^p becomes a vector space. If a “length” or norm

is defined for the vectors of the space, the space becomes a normed vector space. Minkowski’s inequality then becomes the triangle inequality

and Holder’s inequality becomes

providing f is of class L^p , g is of class L^q and

If a distance function

is defined for the space, the space then becomes a metric space. As a metric space, the following inequality holds

The functions are called vectors if the space is being regarded as a vector space or points if it is being regarded as a metric space.

It is assumed that two functions f and g which are equal almost everywhere represent the same point in space since the distance d(f, g) between them as computed by 1) will be zero.

Theorems

Theorem 1. If f(x) ε L^p where p > 1, then f(x) ε L. In other words, L^p L. More generally if p > n 1, then L^p Lⁿ_.

Theorem 2. If g(x) ε L^p and |f(x)| |g(x)|, then f(x) ε L^p.

Theorem 3. If f(x) ε L^p and g(x) ε L^p, then f(x)g(x) ε L^p/2_. In particular if f(x) ε L² and f(x) ε L² then f(x)g(x) ε L.

Theorem 4. If f(x) ε L^p and g(x) ε L^q where

then f(x)g(x) ε L_.

Theorem 5. If f(x) ε L^p and g(x) ε L^p, then f(x) g(x) ε L^p.

Def. Convergence in the mean. Let {f_n(x)}be a sequence of functions which belong to L^p[a, b]. If there exists a function f(x) ε L^p such that

we say that the sequence {f_n(x)} converges in the mean or is mean convergent to f(x) in the space L^p.

If a sequence {f_n(x)} converges in the mean to f(x) we often write this as

which is read “the limit in mean of f_n(x) as n → ∞ is f(x)”

Equivalently we can say that f_n(x) approaches f(x) in the mean if for every ε > 0 there exists a number n₀ > 0 such that

||fn(x) - f(x)|| < ε whenever n > n₀

Theorem 6. If

exists, it is unique.

Cauchy sequences in L^p spaces. A sequence of functions {f_n(x)} is said to be a Cauchy sequence if

or, in other words, if given ε > 0, there exists a number n₀ > 0 such that

whenever m > n₀, n > n₀.

Theorem 7. If a sequence {f_n(x)} converges in the mean to a function f(x) in L^p, then {f_n(x)} is a Cauchy sequence.

Completeness of an L^p space. An L^p space is said to be complete if every Cauchy sequence in the space converges in the mean to a function in the space.

Riesz-Fischer theorem. Any L^p space is complete.

Thus we see that in an L^p space every Cauchy sequence converges in the mean to a function in the space. This function is unique apart from a set of measure zero.

Def. Convergence in measure. Let {f_n(x)} be a sequence of measurable functions defined almost everywhere. Then {f_n(x)} is said to converge in measure to f(x) if

for all δ (where m refers to the measure of the indicated set).

Theorem 8. If the sequence {f_n(x)}converges almost everywhere to f(x), then it converges in measure to f(x).

Theorem 9. If the sequence {f_n(x)}converges in the mean to f(x), then it converges in measure to f(x).

Theorem 10. If the sequence {f_n(x)}converges in measure to f(x) on a set E, then there exists a subsequence which converges almost everywhere to f(x).

Theorem 11. If the sequence {f_n(x)}converges in the mean to f(x) on a set E, then there exists a subsequence which converges almost everywhere to f(x).

We note that if a sequence {f_n(x)}converges to f(x) everywhere, it does not necessarily converge in the mean to f(x). Conversely, if a sequence {f_n(x)}converges in the mean to f(x), it does not necessarily converge almost everywhere to f(x).

References

James and James. Mathematics Dictionary

Spiegel. Real Variables (Schaum)