Inequality that established Lp spaces are normed vector spaces
In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let
be a measure space, let
and let
and
be elements of
Then
is in
and we have the triangle inequality
![{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p))](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d3de762058656808721fc899c4b223914c6c3f)
with equality for
if and only if
and
are positively linearly dependent; that is,
for some
or
Here, the norm is given by:
![{\displaystyle \|f\|_{p}=\left(\int |f|^{p}d\mu \right)^{\frac {1}{p))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d98b372b16f4185ffc2c0c58aafcd0ec93589593)
if
or in the case
by the essential supremum
![{\displaystyle \|f\|_{\infty }=\operatorname {ess\ sup} _{x\in S}|f(x)|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f514691a17ada8a419574d4cfc813f4b765d4e9)
The Minkowski inequality is the triangle inequality in
In fact, it is a special case of the more general fact
![{\displaystyle \|f\|_{p}=\sup _{\|g\|_{q}=1}\int |fg|d\mu ,\qquad {\tfrac {1}{p))+{\tfrac {1}{q))=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d56ccda83307e558835e221c57955bf9909a227)
where it is easy to see that the right-hand side satisfies the triangular inequality.
Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:
![{\displaystyle {\biggl (}\sum _{k=1}^{n}|x_{k}+y_{k}|^{p}{\biggr )}^{1/p}\leq {\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{1/p}+{\biggl (}\sum _{k=1}^{n}|y_{k}|^{p}{\biggr )}^{1/p))](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa53e54b15011c268df0c9885c5a9dd450e2b9ba)
for all real (or complex) numbers
and where
is the cardinality of
(the number of elements in
).
The inequality is named after the German mathematician Hermann Minkowski.
Proof
First, we prove that
has finite
-norm if
and
both do, which follows by
![{\displaystyle |f+g|^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09381447aedbbd09c375faf0b939561316b6abb0)
Indeed, here we use the fact that
is convex over
(for
) and so, by the definition of convexity,
![{\displaystyle \left|{\tfrac {1}{2))f+{\tfrac {1}{2))g\right|^{p}\leq \left|{\tfrac {1}{2))|f|+{\tfrac {1}{2))|g|\right|^{p}\leq {\tfrac {1}{2))|f|^{p}+{\tfrac {1}{2))|g|^{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3bcc20e05c6db9c56fe7377bb3b34cdfa6ebd31)
This means that
![{\displaystyle |f+g|^{p}\leq {\tfrac {1}{2))|2f|^{p}+{\tfrac {1}{2))|2g|^{p}=2^{p-1}|f|^{p}+2^{p-1}|g|^{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05a16c801672e5b5e6546958e5949853ddf7e62d)
Now, we can legitimately talk about
If it is zero, then Minkowski's inequality holds. We now assume that
is not zero. Using the triangle inequality and then Hölder's inequality, we find that
![{\displaystyle {\begin{aligned}\|f+g\|_{p}^{p}&=\int |f+g|^{p}\,\mathrm {d} \mu \\&=\int |f+g|\cdot |f+g|^{p-1}\,\mathrm {d} \mu \\&\leq \int (|f|+|g|)|f+g|^{p-1}\,\mathrm {d} \mu \\&=\int |f||f+g|^{p-1}\,\mathrm {d} \mu +\int |g||f+g|^{p-1}\,\mathrm {d} \mu \\&\leq \left(\left(\int |f|^{p}\,\mathrm {d} \mu \right)^{\frac {1}{p))+\left(\int |g|^{p}\,\mathrm {d} \mu \right)^{\frac {1}{p))\right)\left(\int |f+g|^{(p-1)\left({\frac {p}{p-1))\right)}\,\mathrm {d} \mu \right)^{1-{\frac {1}{p))}&&{\text{ Hölder's inequality))\\&=\left(\|f\|_{p}+\|g\|_{p}\right){\frac {\|f+g\|_{p}^{p)){\|f+g\|_{p))}\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a318b7b6967ffd2aa0f4ffa4335cf08c6cc88181)
We obtain Minkowski's inequality by multiplying both sides by
![{\displaystyle {\frac {\|f+g\|_{p)){\|f+g\|_{p}^{p))}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/188653a338e08015f0d6a85bcb099a129608bc05)
Minkowski's integral inequality
Suppose that
and
are two 𝜎-finite measure spaces and
is measurable. Then Minkowski's integral inequality is:
![{\displaystyle \left[\int _{S_{2))\left|\int _{S_{1))F(x,y)\,\mu _{1}(\mathrm {d} x)\right|^{p}\mu _{2}(\mathrm {d} y)\right]^{\frac {1}{p))~\leq ~\int _{S_{1))\left(\int _{S_{2))|F(x,y)|^{p}\,\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p))\mu _{1}(\mathrm {d} x),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74694688ad2e3f6e8169d4b150b30e9daa7ba18a)
with obvious modifications in the case
If
and both sides are finite, then equality holds only if
a.e. for some non-negative measurable functions
and
If
is the counting measure on a two-point set
then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting
for
the integral inequality gives
![{\displaystyle \|f_{1}+f_{2}\|_{p}=\left(\int _{S_{2))\left|\int _{S_{1))F(x,y)\,\mu _{1}(\mathrm {d} x)\right|^{p}\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p))\leq \int _{S_{1))\left(\int _{S_{2))|F(x,y)|^{p}\,\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p))\mu _{1}(\mathrm {d} x)=\|f_{1}\|_{p}+\|f_{2}\|_{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36fd6b33d9a5711e2ee4f70322405c2848ffda44)
If the measurable function
is non-negative then for all
![{\displaystyle \left\|\left\|F(\,\cdot ,s_{2})\right\|_{L^{p}(S_{1},\mu _{1})}\right\|_{L^{q}(S_{2},\mu _{2})}~\leq ~\left\|\left\|F(s_{1},\cdot )\right\|_{L^{q}(S_{2},\mu _{2})}\right\|_{L^{p}(S_{1},\mu _{1})}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a987a6a478046a49212bebb65df7d8aba33a782f)
This notation has been generalized to
![{\displaystyle \|f\|_{p,q}=\left(\int _{\mathbb {R} ^{m))\left[\int _{\mathbb {R} ^{n))|f(x,y)|^{q}\mathrm {d} y\right]^{\frac {p}{q))\mathrm {d} x\right)^{\frac {1}{p))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/378fb353fca1c3274d623037524f0b25d84d29f4)
for
with
Using this notation, manipulation of the exponents reveals that, if
then