Encyclopedia > Minkowski inequality

  Article Content

Minkowski inequality

In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of Lp(S). Then f + g is in Lp(S), and we have
<math>\|f+g\|_p \le \|f\|_p + \|g\|_p</math>
with equality if and only if f and g are linearly dependent[?].

The Minkowksi inequality is the triangle inequality in Lp(S). Its proof uses Hölder's inequality.

Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:

<math>\left( \sum_{k=1}^n |x_k + y_k|^p \right)^{1/p} \le \left( \sum_{k=1}^n |x_k|^p \right)^{1/p} + \left( \sum_{k=1}^n |y_k|^p \right)^{1/p}</math>
for all real (or complex) numbers x1,...,xn, y1,...,yn.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Anna Karenina

... the novel's first complete appearance was in book form. Related Topics Madame Bovary Molly Bloom's Soliloquy Warning: Wikipedia contains spoilers Outline of ...

 
 
 
This page was created in 1393.5 ms