Space continuations LP

In Mathematical, space $\backslash \; ell^p$ is a space of continuation S with actual values or complexes which has a structure of Espace of Banach.

Motivation

Let us consider the space of the real vectors $\backslash \; mathbb\; \{R\}\; ^n$. The sum of vectors in $\backslash \; mathbb\; \{R\}\; ^n$ is given by:

$(x\_1,\; x\_2,\; \backslash \; dowries,\; x\_n)\; +\; (y\_1,\; y\_2,\; \backslash \; dowries,\; y\_n)\; =\; (x\_1+y\_1,\; x\_2+y\_2,\; \backslash \; dowries,\; x\_n+y\_n),$

And the multiplication by a scalar is given by:

$\backslash \; lambda\; (x\_1,\; x\_2,\; \backslash \; dowries,\; x\_n)\; =\; (\backslash \; lambda\; x\_1,\; \backslash \; lambda\; x\_2,\; \backslash \; dowries,\; \backslash \; lambda\; x\_n).$

The standard of a vector $x=\; (x\_1,\; x\_2,\; \backslash \; dowries,\; x\_n)$ is often given by:

$\backslash |X\; \backslash |=\; \backslash \; left\; (x\_1^2+x\_2^2+\; \backslash \; dots+x\_n^2\; \backslash \; right)\; ^\; \{1/2\}$

But is not the only way of defining a standard, if p is a Real number and p≥ 1 we can define:

$\backslash |X\; \backslash |\_p=\; \backslash \; left\; (|x\_1|^p+|x\_2|^p+\; \backslash \; dots+|x\_n|^p\; \backslash \; right)\; ^\; \{1/p\}$

For each vector $x=\; (x\_1,\; x\_2,\; \backslash \; dowries,\; x\_n)$. It proves that this definition satisfies the properties of a standard. Thus for each p≥ 1 , $\backslash \; mathbb\; \{R\}\; ^n$ together and the p-standard which we have just defined we let us form a Espace of Banach.

Space $\backslash \; ell^p$

The p-standard can be wide with the vectors having an infinity of components what gives us space $\backslash \; ell^p$ . For $x=\; (x\_1,\; x\_2,\; \backslash \; dowries,\; x\_n,\; x\_\; \{n+1\},\; \backslash \; dowries)$, an infinite sequence of realities number or complexes we define the sum:

$(x\_1,\; x\_2,\; \backslash \; dowries,\; x\_n,\; x\_\; \{n+1\},\; \backslash \; dowries)\; +\; (y\_1,\; y\_2,\; \backslash \; dowries,\; y\_n,\; y\_\; \{n+1\},\; \backslash \; dowries)\; =\; (x\_1+y\_1,\; x\_2+y\_2,\; \backslash \; dowries,\; x\_n+y\_n,\; x\_\; \{n+1\}\; +y\_\; \{n+1\},\; \backslash \; dowries),$

And multiplication by a scalar:

$\backslash \; lambda\; (x\_1,\; x\_2,\; \backslash \; dowries,\; x\_n,\; x\_\; \{n+1\},\; \backslash \; dowries)\; =\; (\backslash \; lambda\; x\_1,\; \backslash \; lambda\; x\_2,\; \backslash \; dowries,\; \backslash \; lambda\; x\_n,\; \backslash \; lambda\; x\_\; \{n+1\},\; \backslash \; dowries).$

We define the p-standard:

$\backslash |X\; \backslash |\_p=\; \backslash \; left\; (|x\_1|^p+|x\_2|^p+\; \backslash \; dots+|x\_n|^p+|x\_\; \{n+1\}|^p+\; \backslash \; dowries\; \backslash \; right)\; ^\; \{1/p\}.$

But here a problem occurs, it is that the series of right-hand side is not always convergent, for example, the series $(1,1,1,\dots )$ has an infinite p-standard for any p . Thus space $\backslash \; ell^p$ east defines as the whole of the infinite sequences of real numbers or complexes whose standard is defined.

One defines also the ∞ - standard like:

$\backslash |X\; \backslash |\_\backslash infty=\backslash sup(|x\_1|,\; |x\_2|,\; \backslash \; dowries,\; |x\_n|,|x\_\; \{n+1\}|,\; \backslash \; dowries)$

and spaces it corespondant $\backslash \; ell^\; \backslash \; infty$ of all the vectors or limited sequences. Moreover one a:

$\backslash |X\; \backslash |\_\; \backslash \; infty=\; \backslash \; lim\_\; \{p\; \backslash \; to\; \backslash \; infty\}\; \backslash |X\; \backslash |\_p$

See too

• Space L^p of functions

Random links:

Resh (letter) | Georg Stengel | National park of Waza | Japanese plum | Tangerine (music) | Salaire_égal

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org