Question

Show that the space L^p is a normed linear space

Answer 1

Let ,$1\leq p <\infin .$ and $(X,\mathcal{F},\mu)$ be a measurable set .Where $X$ denote the under lying space , $\mathcal{F} \ the \ \sigma$ -algebra of measurable set and $\mu$ the measure.

Then the space $L^{p} (X,\mathcal{F},\mu)={L}^p$ ,consists of all complex valued measurable function on X that satisfies

$$\int_X{|f(x)|^p d\mu (x) } \ \le \infin$$

The set of such function forms a vector space ,with the following natural operation

$(f+g)(x)=f(x)+g(x)$

$(\lambda f)(x)=\lambda f(x)$

For the scalar $\lambda$ .

Now ,if $f\in L^p$ then we define the $L^p$ norm of $f$ by

$$\vert\vert f\mid\mid_{L^p}=(\int_X|f(x)|^pd\mu(x) )^{\frac{1}{p} }$$

We also abbreviate this to $||f||_p$ .

Now ,we show that $L^p$ is a normed linear space under the norm $||.||_p$

Clearly ,$||f(x)||_p\geq0$ ,since $|f(x)|^p\geq0$ .

since $(\lambda f)x=\lambda f(x)$ $\implies ||(\lambda f)x||=\lambda ||f(x)||$ .

Again we known from Minkowski inequality that if $1\le p <\infin$

and $f,g\in L^p \ then \ f+g\in L^p$

and $||f+g||_p \leq ||f||_p+||g||_p$ .

Hence,$L^p \ together \ with \ ||.||_p$ is a semi normed vector space.

Which is denote by ${\mathcal L}^p(X,\mu)$

wayThis can be made into a normed vector space in a standard way,one simply takes the quotient space with respect to the kernel of $||.||_p$ .

Since, for any measurable function f we have

$||f||_p=0 \iff f=0 \ almost \ everywhere$ ,the kernel of $||.||_p$

does not depends upon p .

Let $\mathcal N =\{ f:f=0 \ almost \ everywhere \}=ker(||.||_p) , \ \forall \ 1\leq p<\infin$

In the quotient space ,two function f and g are identical if $f=g$

almost everywhere.

The resulting normed linear space is by definition,

$$L^p\equiv {\mathcal L^p(X,\mu)}/{\mathcal N}$$

Hence ,$L^p$ is a normed linear space.