Limit

**surjective** · April 5th 2011, 05:37 PM

I have a function defined as:

$\begin{displaymath}f(x) = \left\{\begin{array}{lr}\frac{1}{k} & : x \in [0,k]\\0 & : x \notin [0,k]\end{array}\right.\end{displaymath}$

I am asked to show that $f_{k} \to 0$ in $L^{\infty}(\mathbb{R})$ as $k \to \infty$

To be honest, I don't understand the question completely. When I make a skecth of the first few functions it's clear that $f_{k} \to 0$ . It's the "in $L^{\infty}(\mathbb{R})$ " which is confusing me.

Could someone clarify?

Thanks a bunch.

**tonio** · April 5th 2011, 06:35 PM

Originally Posted by surjective

I have a function defined as:

$\begin{displaymath}f(x) = \left\{\begin{array}{lr}\frac{1}{k} & : x \in [0,k]\\0 & : x \notin [0,k]\end{array}\right.\end{displaymath}$

I am asked to show that $f_{k} \to 0$ in $L^{\infty}(\mathbb{R})$ as $k \to \infty$

To be honest, I don't understand the question completely. When I make a skecth of the first few functions it's clear that $f_{k} \to 0$ . It's the "in $L^{\infty}(\mathbb{R})$ " which is confusing me.

Could someone clarify?

Thanks a bunch.

Please define $L^\infty(\mathbb{R})$ ...didn't you mean $l^\infty(\mathbb{R})$ ?

Tonio

**surjective** · April 5th 2011, 06:50 PM

Hey,

No, the exercise says $L^{\infty}(\mathbb{R})}$ and is defined as:

$L^{\infty}(\mathbb{R})}= \left\lbrace f: \mathbb{R} \to \mathbb{C} | \text{f is bounded} \right\rbrace$

**Drexel28** · April 5th 2011, 10:27 PM

Originally Posted by surjective

Hey,

No, the exercise says $L^{\infty}(\mathbb{R})}$ and is defined as:

$L^{\infty}(\mathbb{R})}= \left\lbrace f: \mathbb{R} \to \mathbb{C} | \text{f is bounded} \right\rbrace$

Presumably what they mean is that $f_k\to 0$ in whatever metric is placed on $L^{(\infty)}\left(\mathbb{R}\right)$ ( think $\mathcal{B}\left(\mathbb{R},\mathbb{C}\right)$ is a more common notation--at least it is the notation used by such authors as Simmons). Since $L^\infty\left(\mathbb{R}\right)$ is evidently a vector space(algebra) one can guess that the metric is induced by a norm/inner product. Did they give you one?

**tonio** · April 6th 2011, 01:12 AM

Originally Posted by surjective

Hey,

No, the exercise says $L^{\infty}(\mathbb{R})}$ and is defined as:

$L^{\infty}(\mathbb{R})}= \left\lbrace f: \mathbb{R} \to \mathbb{C} | \text{f is bounded} \right\rbrace$

Ok...never saw this notation, and I presume the norm or distance function here is $||f||:=\sup\limits_{x\in\mathbb{R}} |f(x)|$ ?

If this is so, we have $||f_k||=\sup\limits_{x\in\mathbb{R}}|f_k(x)|=\frac {1}{k}\xrightarrow [k\to\infty]{}0$ ...piece of cake, or not.

Tonio

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