For 1≤p<∞, let us consider the following set of sequences in the linear space K, denoted by l^{p}:

l^{p} = {(x_{1}, x_{2}, … ): x_{i} ∈ K, $\sum_{i=1}^\infty$ |x_{i}|^{p}<∞}.

For x = (x_{1}, x_{2}, …) and y = (y_{1}, y_{2}, …) ∈ l^{p}, define

d_{p}(x, y) = ($\sum_{i=1}^\infty$ |x_{i} – y_{i}|^{p})^{1/p}.

Next, we prove that d_{p} is a metric on the space l^{p}.

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## l^{p} is a Metric Space

**Theorem:** (l^{p}, d_{p}) is a metric space.

*Proof:*

**Step 1:** To show d_{p}(x, y) is finite.

In the Minkowski’s inequality, let a_{i} = x_{i} and b_{i} = -y_{i}, and then taking n**→**∞ we deduce that d_{p}(x, y) < ∞ for all x, y ∈ l^{p}.

**Step 2:** The first two conditions of the definition of a metric space are easily satisfied.

**Step 3:** Check triangle inequality.

Let x, y, z ∈ l^{p}.

For i=1, 2, … we set a_{i} = x_{i}-z_{i} and b_{i} = z_{i}-y_{i} in the Minkowski inequality. Letting n**→**∞ we get that d_{p}(x, y) ≤ d_{p}(x, z) + d_{p}(z, y), so the triangle inequality is satisfied.

As d_{p} satisfies the three conditions of being a metric, we conclude that l^{p} is a metric space with respect to the metric d_{p}.

## l^{∞} Space

Now, we consider the set of all bounded sequences in K, denoted by l^{∞}.

l^{∞} = {(x_{1}, x_{2}, … ): x_{i} ∈ K, sup_{i=1,2,…} |x_{i}|<∞}.

For x = (x_{1}, x_{2}, …) and y = (y_{1}, y_{2}, …) ∈ l^{∞}, define

d_{∞}(x, y) = sup_{i=1,2,…} |x_{i} – y_{i}|.

It can be easily verified that d_{∞} is a metric on the space l^{∞}, that is l^{∞} is a metric space with respect to the metric d_{∞}.

**ALSO READ:**

Metric space: definition, examples, properties

## Metric space of bounded functions

Let T be a set and let B(T) denote the set of all K-valued bounded functions defined on T. The set B(T) is given below.

B(T) = {x: T **→** K, sup_{t∈T} |x(t)|<∞}

Note that l^{∞} is a special case of B(T). For x, y ∈ B(T), define

d_{∞}(x, y) = sup_{t∈T} |x(t) – y(t)|.

The above function d_{∞} is a metric on B(T), that is (B(T), d_{∞}) is a metric space. This metric is knnown as **sup metric**.

## FAQs

**Q1: Is l**

^{p}a metric space when 1≤p≤∞?Answer: Yes, l^{p} is a metric space.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.