Notes on l^p space [1≤p≤∞]: Definition, Examples, Metric

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

lp = {(x1, x2, … ): xi ∈ K, $\sum_{i=1}^\infty$ |xi|p<∞}.

For x = (x1, x2, …) and y = (y1, y2, …) ∈ lp, define

dp(x, y) = ($\sum_{i=1}^\infty$ |xi – yi|p)1/p.

Next, we prove that dp is a metric on the space lp.

lp is a Metric Space

Theorem: (lp, dp) is a metric space.


Step 1: To show dp(x, y) is finite.

In the Minkowski’s inequality, let ai = xi and bi = -yi, and then taking n∞ we deduce that dp(x, y) < ∞ for all x, y ∈ lp.

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 ∈ lp.

For i=1, 2, … we set ai = xi-zi and bi = zi-yi in the Minkowski inequality. Letting n∞ we get that dp(x, y) ≤ dp(x, z) + dp(z, y), so the triangle inequality is satisfied.

As dp satisfies the three conditions of being a metric, we conclude that lp is a metric space with respect to the metric dp.

l Space

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

l = {(x1, x2, … ): xi ∈ K, supi=1,2,… |xi|<∞}.

For x = (x1, x2, …) and y = (y1, y2, …) ∈ l, define

d(x, y) = supi=1,2,… |xi – yi|.

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.


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, supt∈T |x(t)|<∞}

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

d(x, y) = supt∈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.


Q1: Is lp a metric space when 1≤p≤∞?

Answer: Yes, lp is a metric space.

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