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 l^p:

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

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

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^∞ = {(x₁, x₂, … ): x_i ∈ K, sup_i=1,2,… |x_i|<∞}.

For x = (x₁, x₂, …) and y = (y₁, y₂, …) ∈ 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_∞.

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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.

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