Page 182 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 15: Banach Space: Definition and Some Examples

(iii) n ( x) = | | n (x) Notes
It is customary to denote n (x) by
n (x) = x (read as norm x)
With this notation the above conditions (i) – (iii) assume the following forms:

(i) x 0, x = 0 x = 0;
(ii) x + y x + y ; and
(iii) x = x .
A linear space N together with a norm defined on it, i.e., the pair (N, ) is called a normed linear
space and will simply be denoted by N for convenience.

Notes
1. The condition (ii) is called subadditivity and the condition (iii) is called absolute
homogeneity.

2. If we drop the condition viz. x = 0 x = 0, then is called a semi norm (or pseudo
norm) or N and the space N is called a semi-normed linear space.

Theorem 1: If N is a normed linear space and if we define a real valued function d : N × N R by
d (x, y) = x – y (x, y N), then d is a metric on N.
Proof: We shall verify the conditions of a metric
(i) d (x, y) 0, d (x, y) = 0 x – y = 0 x = y;
(ii) d (x, y) = x – y = (–1) (y – x) = |–1| y – x = y – x = d (y, x);

(iii) d (x, y) = x – y = x – z + z – y (z = N)
x – z + z – y = d (x, z) + d (z, y)
Hence, d defines a metric on N. Consequently, every normed linear space is automatically a
metric space.
This completes the proof of the theorem.

Notes
1. The above metric has the following additional properties:
(i) If x, y, z N and is a scalar, then
d (x + z, y + z) = (x + z) – (y + z) = x – y = d (x, y).

(ii) d ( x, y) = x – y = (x – y)
= | | x – y = | | d (x, y).
2. Since every normed linear space is a metric space, we can rephrase the definition of
convergence of sequences by using this metric induced by the norm.

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