Show that every normed liner space which is linearly isometric with a separable space is separable.
A normed linear space or normed vector space is a vector space over the real or complex numbers, together with a function (called a norm) that assigns to each vector a non-negative scalar, satisfying certain axioms.
We can denote a normed linear space by
A space is called separable if it contains a countable dense subset. In other words, there exists a countable subset D of the space such that the closure of D is the whole space.
Two normed linear spaces X and Y are said to be linearly isometric if there exists a bijective linear transformation
Given these definitions, our goal is to show that if X is a normed linear space and is linearly isometric to a separable space Y, then X must also be separable.
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