My question is about the use of English language in mathematics.
Should I write "Given a basis $b_n$ of the linear space $B_n := span\,b_n$..." or "Given a basis $b_n$ of a linear space $B_n := span\,b_n$...", i.e., should I use "a" or "the" when referring to $B_n$? Here $b_n$ denotes any basis and we define $B_n$ so it is mentioned for the first time.
Use the word "the" here; you are referencing the unique space that is the span of your basis.