In the paper the Bernstein basis of the vector space of polynomials of degree at most $n$ and its key properties are presented. The Bézier curves and their use in computer aided geometric design and the de Casteljau's algorithm as a stable method for finding value of polynomial in a given point are briefly discussed. An explicit formula for the dual basis functions expressed as linear combinations of Bernstein polynomials is derived and a matrix form of the relation between these two bases is given. With the Bernstein basis functions we introduce dual functionals which span the dual vector space. The vector space of polynomials of degree at most $n$ with boundary constraints is defined and both the Bernstein and the dual Bernstein basis of such space are derived. Finally, we provide a practical application of the results, the continuous least squares approximation, where, for a given function $f$, we search the polynomial $p^*$ that minimizes the second norm $\norm{f-p}.$
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