Hilbert Spaces can be FINITE dimensional or INFINITE Dimensional

Finite Dimensional Hilbert Spaces

n-Tuples of real numbers – R^n – e.g. R^5
Inner Product would be just the dot product of two vectors
\vec{x} = \begin{pmatrix}x1\\x2\\x3\\x4\\x5 \end{pmatrix} and \vec{y} = \begin{bmatrix}y1 & y2 & y3 & y4 & y5 \end{bmatrix}

n-Tuples of complex numbers – C^n – e.g. C^2

\vec{c1} = \begin{pmatrix}a1+ib1 \\a2+ib2 \end{pmatrix}

The inner product here would be the complex inner product: $ (complex  conjugate of c1)^T (c2)

Infinite Dimensional Hilbert Spaces

Space of all complex valued functions with inner product defined in a particular way (square integrable functions)