I am new to functional analysis, and I am trying to develop a sample construction to aid my understanding. If someone could check the rigor of my construction, as well as the validity of logical deductions I make, I would greatly appreciate it!
The construction:
Let $b-a$ be a positive integer, and $a,b \in \mathbb{R}$. Let $V$ be a set whose elements consist of all finite-termed, bounded, complex-valued sequences indexed from $a$ to $b$ as follows: $V = \{ u(a), u(a+1), ... , u(b-1), u(b) \}$, where $u$ is a function $u: \{ a, a+1, a+2, ... , b-1, b \} \rightarrow \mathbb{C}$. We also equip $V$ with the field of complex number scalars, and an inner product as follows: \begin{align*} \langle u(t),v(t) \rangle = \sum^{b}_{n=a} \overline{u(n)} v(n) \end{align*} We note that $V$ is a vector space over $\mathbb{C}$ together with the inner product mapping $\langle \cdot , \cdot \rangle : V \times V \rightarrow \mathbb{C}$. This inner product mapping is standard for such constructions, and hence we have a sufficient construction of a complex (Hermitian) inner product space.
We now observe that elements of $V$ can be formed from $b-a+1$ basis vectors, and hence $V$ is $b-a+1$-th dimensional. Since this is a finite term by construction, $V$ is finite-dimensional. By a standard result of linear algebra, we know $V$ is isomorphic to all other $b-a+1$-th dimensional vector spaces, including the complex Euclidean space $\mathbb{C}^{b-a+1}$ (is this true for inner product spaces?), which is a Hilbert space. Thus, $V$ is also a Hilbert space.
Therefore, we can take any subset (not necessarily a subspace) of sequences (elements) from $V$ and apply the properties of Hilbert space to deduce properties of the said subset.