Hilbert Spaces

• Vector space ${\mathbb{C}}^n$ extends ${\mathbb{C}}^{\infty }$
• The inner product in ${\mathbb{C}}^{\infty }$ is defined analogously to ${\mathbb{C}}^n$. For two vectors $u,v\in {\mathbb{C}}^{\infty }$, their inner product is:

$$⟨u,v⟩=\sum _{i=1}^{\infty }{u}_i^{\ast }{v}_i$$

For the inner product to be well defined, the series must converge, leading to the concept of square-summable sequences (${\ell }^2$)

Functional inner spaces

• Functions equipped with inner product. E.g.,

$$⟨f,g⟩={\int }_a^b{f}^{\ast }(x)g(x)dx$$

• Space of all square-integrable functions on $[a,b]$:

$$L^2([a,b])=\left\{f:[a,b]\to \mathbb{C}|{\int }_a^b|f(x){|}^{2}dx<\infty \right\}$$

$L^2$-norm

$$\Vert f\Vert =\sqrt{{\int }_a^b|f(x){|}^{2}dx}$$

E.g. Fourier Series

On the interval $[0,2\pi ]$, the basis functions are the complex exponentials:

$${\varphi }_k(x)=\frac{1}{\sqrt{2\pi }}{e}^{ikx},k\in \mathbb{Z}$$

The basis satisfies the orthogonality condition:

$$⟨{\varphi }_k,{\varphi }_j⟩={\int }_0^{2\pi }{\varphi }_k^{\ast }(x){\varphi }_j(x)dx={\delta }_{kj}$$

where ${\delta }_{kj}$ is the Kronecker delta.

Any square-integrable function $f(x)\in L^2([0,2\pi ])$ can be expressed as $f(x)={\sum }_{k=-\infty }^{\infty }{c}_k{\varphi }_k(x)$, where the Fourier coefficients $c_k$ are computed using the projection formula:

$$c_k=⟨{\varphi }_k,f⟩={\int }_0^{2\pi }\frac{1}{\sqrt{2\pi }}{e}^{-ikx}f(x)dx$$

Every finite-dimensional real or complex inner product space is a Hilbert space.

Convergence $\Rightarrow$ a sequence converges to a limit $L$.

Completeness $\Rightarrow$ every Cauchy sequence converges to a point in the space