Special vector spaces

Here is a list of special vector spaces

W.r.t function support
There are two special vector spaces w.r.t. support.
 * $$C_c(\mathbb{R})$$, the space of all continuous functions with compact support: $$C_c(\mathbb{R}):=\left\{f:\mathbb{R}\to\mathbb{C} : f \text{ is continuous and has compact support}\right\}$$
 * $$C_0(\mathbb{R})$$, the space of all continuous functions that tend to zero as $$x\to\pm\infty$$: $$C_0(\mathbb{R}):=\left\{f:\mathbb{R}\to\mathbb{C} : f \text{ is continuous and } f(x)\to 0 \text{ as } x\to\pm\infty\right\}$$

Some Banach spaces

 * $$L^p(\mathbb{R}):=\left\{f:\mathbb R \to \mathbb C \mid \int_{-\infty}^{\infty}|f(x)|^p dx < \infty \right\}$$ for $$p \in [1,\infty)$$
 * $$L^\infty(\mathbb R) := \left\{f : \mathbb R \to \mathbb C | f \text{ is bounded } \right\}$$
 * The space of continuous functions on a bounded interval $$[a,b]$$: $$C[a,b]:=\{f[a,b]\to \mathbb C | f \text{ is continuous }\}$$ (w.r.t. norm $$||f||_\infty = max_{x\in[a,b]} |f(x)|$$ --- see Th. 3.1.6 )
 * $$C_0(\mathbb{R})$$

Some Hilbert spaces

 * $$L^2(\mathbb{R})$$

Advantages/disadvantages of some spaces
Those notes are from Week 6 notes of Real Analysis course of Ole Christensen.

The space $$C_c(\mathbb{R})$$:

 * + The assumption of compact support is realistic for applications
 * + $$C_c(\mathbb{R})$$ is dense in $$L^p(\mathbb{R})$$ for any $$p\in[1,\infty)$$ (T.5.4.2 )
 * - Not all signals are continuous
 * - Not a Banach space

The space $$C_0(\mathbb{R})$$

 * + Is a Banach space
 * + The assumption of $$f(x)\to 0$$ as $$x\to\pm\infty$$ is realistic for applications
 * - Not all signals are continuous

The space $$L^1(\mathbb{R})$$

 * + Is a Banach space
 * + Contains non-continuous functions too
 * + Realistic for applications
 * - Not a Hilbert space
 * - Not a space of equivalence classes

The space $$L^2(\mathbb{R})$$

 * + Is a Banach space
 * + Contains non-continuous functions too
 * + Realistic for applications
 * + Is a Hilbert space
 * - Not a space of equivalence classes