Subjects: Metric and Normed Spaces
Links: Continuity on R, Normed Vector Spaces, Vector Spaces, Bounded Function Spaces, Continuity on Metric Spaces, Compactness in Metric Spaces

Space of Continuous Functions

We usually work with two types of functions spaces, the bounded ones and the continuous ones. We actually have a special notation for the set of all continuous functions from a metric space $X$ to another one $Y$

But this might not be good enough and we actually work with a subset

This more restricted space is usually better behaved.

If we make it such that $X$ is a compact metric space, we actually get that

Meaning that this is the most well behaved. If we have that $Y$ is a normed space then any ${\mathcal{C}}^0(X,Y)$ is also a normed space, and we usually endow it with the uniform norm, or uniform metric as the Bounded Function Spaces

Lastly, if we only denoted it as

Since it is so common to send them to $\mathbb{R}$

Continuous Functions from $[a,b]$ to $\mathbb{R}$

We will look at ${\mathcal{C}}^0[a,b]$ be the set of continuous functions $f:[a,b]\to \mathbb{R}$. Then we can see that ${\mathcal{C}}^0[a,b]$ is a vector space. Similarly, that in the case of the ${\ell }^p$ spaces we will define a $\parallel \cdot {\parallel }_p:{\mathcal{C}}^{\mathcal{0}}\mathcal{[}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{]}\to \mathbb{R}$ with $p\in [1,\mathrm{\infty })$ having

and in the case that $p=\mathrm{\infty }$ we get that

depending on what we are working on we might even work on $L^p$ spaces which depend on Measure Theory and stuff

Hölder’s Inequality for Integrals

Let $p,q$ be harmonic conjugates. Let $f,g\in {\mathcal{C}}^0[a,b]$, then we get that

Minkowski’s Inequalities for Integrals

For $p\in [1,\mathrm{\infty }]$, and $f,g\in {\mathcal{C}}^{\mathcal{0}}\mathcal{[}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{]}$, then

Then for $p\in [1,\mathrm{\infty }]$, then $({\mathcal{C}}^{\mathcal{0}}\mathcal{[}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{]}\mathcal{,}\parallel \cdot {\parallel }_{\mathcal{p}}\mathcal{)}$ is a normed space. We can compact the notation to ${\mathcal{C}}_{\mathcal{p}}^{\mathcal{0}}\mathcal{[}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{]}$ to represent $({\mathcal{C}}^{\mathcal{0}}\mathcal{[}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{]}\mathcal{,}\parallel \cdot {\parallel }_{\mathcal{p}}\mathcal{)}$, and if we write ${\mathcal{C}}^{\mathcal{0}}\mathcal{[}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{]}$ referers to $({\mathcal{C}}^{\mathcal{0}}\mathcal{[}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{]}\mathcal{,}\parallel \cdot {\parallel }_{\mathrm{\infty }}\mathcal{)}$

We can see some properties of these $p$-norms

• $\parallel f{\parallel }_s\le (b-a{)}^{\frac{r-s}{sr}}\parallel f{\parallel }_r$ for $1\le s<r<\mathrm{\infty }$
• $\parallel f{\parallel }_s\le (b-a{)}^{\frac{1}{s}}\parallel f{\parallel }_{\mathrm{\infty }}$ for $1\le s<\mathrm{\infty }$

One important characteristic to note about ${\mathcal{C}}_p^0[a,b]$ is that for $p<\mathrm{\infty }$, then ${\mathcal{C}}_p^0[a,b]$ it is not complete, and the completion of this space is actually closely related to the Lp spaces