# Questions tagged [function-spaces]

The tag has no usage guidance.

40 questions
Filter by
Sorted by
Tagged with
93 views

### About the normability of the space of continuous functions

Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that ...
155 views

### Relationship between $C(X\times Y,Z)$ and $C(X,C(Y,Z))$

Let $X$, $Y$, and $Z$ be locally-compact, complete, and separable metric spaces and suppose that $X$ is compact; all non-empty. Consider the spaces $C(X,C(Y,Z))$ and $C(X\times Y,Z)$ both equipped ...
121 views

### Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?

I know that $\ell_2$ is isomorphic to a subspace of $L_p(0,1)$ for any $1\le p<\infty$. However, I haven't seen anything about $L_\infty$. Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?...
47 views

142 views

### Has anybody studied continued fractions in function spaces?

For the text below, define $f^\infty(x) = \lim_{n\to\infty} f^n(x)$ where $f^n = \underbrace{f \circ \ldots \circ f}_{n}$. Usually 'continued fraction' means continued fraction in $\mathbb{R}$. For ...
477 views

5k views

### Compact open topology

What is the intuition behind using compact open topology for eg. in the case of Pontryagin dual ?
I would like to have a reference to the following statement which I think is true: $$h^1 \hookrightarrow L^1.$$ The closest I came to this is in D. Goldberg's paper, "A local version of real Hardy ...
Question: What are the connected components of the familiar spaces of functions between two (let's say compact and smooth, for simplicity) manifolds $M$ and $N$? Specifically, I'm thinking of the ...