Is there a symbol for the set of twice differentiable functions (2nd derivative not necessarily continuous)? I believe the symbol for twice continuously differentiable functions is $C^2(\mathbb F)$? Thanks
[Math] Symbol for twice differentiable functions
functional-analysisnotation
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Being isomorphic to $\mathbb{R}^n$ for some $n$ for a real vector space is the same as being finite dimensional. So is $C[a,b]$ (continuous functions $[a,b]\to \mathbb{R}$) finite dimensional for $a<b$? No, because it is possible to find an infinite linearly independent subset, for example, choose a sequence of functions $(f_m)_m$ where $f_m$ is nonzero in $[2^{-(m+1)},2^m]$ and zero otherwise, and then these functions form a linearly independent subset of $C[0,1]$ (obviously the case for general $a,b$ follows by scaling this example).
This post serves to elaborate on the comments made by copper.hat and Branimir Ćaćić above. In what follows, by a continuous function on a topological space, we shall mean an $ \mathbb{R} $- or $ \mathbb{C} $-valued continuous function on the space.
Part 1
The space of continuous functions is an interesting object of study. Indeed, we have a complete classification of commutative C*-algebras in terms of such spaces. This classification result is known as the Commutative Gelfand-Naimark Theorem, and its precise statement is as follows.
Theorem 1 (Gelfand-Naimark) Let $ \mathcal{A} $ be a commutative C*-algebra. If $ \mathcal{A} $ is unital (i.e. it has an identity element), then $ \mathcal{A} $ is isometrically *-isomorphic to $ C(X,\mathbb{C}) $ for some compact Hausdorff space $ X $. If $ \mathcal{A} $ is non-unital, then $ \mathcal{A} $ is isometrically *-isomorphic to $ {C_{0}}(X,\mathbb{C}) $ for some non-compact, locally compact Hausdorff space $ X $.
Hence, C*-algebras, which are abstract algebro-topological objects, can be concretely realized as algebras of continuous $ \mathbb{C} $-valued functions on locally compact Hausdorff spaces.
We also have the following result stating that compact Hausdorff spaces are classified, up to homeomorphism, by their ring of continuous $ \mathbb{R} $-valued functions.
Theorem 2 (Gelfand-Kolmogorov) If $ X $ and $ Y $ are compact Hausdorff spaces, then $ X $ and $ Y $ are homeomorphic if and only if $ C(X,\mathbb{R}) $ and $ C(Y,\mathbb{R}) $ are ring-isomorphic.
Yet another strong classification result is the Serre-Swan Theorem, which says that for compact Hausdorff spaces $ X $, there is an equivalence of categories between
the category of $ \mathbb{R} $- or $ \mathbb{C} $-vector bundles on $ X $ and
the category of finitely-generated projective modules over (resp.) $ C(X,\mathbb{R}) $ or $ C(X,\mathbb{C}) $.
Hopefully, the array of results presented here will imbue the reader with a deeper appreciation of the importance of spaces of continuous functions.
Part 2
Being important and being computationally useful are two different things, which is what the OP may have had in mind. This is especially so when we consider topological spaces that are equipped with a differential structure, which makes them differentiable manifolds. If we only use the space of continuous $ \mathbb{R} $-valued functions to study these manifolds, then we are doing ourselves a great disservice by failing to exploit their differential structure, which can be rich in topological information. Before giving a simple illustration, let us provide a definition.
Definition Let $ X $ be a topological space and $ \mathcal{A} $ an algebra of $ \mathbb{R} $-valued functions on $ X $. Given an $ x \in X $, we say that a mapping $ \delta: \mathcal{A} \to \mathbb{R} $ is a point-derivation on $ \mathcal{A} $ at $ x $ if $ \delta $ is an $ \mathbb{R} $-linear homomorphism and $$ \forall f,g \in \mathcal{A}: \quad \delta(fg) = f(x) \cdot \delta(g) + \delta(f) \cdot g(x). $$ The set of all point-derivations on $ \mathcal{A} $ at $ x $, which happens to form an $ \mathbb{R} $-vector space, is denoted by $ {\text{Der}_{x}}(\mathcal{A}) $.
Let $ X $ be any topological space. Then $ C(X,\mathbb{R}) $ is an algebra of $ \mathbb{R} $-valued functions on $ X $. We can therefore ask ourselves: What is $ {\text{Der}_{x}}(C(X,\mathbb{R})) $ for a given $ x \in X $? Well, it turns out that $$ \forall x \in X: \quad {\text{Der}_{x}}(C(X,\mathbb{R})) = \{ 0_{C(X,\mathbb{R}) \to \mathbb{R}} \}, $$ which is just the trivial vector space! Not very interesting indeed. (Click here to see a proof.)
Suppose this time that $ X $ is a differentiable $ n $-dimensional manifold. Then $ {C^{1}}(X,\mathbb{R}) $ is also an algebra of $ \mathbb{R} $-valued functions on $ X $. However, $ {\text{Der}_{x}}({C^{1}}(X,\mathbb{R})) $ will be an $ n $-dimensional vector space for any $ x \in X $ (the previous link also contains an explanation of this).
Note: The point-derivations on $ {C^{1}}(X,\mathbb{R}) $ at $ x $ can be used to define the tangent space of $ X $ at $ x $.
In summary, for a differentiable manifold $ X $, the vector space of point-derivations on $ C(X,\mathbb{R}) $ at any point gives us no information whatsoever. However, by considering the algebra of differentiable functions on $ X $ instead, this very same vector space can tell us the dimension of the manifold, which is an important topological invariant.
Morse Theory takes this concept to a new level of sophistication by using smooth functions on a smooth manifold to extract even more topological information. As an example, let us take a look at the following result, which is one of the jewels of Morse Theory.
Theorem 3 Let $ M $ be a compact smooth $ n $-dimensional manifold, and suppose that there exists a smooth $ \mathbb{R} $-valued function on $ M $ having exactly two critical points, both of which are non-degenerate. Then $ M $ is homeomorphic to $ \mathbb{S}^{n} $.
This result is truly amazing, for it gives us a criterion by which the differential structure of a compact smooth manifold can actually determine its topology. One can also use smooth functions to construct a new homology theory for smooth manifolds, called Morse homology, which turns out to be isomorphic to singular homology.
Many deep theorems can be proven using Morse Theory, such as the Bott Periodicity Theorem for the homotopy groups of classical Lie groups and the existence of exotic smooth structures on $ \mathbb{S}^{7} $ (a result for which John Milnor was awarded the Fields Medal).
Conclusion: The continuous functions on a topological space are important, but if you are given a differential structure, then you might want to exploit this structure by studying the differentiable functions instead because they can provide you with valuable topological information.
Best Answer
Given your tag of functional analysis, the following might be of interest for you:
The Sobolev space $W^{k,p}(\Omega)$ is the set of all functions $f \in L^p(\Omega)$ such that for every multi-index $\alpha$ $|α| \leq k$, the weak partial derivative $D^\alpha f$ belongs to $L^p(\Omega)$.
More details in Wikipedia.