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I read somewhere recently (in a general topology book), that the set of all continuous functions $f : [a,b] \rightarrow R$ forms a vector space. Here $[a,b]$ is just an interval on the real line.

I only vaguely recall that trick from my linear algebra classes some N years ago.
I mean... it sounds familiar but I want to know a bit more.

So I wonder now:
1) is this true?
2) what is the dimension of this vector space?
3) what is its basis?

I guess the dimension is infinite but ... I mean... is it countable infinite or maybe having same cardinality as $R$ or maybe "bigger" cardinality?

peter.petrov
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1 Answers1

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The space $C[a,b]$ of continuous real functions on the interval $[a,b]$ is a vector space with the pointwise-defined operations of sum and product by scalars: $$ (f+g)(x)=f(x)+g(x),\quad (\lambda g)(x)=\lambda g(x) $$ for $f,g \in C[a,b]$ and $\lambda\in \mathbb R$.

The dimension of the space is of course infinite. Unfortunately, the proof of existence of a basis in the infinite-dimensional case is not constructive, so we cannot display a basis. The cardinality of any basis is at least $\mathbb R$. In fact, you can find an uncountable set of functions $\mathcal F$ such that any finite subset of $\mathcal F$ is made of linearly independent functions. For example, consider the set of functions: $$ \mathcal F =\{\sin(tx)|, t \in [0,1]\}. $$ On the other hand, the cardinality of a basis cannot exceed $\mathbb R$, since the set $ C[a,b]$ itself has the cardinality of $\mathbb R$, see: Cardinality of continuous functions $f:\Bbb R\to \Bbb R$

DiegoG7
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