# Is Lipschitz space finite dimensional?

Let $$X=C([a,b])$$ equipped with the norm $$\Vert.\Vert_{\infty}$$. The closed subspace $$F$$ of $$\alpha$$-Holder continuous ($$0\lt\alpha\le1)$$ functions $$F\subset X$$ is finite dimensional because if I take a closed unit ball in it, the ball is compact (using Ascoli-Arzelà). What can we say about Lipschitz continuous functions space? Can we argue this way and conclude that it is finite dimensional?

• I think the same proof should work. – pitariver Jun 26 '19 at 13:19

Your argument in the case $$\alpha \in (0,1)$$ is not true. Arzela-Ascoli asserts that the ball $$\{ f | \|f\|_{0,\alpha} \le 1\},$$ where $$\|\cdot\|_{0,\alpha}$$ denotes the Hölder norm, is compact in the space $$(C([a,b]), \|\cdot\|_\infty)$$. It does not implies compactness in the Hölder space $$(C^{0,\alpha}([a,b]), \|\cdot\|_{0,\alpha})$$!

In fact, $$C^{0,\alpha}([a,b])$$ is infinite dimensional and, thus, cannot have a compact unit ball.

$$1,x,x^{2}/2,x^{3}/3,...$$ are linearly independent elements of this space when $$a=0$$ and $$b=1$$.

For equicontinuity you need a condition like $$|f(x)-f(y)| \leq M|x-y|$$ for all $$f$$ in $$F$$ with $$M$$ independent of $$f$$. Since such a condition is not available your argument fails. For no value of $$\alpha$$ is your space finite dimensional.

PS: linear independence of the sequence I have defined is obvious since no non-zero polynomial can have infinitely many zeros.

• Why the proof doesn't work with $\alpha=1$? technically for $\alpha$ in that range Lipschitz functions are a subspace of $\alpha$-Holder continuous functions – banach-alaoglu-zielony Jun 26 '19 at 13:43
• Equicontinuity is not true. You are considering functions with sup norm bounded by 1 but the constant in Lipschitz condition is not bounded. – Kavi Rama Murthy Jun 26 '19 at 13:56

No. $$\{x\mapsto \lvert x-y\rvert: y\in(a,b)\}$$ are linearly independent 1-Lipschitz functions in $$C([a,b])$$.

• Why the proof doesn't work with α=1? technically for α in that range Lipschitz functions are a subspace of α-Holder continuous functions – banach-alaoglu-zielony Jun 26 '19 at 13:49