Questions tagged [compactness]
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56 questions with no upvoted or accepted answers
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Are all infinite-dimensional Lie groups noncompact?
Basically what the title says — if a Lie group is infinite-dimensional, is it necessarily noncompact?
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Implicit function theorem on curves
I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\...
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A question about a class of pro-$\mathcal X$-group
This question concerns the following lemma of this paper:
Lemma 2. Let $\mathcal X_1,\ldots,\mathcal X_n$ be classes of finite groups closed with respect to normal subgroups and subdirect
products ...
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Prove or disprove the compactness of an operator
Consider $X=L^{2}(0,\pi, \mathbb{R})$.
Let $X_{\frac{1}{2}}$ be the domain of $(\Delta)^\frac{1}{2}$ where $\Delta$ is the laplacien operator.
We define the operator $K:C([0,a],X_{\frac{1}{2}})\...
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Compact embedding of anisotropic Sobolev space
I am wondering if the embedding from $W^{2,1}_p(\Omega \times [0,T])$ to $C^{\alpha,\alpha/2}(\Omega \times [0,T])$ is compact, for some suitable domain, $p$ and $\alpha$. I have found some results. I ...
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Compact operator
Let $k:[0,1]^2 \to [0,1]$ be a measurable function. Define $K:L^2([0,1])\to L^2([0,1])$ to be the operator:
$$
(Kf)(x) = \int_0^1\int_0^1 f(z) k(x,y) \mathbf{1}_{x\leq z\leq y} \ \mathrm{d}z \mathrm{d}...
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