I am trying to understand the concepts of coverage and packaging of the collection of functions in a metric space. Thanks in advance for any contribution.

Leave $ mathcal H $ be a collection of functions $ mathcal X rightarrow (0, M) $, in a metric space $ mathcal X = ( mathcal X, d) $. by $ h in mathcal H $ and $ lambda ge 0 $, define $ h_ lambda: mathcal X rightarrow mathbb R $ by $ h_ lambda (x): = sup_ {x & # 39; in mathcal X} h (x & # 39;) – lambda d (x & # 39 ;, x) $. Leave $ L> 0 $ and define the collection derived from functions $ mathcal H_L: = {h _ { lambda} mid h in mathcal H, ; lambda in (0, L) } $. I am interested in complexity measures. For example, I would like to estimate the coverage number of $ mathcal H_L $that is to say

$$

mathcal N_p ( mathcal H_L, varepsilon, n): = sup_ {x_1, ldots, x_n in mathcal X ^ n} mathcal N_ {n, p} ( mathcal H_L (x_1, ldots, x_n), varepsilon),

$$

where $ mathcal H_L (x_1, ldots, x_n): = {(h_ lambda (x_1), ldots, h_ lambda (x_n) mid h in mathcal H } subseteq mathbb R ^ n $ and $ mathcal N_ {n, p} (A, varepsilon) $ is the minimum number of radio balls $ varepsilon $ need to cover a subset $ A $ of metric space $ ( mathbb R ^ n, ell_p) $.

Now, it's easy to see that, every $ h _ { lambda} in mathcal H_ lambda $ is $ lambda $-Lipschitz, because

$$

begin {split}

| h _ { lambda} (x) -h _ { lambda} (x_0) | &: = | sup_ {x & # 39;} h (x & # 39;) – lambda d (x & # 39 ;, x) – sup_ {x & # 39;} h (x & # 39;) – lambda d (x & # 39 ;, x_0) | \

& le sup_ {x & # 39;} | h (x & # 39;) – lambda d (x & # 39 ;, x) – (h (x & # 39;) – lambda d (x & # 39 ;, x_0)) | \

& = lambda sup_ {x & # 39;} | d (x & # 39 ;, x) -d (x & # 39 ;, x_0) | le lambda d (x, x_0)

end {split}

$$

A) Yes $ mathcal H_L subseteq text {BLip} _ {M, L} ( mathcal X rightarrow mathbb R) $, where

$$

text {Lip} _ {M, L} ( mathcal X rightarrow mathbb R): = {h: mathcal X rightarrow (0, M) mid f text {is} L text {- Lipschitz continued} }.

$$

It follows then (for example) that

$$

mathcal N_ infty ( mathcal H_L, varepsilon, n) le mathcal N_ infty ( text {BLip} _ {M, L} ( mathcal X rightarrow mathbb R), varepsilon, n) le mathcal N_ {n, p} ((0, M), varepsilon / L) = (LM / varepsilon) ^ n.

$$

- Is my previous reasoning correct?
- I am intrigued by the fact that the previous limit is independent of the complexity of $ mathcal H $ or the "dimension" of $ mathcal X $. I'm missing something.