Functional Analysis – Show that a closed unit sphere is not compact

To let $ B (0, 1) $ be the space of all limited functions $ (0, 1) $, Show that the closed unit
ball $ D = {f in B (0, 1): | f | leq 1 } $ is not compact in $ (B (0, 1), |. | _ { Infty}) $,

we can argue as follows. When a closed unit sphere bounded by a normalized linear space is compact, the space should finally be dimensioned. But $ X = B (0,1) $ is not finally dimensional.
Therefore it is not compact?
If that works?

Manifolds – What is a Lipschitz continuous map between Riemann surfaces in Jost's book Compact Riemann Surfaces?

This appears in section 3.7 of the book Compact Riemann surfaces by Jürgen Jost directly to Lemma 3.7.3. The exact words are

Now let it go $ v: Sigma_1 to Sigma_2 $ be a Lipschitz endless card. cover $ Sigma_1 $ by coordinating neighborhoods. Choose $ R_0 <1 $ with it for everyone $ z_0 in Sigma_1 $, a slice of form
$$ B (z_0, R_0) = {z: | z-z_0 | <R_0 } $$
is contained in a coordinate environment.

Note that the only mention of $ Sigma_1 $ before the quoted paragraph assumes it $ Sigma_1 $ to be a compact surface and to judge how these arguments should be used later, $ Sigma_1 $ is undoubtedly a compact Riemann surface (without a given metric).

My questions are:

(1) What is a Lipschitz endless map of a surface in this context?

(2) What absolute value is used for the representation of? $ B (z_0, R_0) $?

(3) It seems that the author is trying to identify (locally) $ Sigma_1 $ with its local coordinates. However, there are two problems when we do this: First, since we can multiply a graph by a constant, we can subtract any two points (covered by the same graph, where the distance is given by the absolute value) to any desired one Change distance and the Lipschitz border is not well defined; second, for the same reason as above, no definition needs to be made $ R_0 $,

I have searched this section again and again for the absolute value $ Sigma_1 $ is, but I can not come up with a plausible explanation.

Any help is greatly appreciated.

General Topology – Embedding of regular rooms in countably compact Hausdorff rooms

It is well known that every ordinary room is embedded in a compact Hausdorff room.

Problem. Is it true that every regular space is embedded in a countably compact Hausdorff space?

The problem was posed on August 31, 2017 by Serhii Bardyla ( on page 161 of Volume 2 of the Lviv Scottish Book.

Price: Beer + Pizza in the Cosa Nostra.

ct.category theory – Functor From rings to compact Hausdorff spaces

There is a supplement $ text {Bool} ^ {op} leftrightarrow text {Set} $ between Boolean algebras and sets that send a Boolean algebra to the set of their primary ideals and a set $ X $ to $ (X, mathbb {F} _2) _ { text {Set}} $, This link factors as $ text {bool} ^ {op} cong text {PF} leftrightarrow text {set} $, from where $ text {PF} $ is the category of profinite quantities.

There is a supplement $ C ^ * text {-alg} ^ {op} leftrightarrow text {Top} $ between commutative unitial $ C ^ * $-algebras and top sending a commutatives unitial $ C ^ * $-Algebra on the set of their most important ideals and a lot $ X $ to $ C ^ * (X, mathbb {R}) $, This link factors as $ C ^ * text {-alg} ^ {op} cong text {CH} leftrightarrow text {Top} $, from where $ text {CH} $ is the category of compact Hausdorff rooms.

These two additions seem to be related to each other in a certain sense. My question is whether, for compact hausdorff spaces, instead of profinite sets, there is an analog of the following extension:

Sentence: Define a functor $ R text {-alg} rightarrow text {Bool} $ Sending one $ R $-algebra on the boolean algebra given by the set of its idempotents (see Pierce Spectrum Functor). This functor is on the right side. So we have a supplement $ R text {-alg} leftrightarrow text {Set} $,

Now I'm looking for a functor $ text {top-} R text {-alg} rightarrow C ^ * text {-alg} $ or maybe a functor $ text {top-} R text {-alg} rightarrow text {CH} $, or a fine-tuning thereof, which would be analogous to the above theorem.

Could someone also clarify whether these additions are monadic for me? It seems like the same monad is over the set. Could not both be monadic? But several sources that I know have told me.

ct.category theory – compact spaces from the perspective of this adjunction

A compactly generated space is a space $ X $ so that $ f: X rightarrow Y $ is just then continuous, though $ K rightarrow X stackrel {f} { rightarrow} Y $ is steady for every compact hausdorff room $ K $,

To let $ I $ be the category of compact hausdorff topological rooms with continuous maps. Being compact generates, in my opinion, the canonical map $ epsilon_X: int_ {K in I} (K, X) _ { text {Top}} times K rightarrow X $ to be an isomorphism. $ epsilon_X $ is the point of an agreement between functors $ L: (I ^ {op}, text {Set}) _ { text {Cat}} rightarrow text {Top} $ send $ S $ to $ int_ {K in I} S (K) times K $ and $ R: text {Top} rightarrow (I ^ {op}, text {Set}) _ { text {Cat}} $ send a space $ X $ to the preliminary bundle $ S: I ^ {op} rightarrow text {Set} $ send $ K $ to $ (K, X) _ { text {Top}} $,

If I am not mistaken, this addition is at the center of the condensed mathematics of Peter Scholze and Dustin Clausen. Surprisingly, no information is lost if we take instead $ I $ the smaller category of profinite quantities. There is a completely reliable functor that ranges from compact spaces to condensed sets (sheaves on the étale location of a point), and thus arises.

My question is, has anyone chosen this perspective to obtain a slim, categorical proof that the category of compact spaces is Cartesian closed? We could start by observing that the category of presheaves continues $ I $ has all the small limits and colimits, $ L $ preserves colimits, and $ R $ To preserve borders. Could this allow us the Cartesian unity of $ (I ^ {op}, text {set}) _ { text {cat}} $ in the category of compact spaces?

Compact representation of a progress in many sub-operations?

I have the following option of presenting a long-term operation that is detailed in the sense that each progress of the sub-operation is displayed:

Enter image description here

(It is currently displayed in a text-mode application, but the final shape is displayed in a desktop application.) Implementing it in graphical mode is just a bit more attention, but still occupies as much screen space as in this text-mode view.)


Although this is a very detailed view of the current progress, it is not really compact. In general, the presentation of some advances should be compact:

Enter image description here

But in my case, it takes a non-negligible part of the screen.

This is for a video game where the loading times are quite long and I would like to give some information on what is being loaded, eg. Graphics, sounds, etc


I've been thinking about a better way and had the following idea to use a circular progress bar like the following:

Enter image description here

It's definitely more compact, but there are some issues with this approach:

  • I can not view tasks that are at the same hierarchical level
  • I have to sacrifice deeply nested operations, though that's acceptable to me

Is this the right approach or is there an even better approach?

Focal length – How do I calculate the difference in lens reach between a super zoom compact lens and a DSLR zoom lens?

In photography, especially the angle of view (AOV) is interesting. The AOV is the angle that a lens provides for a sensor – it can be specified horizontally, diagonally or vertically.

AOV (°) = 2 * arctan ( sensor_height|width|diagonale (mm) / (2 * focal_length (mm)) )

The formula for getting from a certain focal length (FL) on a non-frame sensor to the frame equivalent focal length is:

equivalent_FL (mm) = true_FL (mm) * crop_factor

The harvest factor can be determined by comparing the diagonals:

crop_factor = full_frame_diag (mm) / your_sensor_diag (mm)

This means:

  • At the same focal length, a larger sensor (but equal aspect ratio) will give a larger AOV
  • For the same sensor dimensions, a smaller focal length gives a larger AOV
  • AOV differs in vertical, horizontal and diagonal axes (except for a quadratic sensor where vert and h are the same)

Or in practical terms:

  • With a 10mm lens on your 5.6 crop factor sensor, you get an AOV equivalent to that of a 56mm lens on a full screen sensor.
  • The same 10mm lens on a 1.6 Crop Factor sensor provides an AOV equivalent to that of a 16mm lens on a full screen sensor.
  • A 1600mm lens with a full-frame sensor provides the same focal length as a 1000mm APS-C (1.6-inch cropped) lens or a ~ 285mm lens for your shot.
  • A 16mm lens on a full-frame sensor has the same focal length as a 10mm lens on an APS-C camera or a ~ 2.85mm lens at your pick-up point.
  • For all other factors, smaller sensors prefer smaller AOVs /
    Higher range while larger sensors prefer wider AOVs.

    • Among the ignored factors are:
    • Pixel density (a 20mm² sensor with 20MP has pixels half as big as
      a 40mm² sensor with 20MP) which influences the noise (smaller pixels)
      are usually worse at collecting light and therefore contain more noise
    • Aperture (f / 4 at a 5.6 crop factor is approximately f / 24 at full aperture
    • Physical limitations (eg negatively weighted focal lengths
      (-1mm) are not possible)

Why do we use focal lengths (in mm) for lenses? Because AOV is not a function of the lens, but the sensor-lens combination. A lens retains its focal length forever, but depending on which sensor it is mounted on, its AOV varies. (Of course, the image circle that a lens can provide limits its capabilities at some point, so attaching a 3mm smartphone to a medium format sensor does not do much good ;-))

Oh and Why compare it to full screen? Since we needed a metric to compare, we could also use IMAX or Super35 or 1 / (⅔ * π) (inches) If we like.

Now for the actual answer to the question:

Their formula was:

(1365 / 5.6) * 1.6 = 390

Which would mean:

effective_FL / crop_factor_PnS = real_FL_PnS
real_FL_PnS * crop_factor_APS-C = ??

What you calculate is therefore the effective focal length of the lens of your Point & Shoot camera on the sensor of your new camera.

Its 1365mm is already full-screen equivalent, so you can already calculate the APS-C-related true focal length with this value.

This means:

1365 / 1.6 = 853.125 (mm)

So you need a lens with this focal length to achieve the same tight AOV with a 1.6 Crop Factor sensor.

Note that the AOV difference is between 100 and 200 mm larger than that between 500 and 600 mm!

Note that as mentioned earlier, 400mm lenses are usually very expensive and are primarily limited to primes (and / or using teleconverters that can disable the AF of your camera if your lens is not fast enough ). This is because it is usually a niche market designed for professionals who want / want the highest image quality most 15,000-euro lenses on 5,000-euro bodies offer in the worst circumstances, a better picture quality than any 500-euro camera. Does that mean that you are a better photographer or that you need this setup? No!

I'm not interested in it, but if you want a modular system with this range, I think μ4 / 3 might be the better choice if you have a limited budget – it offers a 2x crop and 100-400 mm lenses not quite as expensive as a 800mm prime from Canon 😉

real analysis – dual from $ C (K) $ to $ K $ compact

What can we say about double? $ C (K) $, the set of steady functions on a compact Hausdorff space? Can we say that it is isometric to the amount of all regular drill measurements? I know that when $ K $ If only LCH, then the dual is the space of radon measurements. But is a Radon measure in a small space the same as a regular Bohrmaß?

Thank you in advance for your answers.