Beyond Facebook borders

    Really I couldn't offer beyond the echoing of articles, but I would suspect, give it or a few years or more, or maybe less, that we have yet another scandal regarding potential endemic cases of social manipulation having gone well outside the  confines of Facebook alone.  The problems that I have perceived are as I've imagined and witnessed at any times past, one confined either exclusively to Facebook, and its likely not a case confined to merely a scant week in a given isolated year for limited purposes, but more amply something designed for the purposes of producing longer term effects.  I've seen it potentially it occur around a cross of major mmo platforms (in the gaming industry), I've seen it potentially in social networking on various platforms outside of Facebook.  Thus while there is supposedly some potential ire for a weeks work of potentially bad business practice, this story isn't a high ranking offender in my mind when it has comes to any number of sad social practices that I have witnessed.  Rather sad that this sort of drip the faucet story bit is the at times extent of social dialogue concerning privacy and personal data concerns.   Some day hopefully soon a bit more to history to be added.

Facebook secretly experimented with the moods of 700000 of its users

Facebook secretly experimented with the moods of 700000 of its users

So I wonder how it bodes well for the contingent theory that the ultra violent so much as engage in the daily lifting of a pen or mouse in hand at best,  are left with the psychopathic eminent domain so to as to protect the public as a whole, bolster the likes of the NSA, boost private securities black lists, and so forth.   Apparently in terms of theory running these days towing informally what is likely in informal circles 'degenerate' art.  I think this has been obvious with Facebook for a time, this sort of hit and run is not completely un common in other social networks.

Unfortunately, an associated picture does little to explain or really justify itself otherwise in a given article, or really make light, or at least one is to wonder where the money came for any study such as this.

 A correction to the subsidiary image associated with the article leads in time, not only being something of a poster child, but being something substantially more for a given society that bore itself to disgustingly high levels of hypocrisy.

Over any number of years, to suggest in dealing in the sense of refrain or avoidance to social situations such as these invokes the theory, much laughter here, as to being the anti social, cold, detached, disassociated sort of individual, or strategic use of social networking as a whole, hardly a crime there. :).

Other problems one should imagine, a bit overlooked, outside of the existing, that if you hadn't picked up a nasty bug redirecting your links to a fake Facebook site which attempted to co op likely greater extremes of social manipulation (if not engaged in some form of outright malicious behavior to a more extensive degree), in dealing with the likes of any other private and governmental groups that should use Facebook to the extent of social manipulations, or it seems for all this, the better solution, drop all the problem links, heavily censor your personal content, generally minimize content use of this type, redirect social activities, or at least don't heavily invest yourself in certain directions beyond the time that you desire investing here.  As to time spent, its yours, as to any discourse, I'd hope to personally keep the time spent bolstered.  Not to pick on Facebook here exclusively here, I've seen something of this problem occurring in other social networking venues.  Facebook just happens to be the largest of the bunch, and the easiest likely, in using with wider audience appeal, and in so far as creating social theater, the more likely to be used by predatory groups phishing for information or worse.  In retrospect though I wonder how much a drop in the bucket such a situation were?!

The sorts of social manipulation these days might have you convinced the food supply weren't exactly safe, nor drinking water, and that individuals were even corrupting it by tainting the 'cat chow', and all this were the aims of public safety?   Here if necessary to this extent there were supposedly even enough warrant to make it so, or that few should care, or that with some effort your condition could even be a self induced somatic condition.  Certainly all of these perceptions, experiences must go beyond Facebook, I mean Facebook couldn't be responsible for, for instance, something like mutilated cattle, or black helicopters in the night suspiciously flying in the vicinity of a given location, and here you have the ugly disgrace, of tortured animals coupled with little compensation provided, the puzzled rancher in us scratches his head.  After all it had seemed a bit puzzling the idea of any number of widespread collusion to be more than happenstance annoyance.  Here, given to the idea of reacting, hadn't given so much the telephone hard dialings with list of emergency contacts in tote, or having evaded much of this with an overnight succession that should appear as good as any Roanoke or Mary Celeste fly by night venture, you were as in the previous day, should exist more easily in alms of Stockholm which must in itself be part of the calculus of captivity.  There really weren't any backup money, backup plans, phone numbers, secret networks to shield and aid one here, nor really anyone to provide kind guidance, counseling, shelter, underground networks.  To this extent, the silent sufferer lives out the same sorts of existence, and lives by some sort of mutual assurance it is then reasoned, something exists which prevents anything further from being any more true.  After all the clearer signs should be anything of obvious warrant less searches, daily personal intrusions, forced labor, forced educational camps, something more squeamish exists on the part of a given society?!  Bridge scandals and diarrhea are inconveniences, and surely the extent or rather that personal conquests of sorts in the proof, might have asserted something like imminent domain to the extent of state seizure of children to be held for an apparently undefined indefinite amount of time to try and prove its case, here you wondering if trained doctors really existed in hospitals, or at least something of the focus came from some shadowy individual with a red phone, arrogance, ignorance, and/or both to say one thing over the other and make things so, despite the better advice provided by any well trained jury.  The problem is that when the phone call were made, as likely as it could have been instigated from someone with a recording in tote (or something else), to suggest the untimeliness of things, that it produces exactly the same sets of conditions leading to the same sets of likely outcomes, or that mere suspicion were the warrant of repetitions of things and its inclination, but likely less so the admittance of failure in a given theory, or when enough is enough is more likely the political circus of social ignorance, as much as that sort of captivity has bred, likely the captive subject is as captive as to be expected, and in the adulteration of literature, begs the question 'When do the algorithms of routine human existence, indicate more so when someone is not human?'  It is the sort of question that the torturer asks to make himself feel at home to his own position, his own power, his own justification, his own reasoning...on the other hand, one should find oneself making due with absurdities, surely a stomach ailment were a stomach ailment that should get better in time.  Sure one might have suffered but always to some extent of reasoned injury, and therein the problem with the experiment it must likewise be reasoned, and it weren't like one were something of a gun nut, or it would seem less likely these days the sorts of provocation of this sort were aimed at individuals such as these, more likely someone less likely to inflict harm or injury, that were to serve as the lynch pins of society, as to its state of existential decay, as to its malaise, its problems, its injury, and so forth.  Everyone likely knows this: rewards are bestowed to the problems of this world, and taken away from the less likely of problems in this world.  Or at least its the sort of maxim of the obvious sort, you always need an alibi when you have a hard time living with yourself, and that itself is enough to the warrant of suspicion in having observed it.    

Here's a given recipe for social manipulation:
It would be noting that mirroring social networking information in theory shouldn't be so hard, after all if you could obtain account information knowing someone's account, you'd simply need to duplicate the information from a given account to a given user, and this ensures likely that you aren't held to the believable world simulations problem, that is where you were held to the responsibility of creating any number of posts that should sufficiently seem like the things the users friends may or may not say...after all, if you step out of bounds with out of character things too often too far, it might be more obvious to a given user that something is up, maybe the user is actually more likely to talk to someone in person about something, but its hard to know what sort of note comparison threshold exists here?!  But the mirror world solution deals inherently with these problems not merely by creating from scratch anything, but only slightly modifying user posts as needed.  In this way, you could for instance, have a post slightly more malignant, slightly more needling, but not enough for average note comparison thresholds, i.e., when and whenever a user goes a step further in talking to a person face to face, which seems more likely not true as opposed to being true, unless you might have sensed you were dealing with people likely to interact in any frequent sense face to face, e.g., soccer moms :).  And even so in case of note comparisons, who wants to believe that third party actors were actively manipulating posts and posing as their friends, in this case, the user might even think their friends were more liars than they might have thought.  Even so with the case, of more outright and obvious examples, it would seem more likely that one should even so less likely suspect their social networking service to be bad relative to a fake social manipulation.  The other part of this problem, comes by way of omissions, one should imagine modifying given statistical likes for a user post could have slight and subtle effects on those seeking something of social approvals or positive user feedback.  For instance, a user is rewarded with positive feedback for any linked posts, but then over time, I attempt to steer the user in a subtle way by reducing the user friend 'like' feedback provided for any set of posts containing similar socio/political/ideological content  in this way, maybe the user is more likely not to post anything, if a given user friend base, not being required to say anything, neither provide approvals.  Of course, it seems no wonder that psychological differences between individuals are such that some react less and more to these subtle feedback queue systems, and thus it seems for a given refinement, one should articulate feed back by either increasing or diminishing as necessary a given threshold in which someone may or may not respond to a given set of social stimulus.    Reward and punishment systems (positive and negative) feedback systems are critical one should imagine in dictating the frequency in which people may or may not interact in these social systems.  If you could, for instance, hook users into more use of a given social system, and then having changed reward/punishments over time could lead, one should imagine, a greater likelihood of changing content posts.  One could not neglect carrot in this approach either, rewards should be as much in the mix as it were punishment.  Otherwise, one risks losing a given user entirely in the process, or at least the user manages to find other avenues in communicating and/or comes to subconsciously know that a given social network tends to lean more often than not towards the negative.  Few tend to be active seekers of negative rewards stimulation as in human nature it should seem?!   Its hard to imagine this at least, or that a fair number would likely engage in avoidance, or dismissal of a given experience.  Internalizing negative stimulation tends to be something more often than not defined as an issue of emotional maturity one should reason.  At least  adolescents could be more likely to internalize or act upon negative social stimulus, while adults more likely having a diversified sense of esteem grounded in alternative forms of social activity.  For instance, the adult could be more likely to have any number of social activities in a given acquired social experience wisdom to fall back upon in seeking reward stimulation.  Whereas young adults and adolescents less likely to be self resourceful in finding alternative forms of positive social stimulation.  Having recalled a recent public radio segment on the origins of tracking devices used for alternatives to traditional incarceration.  The tracking device's intent was not so much designed in its early days to ensure conformity to  home incarceration, for instance, in the negative sense of something like parole violation alone when the would be prisoner, for instance, had violated the condition of their confinement say going out when they weren't supposed to leave a given premise, but instead, reward systems could be implemented when a prisoner had actually conformed to the condition of achieving something, like completing a voluntary work task, and then getting say something like pizza privileges.  At that time, at least the approach were considered differed relative to what it were today, which often tends towards the punishment ends.  At least if you the manipulator intend to use something like prisoner models, you think actively of ways to achieve adequate levels of social interaction in the prisons that you intend to create, I'd offer here.

As you see I post in a similar theme these days, now it would appear in some ways that exterior virtual social systems have inflected the nature and degree of any number of these posts.  More so when negative stimulus are provided in varying degrees, leads to its likely to a mirrored outcome, or this is to say, if the world seems more negative by the degree and influence of social spheres, one may likely perceive this as negative.  As to the difference between reflection, ideation, and violent behavior.  That is, another matter of nebulous territory, or at least it seems dubious enough with respect to those that do or don't react in ways violently, a vast statistical majority of those having thought in manner of violence never do act upon these thoughts (something like in the single digit several percent territory do), and one should imagine even well in this day and age, a vast statistical majority of America as whole invariably at one time or another having violent thoughts or fantasies that neither acted out on any number of these thoughts.  Again, something of the pop psychology social circus and the great and grand sociological experiment entailed,  prediction of social behavior always a foregone conclusion?!

I'd only leave off that I hadn't thought Facebook is inherently bad despite present and past hand wringing, as it were evident enough in their popular rise, and part of these being proximate to responding to the needs of their user base, in so far as providing user site content control, obviously a less flexible site could lead to the types of online sociological disaster, like I didn't know that about my friend/neighbor and not for the better, which should lead to the sorts of consequences in the long term, that a once higher user population base finds themselves leaving the social networking experience behind.  To be a successful venture means articulating in some ways a balance which gives user a way to implement site controls so that they feel empowered enough in so far as site content (i.e., who wants to be subjected to a barrage of opposing negative partisan chatter), and then theses days I've delved into something of horror (fictional one should hope), but unfortunately, it would seem the problems of social networking may go well outside Facebook alone.  Social experiences here are neither less solitary at times, outside of the incidental or non incidental contacts, being something of an aged male at that of what ever opinion should be hit and miss, or at least this statistically one should gather is a more endemic process of the aging male, while some manage social attractions, such species is more likely (outside of the profession sorts) doomed to social isolation in many ways.  If socially normative expectations were that one had too much to say, too much opinion, hadn't conformed to any number of perceptions of the aged white male, it would seem that something of social conditions is likely reflected by this.  Being quiet, were not enough at times in one's defense, being less than quiet certainly not enough, being too compassionate something else, while the opposite most definitely a problem, it seems in any event there could be much indictment for its own sake here, however, arbitrary all of this were in reality.  Maybe we relate to one another certainly, we relate to our own social aggression, the sort of opposite of sex attractions at once seemingly holding to the paradox of being allured truly to the likeliest perpetrators of crimes in general.

     There is a lot of silence these days, and apparently its dangerous enough so that I shouldn't be saying so much, or supposedly this way, but fortunately (living in America) provides these sorts of freedoms of opinions, at least one should gather, and those deviating from the customary language of dealing with the happenstance matters of life that were perceived neither consequential much to the affairs of anything else, or were disinterested in varying degrees.  On the one hand, it seems something of horror lurks in the nooks but at least neither so obvious as in the case, of heads gruesomely  placed upon pikes surrounding the castle, or the sort of indifference were someone were slow in coming to fetch something or other, these were supposedly and suggested to be embedded and interwoven into the nature of one's active life.  At least if affairs were less obviously disrupted to the degree that outside of the dungeon, everything looks like its in working functional order , all must be right anyways.  Other absurdities entailed were such to the extent of dealing with something of craziness in general, and here I should be reminded of this sort of logic how possibly insane and idiotic could it be, could it get?  Here you resort to the golden standard, that law abiding societies love law and order, and hadn't resorted to barbarism of centuries past (or even decades, or years, days, minutes or seconds ago).   In passing, one could could be called a mess, but really, it isn't because you were locked in some deep cavernous abyss (waiting for things supposedly to subside) but that you were in the same ways as before went about a similar cautious routine of getting out but not a whole lot.  On the one hand, the typical operands of the world, were active enough, the same usual rhythmic sway of traffic (that would be the same likely even if a large meteor were on a collision path with Earth), or at least one should sense things likely in this way, the freeways would be typically congested with those out to club and totally unaware that their world were really coming to an end, and that is the sadness of this mysterious world, filled with its horror, and then seemingly  tranquil and peaceful alike.  Here one couldn't describe that other nature, of the other part of the world that one should scarcely know or want to know really, for that part, even then you refuse to pay service to a great social conditioning project which seemingly would have anything suggested that were natural to one's condition  at that were something sadly of a given societies life blood. 

You wait to hear the words, 'No one really cares, because inevitably enough people have been caught up in tragedy having witnessed, having heard and so forth so much that social conditioning has made any number of horrors common place."  Its why some societies hadn't considered dead bodies floating in a water way a cause for alarm while others had, and at that some people could be drinking and bathing from the same river with enough proximity so as to horrify the average westerner.  But when all has reached the level of supermarket sanitation the objections arise, there is a place for horror aside, and a place for the clean and tidy appearances of things.  You were after all raised in that world, not the other, raised to relate generally to the one not the other.  You are to sense the insanity of the objection runs its course as it had over the past millennium seemingly, more important were institutional doctrines and the upholding of this should mean that cute depictions of angels torturing souls were the right reflection of a truly God less society, they play and tinker with their religion and shape it as they like to reflect the service they would have in their temples.  When social conservative movements have arisen and move as it were to their arbitrary whims, it were the sort of orderliness that have been constructed throughout the ages, in consolidating minds and faith in a given universal fashion, more so blasphemy seems defined upon the precept of violating this social order, this disorderly mess of the individual defined by their lack of patriotism as their religion, and that their desire to clean the world up were part of this...here you might have imagined the psychopath with handle bar mustache in the Gangs of New York that were one of those patriotic founders?!  :)

Years have given way to changing occurrences it should seem, or that at least in previous years, the nature of alienation is neither one and the same in time.  Times change, I'd have to admit on the one hand, things seem more tame, I don't see the sorts of TI intrusions or social theater that I might have once seen, but then I don't frequent any of the same places that I used to, but then social habits have changed, and as to years ago, something of an innocence about the old encounters have changed.  Being conditioned to one thing, has led to the fruition of other things one should imagine.  Who cares when many are supposedly suffering, and this is the problem with conflict driven societies, when it seems that other are conditioned to the notion that old taboos are violated and less see any wrong for that.  At least one is to think that some part of the world cared when a part of that world seemingly changed decades ago, hallmarking new characteristics in a given world.   As to change, it seems there is always something of inherent resistance to be found.  When it seems things could change, technologies are not always adopted overnight, and some despite all the reasons for being adopted are much late in being adopted if being adopted period.  Social orders supposedly slow in changing, but often  resistance built inherently into social systems are those that have permitted for any length in time the mechanisms of governance which have been operating.  It seems more obvious that entire systems of caste/class and social order permeate by the persistence of these systems.   The problem with decades of rule as it were without the impetus for reform and change, leads to the likelihood of social and economic stasis one should imagine.  With little or no reforms in sight, and overtures to decades old talk of internal security, it seems more likely continuance of the same conditions for any length of time.  Sadly, the bit of western dialogue that often emerges from this all too known condition is that involvement should be segregated, or happenstance, hallmarked by the necessity of its own security interests relative to another culture that it supposedly little can comprehend, or too much the reminder of decades old culture thinking of its own view of the world.  Here I wonder to what degree the internet or the possibility of easily accessed information has really changed anything?   It is to wonder what precedent state and condition serves as much the catalyst.  At least, by what precedence:  would massacres in this day and age serve more so relative to the past, or would this generation more likely turn a blind eye (so ensconced in console and fictionalized dramas).  Then I am reminded of old photographs where children literally loaded their snowballs with rocks and stones, and would appear severely beaten around the eyes, or sickly violent souls at times that bore little care or empathy one should possibly discern.  At least these sorts of cultures pop up from time to time in past literature to be castigated by their writers, at least for writings speaking of social times, violent and cruel societies come and go, and usually one should imagine aren't remembered as well...at least outside of history, I could scarcely recall, for instance, what good Andrew Jackson did for this country or for that matter likely care, notably, he did bring about the infamy of the Trail of Tears.       

Absorption into Nonlinear and Integral Equations

For real Friday night action and entertainment!

Some more notes on some Real and Functional Analysis text

Updating this as I am working.  Sort of interested in seeing if these notes are at all useful to myself or anyone else studying.

From pg 87. (Real and Functional Analysis, Serge Lange, Springer Verlag, 1993, 3rd edition)

Notes on Theorem 1.3: Stated ...Let c be a right end point of this interval, namely of the convex and compact set λ(S0) . To show that λ(S0) consists of one point. The the set λ−1(c)⋂S0 is non empty, convex and compact. We contend that it lies in F. Let x be an element in λ−1(c)⋂S0 and suppose that we can write

x = ty1 +(1−t)y2 with y1,y2 ∈ S and 0S

0 ∈ F , we get y1,y2 ∈ S0 . Applying λ we find that λ(x) = c = tλ(y1)+(1−t)λ(y2). Since c is an end point of the interval λ(y1) = λ(y2) = c.

And it is here where I begin a bit of notes since at least to me it hadn’t appeared immediately clear that λ(y1) = λ(y2) = c.

First we’d presume the interval including c is ordered and that c is a maximal element in such interval being a chosen right end point of the interval. Secondly. We could presume for now that λ(y1) ≤ c, λ(y2) ≤ c since by definition c is an end point or maximal element for the set of λ(S0).

Now if λ(y2) < c means t(λ(y1) − λ(y2)) > 0 and since t > 0, then λ(y1) > λ(y2). But c = tλ(y1) +(t −1)λ(y2) < tλ(y1) + (1−t)λ(y1) = λ(y1)or

λ(y1) > c, contradicting our earlier assumption (since c is a maximal element on λ(S0)). Thus λ(y2) = c. and from this relation we see

λ(y1) = λ(y2) = c

Pg. 89 ­90 Lemma 2.2 We have E = C ⋃(−C). To see this, let x ∈ E and suppose x ∈/ C, x ∈/ −C. Since C is maximal, the cone consisting of all elements c+tx with c ∈ C, t ≥ 0. Hence we can write

−a = c1 +t1x = c2 −t2x

with c1,c2 ∈ C and t1,t2 ≥ 0. Consequently

c1 +(t1 +t2)x ∈ C,

However, c1 +t1x = −a is on the line segment between c1and c1 +(t1 +t2)x , and thus lies in C, a contradiction which proves that E = C ⋃(−C). Noting here...that the line defined by points c1 and c1 +(t1 +t2)x, are in C...paying attention to the above C is not to be confused immediately as the cone of elements c+tx which were a point I initially overlooked. Also by the previous lines where −a ∈/ C the cone c+tx and c+t(­x) generates the elements not in C which should include ­a which in turn leads into the contradictions found above where we have x must belong to C and/or ­C. Its worth noting the role that C plays in defining its compliment cone which were necessary in demonstrating contradiction here. Since c2 ∈ C and since t1,t2 ≥ 0 where t1 +t2 form yet another positive real greater than or equal to 0 means that c1 +(t1 +t2)x ∈ C (mostly c2 is the crux for which this is true). Thus with elements of the compliment cone on a given line of elements found in C leads to our given contradiction from this point, or in other words, no such compliment exists as a non empty set.

Further comments on the strategy of this proof. H forming the closed hyperplane also represents the defined kernel of the desired sought after functional on E. In this case the generator of H is found by the convex application of elements of C and the element a, (1−τ)x −τa = h ∈ H.

In application to Mazur’s Theorem, we are looking for a functional λ of our given hyperplane H separating it, such that λ(x) ≥ c where c is a given number determined by the hyperplane and its functional which in turn provides for the definition of the closed half space, and that in defining this functional we had done so by use of intersections of closed half spaces containing A. This is actually done in the remainder of the proof with its other remaining subpart of the proof...along these lines, it would appear that the intersection of closed half spaces containing A are used in defining the functional necessary to supply a given equivalent closed half space.

Hermitian Forms chapter. Theorem 1.6 (pg. 100), Not sure if there is an error in the text. Namely where a reading … “We let α = t < z,y > with t real and not equal to 0. We can then cancel t and get a contradiction for small t, if < y,z >/= 0 . I believe this should correctly read with large negative t as in a previous proof given, but that is apparently if the following logic is true:

First it would be worth noting that under expansions: |z +αy|2 =< z +αy,z +αy >= < z,z >+< αy,z >+< z,αy >+< αy,αy >

=< z,z >+α < y,z >+αˉ < z,y >+ααˉ < y,y > and where |z|2 =< z,z > provides 0 ≤ α < y,z >+αˉ < z,y >+ααˉ < y,y >

using the α = t < z,y >, we have 0 ≤ t < z,y >< y,z >+t < y,z >< z,y >+t2 < y,z >< z,y >< y,y > where dividing by t yields

0 ≤ < z,y >< y,z >+< y,z >< z,y >+t < y,z >< z,y >< y,y > obviously at this point using small t appears to yield nothing obviously of immediate contradiction(?) since 0 ≤ < z,y >< y,z > is positive definite by convention applied for this inequality (recall positive definite means that we examine the real component of the hermitian inner product and this is also assumed to be positive with inner products if not equal to zero, and keep in mind with examination using inequalities, the book is neglecting in notation Re() of all the resultant right hand side of of such inner product and summations, or rather assumes this into the inequalities expression). However, if taking t negative still being real and large we can see contradictory problems arising in the given inequality since the negative t term predominates and over the set of reals we can make t large enough so that the right hand side of the inequality is negative contradicting our initial assumptions regarding the inequality. We could assume here additionally that y is not an element of the null vector space or that < y,y >/= 0 .

pg.103 we have v = v −∑anvn +∑anvn here we apply theorem due to pythagoras as usual assuming v −∑anvn is orthogonal at least to ∑anvn , and

|v|2 = |v −∑anvn +∑anvn|2 = |v −∑anvn|2 +|∑anvn|2 means 0 ≤ |v −∑anvn +∑anvn|2 = |v|2 −|∑anvn|2 means ∑|an|2 = |∑anvn|2 ≤ |v|2 obviously by the

orthogonality of vi = vj where i =/ j which provides for the bessel inequality. It would be noted that an orthonormal vector neither expressed fully in the context of a hilbert basis leaves as in this an orthogonal component of v remaining, and consequently why the bessel inequality is true.

pg.104 Theorem 2.1 Some notes on this proof: Theorem 1.6 provides that there exists z ∈ E, z =/ 0 such that z is perpendicular to F. And further, we contend that some scalar multiple of z achieves our purpose, say αz. A necessary condition on αis that < z,αz > = λ(z) or in other words,

αˉ = λ(z)/ < z,z >. This is also sufficient. Indeed, for any x ∈ E , we can write

x = x −λ(x)/λ(z) z + λ(x)/λ(z) z

and

x −λ(x)/λ(z) z

lies in F. Taking the product with αz , we obtain < x,αz > = λ(x) thus proving our theorem.

Additional notes: First, while Theorem 1.6 says there is a point perpendicular to z ∈F, and from this coupled with another point x ∈ E , we can find with suitable choice of scalar ultimately providing for a vector of desired magnitude in the sense of the unit norm of z such that a resultant vector (from the difference of x and the scalar times the unit vector z) provides for a point in F (or kernel of λ). It would appear outside of additional assumptions or having shown by method what such a scalar should be, beyond abstraction we hadn’t the conveniences of exactly confirming this, although the consequence of this fact, should reside on the existence of any scalar α that should be suitable and that such existence should be inferred from Theorem 1.6 (?), or in other words finding a resultant vector which lands in F is merely picking a suitable magnitude in the direction of unit vector of z such this subtracted from x gives us the appropriate result (our theorem indicates this and this is not outside of reason working in vector algebra, remembering that from the outside z is in direction perpendicular to F, we should have an intercept by choice of scalar on the unit norm of z, thus regardless of where x results on a given coordinate systems relative to F, we should be in the direction of intercepting the line F in the direction of z multiplied by its scalar. More in depth, to convince yourself of this, I recommend graphically sketching this. First we use from a local coordinate system defining some point y0 in F such that y0intercepts our given origin, and having constructed from the point a perpendicular line running from y0 to the set E and intercepting a point z, defines the direction of our vector z, then having this and having chosen any point x in E such that the vector of x is drawn from the origin y0 means that we have a resultant vector intersecting and in the direction of F (parallel and intercept to F), in using suitable x −λ(x)/λ(z) z and thus perpendicular to z. Next from our given x,z ∈ E and x,z ∈/ F we can make use of the the following to aid us. Namely, that

< x −λ(x)/λ(z) z,z >	= 0	since z and x −λ(x)/λ(z) z are perpendicular.
< x −λ(x)/λ(z) z,z >	=	< x,z > −λ(x)/λ(z) < z,z > = < x,z > −λ(x)/(αˉ < z,z >)*< z,z > means < x,z > = λ(x)/αˉ

Then < x,αz > = αˉ < x,z > = λ(x) α/αˉ ˉ = λ(x) .

Some key notes of Chapter 6:

The General Integral

We want two things from an integral which are not provided by the standard Riemann integral of bounded functions:

(1)We want to integrate unbounded functions.

(2)We want to be able to take limits under the integral sign, of a fairly general nature, more general than uniform limits.

To achieve this, we proceed in a manner entirely similar to the manner used when extending the integral to the completion of a space of step functions, except that instead of the sup norm we use the L1 ­ norm.

pg 117 Theorem M7: Let f : X → Y be a mapping of X into a metric space. Let {fn} be a sequence of measurable mappings X into Y which converges pointwise to f . Then f is measurable.

Proof notes: Generally the proof is subdivided into two different parts, and important assumption highlights here {fn} is a sequence of measurable mappings X into Y which converges pointwise to f . Where some element of {fn}is a measurable mapping, and the sequence of these elements converge pointwise to f. It is important also noting, implicitly (and is used in the second part of this proof) that a σ ­algebra of X and Y are used to the definition of a measurable mapping. Here if (X,M) and (Y ,N) are measurable spaces, and f : X → Y is a map, we define f to be measurable if for every B ∈ N the set f−1(B) is in M . Hence for a sequence of measurable maps as indicated above, we should have a σ­algebras defined on the sets X and Y for each measurable map. In this way, if we can show that an element of the inverse mapping of f belongs also to the inverse mapping of an element in {fn}, and vice versa elements in the inverse mapping of {fn} are in the set of elements in inverse mapping of f, then we have that f must be measurable. The first part of the proof, uses convergence of {fn} to f pointwise indicating that from an open set U of Y, f −1(U)is contained in

∞ ∞

m∩=1k∪=mfk−1(U). Conversely, with closed sets showing the opposite is true. For part 2 we can use the following illustration in aiding

Here we can see under successive iterations of n that the dotted circle grows larger and larger to

conform to V as n → ∞. n∪=1∞ V n = n∪=1∞ An = V while

for any n , V n ⊂ An. Now we have open and close set relations formulated so that we can inter relate the open and closed set inclusions shown in the proof (in part 1). While I mentioned formally the definition of a measurable map here, really there weren’t a formal demonstration which showed exactly the definition given for measurable maps and f, but instead by showing equality in the relation of the inclusions above, that we have a link between measurable maps and f means that f is a measurable. (in this case being a countable/denumerable intersection and union of measurable maps)...additionally σ­ algebras on X and Y should make this clear in the given formulation.

Key points in this proof really use the topology of our given metric space, coupled with convergence of f from its sequence of measurable maps to demonstrate the inter relation between f and its sequences of measurable maps, because of this inter relation, f is measurable (coupled with definitions provided by the sigma algebra).

pg. 118 M8 A map f : X → E of X into a finite dimensional space is measurable if and only if it is a pointwise limit of simple maps.

Notes on proof: On simple maps. A simple map f : X → Z into any set Z is said to be a simple map if it takes on only a finite number of values, and if, for each v ∈ Z the inverse image f−1(v) is measurable. The converse assumption is given by M7. So it is to be shown that in the case (because of reducibility. I believe that since the case of the complex plane being nothing more than a R x R problem and by extension any finite dimension of Rn or Cn similarly reducible to one of the same type namely demonstrating in each dimension the problem of the E = R order type) of E = R. For the construction of the simple map, for an integer n ≥ 1 cut up the interval [­n,n] into intervals of equal length 1/n and denote these intervals by J1,...,JN . We take each Jk to be closed on the left and open on the right. We let JN+1consist of all t such that |t| ≥ n. Let

Let Ak = f−1(Jk) for k = 1, …,N+1 so that each Ak is measurable, the sets Ak (k = 1,...,N +1) are disjoint, and their union is X. On each Ak we define a constant map ψn by ψn(Ak) = infAKf if k = 1, …, N. It that these values for the given set Ak should be mapped to the least upper bound of the cut interval (which is the left value set closure element) under ψn . We can write AN+1 = B ⋃B′ where B consists of those numbers t such that f(t) <−n. We define ψn(B) = n and ψn(B′) = −n . Then the sequence {ψn} converges pointwise to f, and each ψn is a simple function. Keeping in mind that as n tends to infinity the given interval size diminishes to zero or namely to each point of X as ψn maps from X into R. The corresponding interval size of Akdiminishes likewise to a point sized interval, or that for any ε > 0 there exists δ > 0 and N>0, such that for m ≥ N so that distance between the sequence ψm and f is less then ε and an corresponding distance interval in X exists less than δwhich is to say nothing more than how convergence is defined for f by a given simple map sequence which converges to f.

A measurable space together with a measure is called a measured space. When we want to specify all data in the notation, we write the full triple (X,M,μ) for a measured space.