Whats Is Going to Be the next Best Online Multilayer Game for XBOX 360?

Actually, we have a non-constructive proof.Lemma : Let $mathcalH$ be a Hilbert space. Then it is impossiblethat $mathcalH$ has a countable dense subset $D$ and an uncountableorthonormal set $e_imid iin I$.Proof: Prove by contradiction. Suppose the contrary that $mathcalH$has a countable dense subset $D$ and an uncountable orthonormal set$e_imid iin I.$ Let $rfrac14$ and for each $iin I$,let $B_iB(e_i,r)$, then open ball centered at $e_i$ withradius $r$. Note that $B_icap B_jemptyset$ whenever $ineq j$.(For, if there exists $xin B_icap B_j$, then $||e_i-e_j||leq||e_i-x||||x-e_j||e_i$, where the sum is independentof ordering. (I leave the proof to you)./////////////////////////////////////////////////////////////I expand the last line, state it as a theorem and give a self-contained proof. This is known as the Fourier expansion in Hilbert space.////////////////////////////////////////////////////////////Theorem: Let $mathcalH$ be a Hilbert space (over $mathbbR$or $mathbbC$) and let $e_imid iin I$ be a maximal (withrespect to $subseteq$) orthonormal set (where the index $I$ maybe uncountable). Then for each $xinmathcalH$, we have $xsum_iin Ilangle x,e_irangle e_i$,where the series converges in unorder sense (explained in Claim (4)below).Proof: Claim (1) (Bessel's inequality): For any finite subset $I_1subseteq I$and any $xinmathcalH$, $sum_iin I_1|langle x,e_irangle|^2leq||x||^2$. Proof of Claim (1): Denote $Pxsum_iin I_1langle x,e_irangle e_i$.Observe that $Pxbot x-Px$, so $||x||^2||Px||^2||x-Px||^2geq||Px||^2sum_iin I_1|langle x,e_irangle|^2$.Claim (2): For any $xinmathcalH$, $iin Imidlangle x,e_irangleneq0$is a countable set.Proof of Claim (2): Let $I'iin Imidlangle x,e_irangleneq0$.Prove by contradiction. Suppose the contrary that $I'$ is uncountable.For each $ninmathbbN$, let $I_niin Imid|langle x,e_irangle|geqfrac1n$.Observe that $I'cup_nI_n$, so there exists $n$ such that $I_n$is uncountable. Choose $k$ such that $frackn^2>||x||^2$.Choose a finite subset $I'_nsubseteq I_n$ that contains $k$elements, then $sum_iin I'_n|langle x,e_irangle|^2geqfrackn^2>||x||^2$,contradicting to claim (1).Claim (3): Given $xinmathcalH$ and let $I_xiin Imidlangle x,e_irangleneq0$.Fix an enumeration for $I_x$, say $I_xi_1,i_2,ldots$(finite or infinite), then the series $sum_klangle x,e_i_krangle e_i_k$is convergent.Proof of Claim (3): If $I_x$ is a finite set, we are done. Supposethat $I_x$ is a countably infinite set. Let $s_nsum_k1^nlangle x,e_i_krangle e_i_k$.By Claim (1), for each $n$, $sum_k1^n|langle x,e_i_krangle|^2leq||x||^2$,so the series $sum_k1^infty|langle x,e_i_krangle|^2$is convergent. We show that $s_n$ is a Cauchy sequence in $mathcalH$.Let $varepsilon>0$. Choose $N$ such that $sum_kN^infty|langle x,e_i_krangle|^20$be arbitrary. Let $U_yB(y,varepsilon)$ be the open ball centeredat $y$ with radius $varepsilon$. Choose $N$ such that $||sum_k1^Nlangle x,e_i_krangle e_i_k-y||leqfracvarepsilon4$.(If $I_x$ in Claim (3) is a finite set, let $N$ be the numberof elements in $I_x$.) We adopt the notation in Claim (3) and continueto work with the enumeration $I_xi_1,i_2,ldots$. Let$I_1i_1,i_2,ldots,i_N$. Clearly $I_1inmathcalC$.Consider the case that $I_x$ is an infinite set (the finite caseis trivial). By continuity of norm (i.e., the continuity of the map$xmapsto||x||$), we have$$lim_n||sum_k1^Nlangle x,e_i_krangle e_i_k-sum_k1^nlangle x,e_i_krangle e_i_k||||sum_k1^Nlangle x,e_i_krangle e_i_k-y||leqfracvarepsilon4.$$On the other hand, for any $n>N$, we have $||sum_k1^Nlangle x,e_i_krangle e_i_k-sum_k1^nlangle x,e_i_krangle e_i_k||^2sum_kN1^n|langle x,e_i_krangle|^2$.Hence $sum_kN1^infty|langle x,e_i_krangle|^2leqleft(fracvarepsilon4right)^2$.Let $I_2inmathcalC$ be arbitrary such that $I_1preceq I_2$.Then$$||theta(I_2)-y||leq||sum_k1^Nlangle x,e_i_krangle e_i_k-y||||sum_iin I_2setminus I_1langle x,e_irangle e_i||leqfracvarepsilon4||sum_iin I_2setminus I_1langle x,e_irangle e_i||.$$Observe that$$||sum_iin I_2setminus I_1langle x,e_irangle e_i||^2sum_iin I_2setminus I_1|langle x,e_irangle|^2leqsum_kN1^infty|langle x,e_i_krangle|^2leqleft(fracvarepsilon4right)^2$$ because for any $iin I_2setminus I_1$, if $inotini_N1,i_N2,ldots$,then $langle x,e_irangle0$.Now we have: $||theta(I_2)-y||leqfracvarepsilon2$. Thisshows that $sum_iin Ilangle x,e_irangle e_iy$ in unorderedsense.Claim (5): The $y$ defined in Claim (3) and Claim (4) is $x$. Thatis $xsum_iin Ilangle x,e_irangle e_i$.Proof of Claim (5): Let $zx-y$. Prove by contradiction. Supposethe contrary that $zneq0$. We adopt the notation in Claim (3) andClaim (4). Recall that for each $ainmathcalH$, the map $xmapstolangle x,arangle$is continuous. Let $iin I$ be arbitrary. Conside the case that $I_x$is infinite (The finite case is trivial.). We have$$langle z,e_iranglelim_nrightarrowinftylangle x-sum_k1^nlangle x,e_i_krangle e_i_k,e_irangle.$$ If $inotin I_x$, we have $langle x,e_irangle0$ and $langle e_i_k,e_irangle0$for all $k$, and hence $langle z,e_irangle0$. Suppose that$iin I_0$, say $ii_k'$ for some $k'inmathbbN$. Then forany $ngeq k'$, we have$$langle x-sum_k1^nlangle x,e_i_krangle e_i_k,e_iranglelangle x,e_irangle-langlesum_k1^nlangle x,e_i_krangle e_i_k,e_iranglelangle x,e_i_k'rangle-langle x,e_i_k'rangle0.$$ Therefore $langle z,e_irangle0$ in all cases. Define $tildezz/||z||$,then $e_imid iin Icuptildez$ is an orthonormal set,containing $e_imid iin I$ as a proper subset. This contradictsto the maximality of $e_imid iin I$.If $Dx_n : nin mathbbN$ is a countable set dense in a Hilbert space $mathcalH$, how can I show that Gram-Schmidt algorithm applied to $D$ (or a subset of $D$) produces an orthonormal numerable basis for $mathcalH$?So far I have been able to prove that every ortonormal basis of $mathcalH$ has to be numerable.

The maid of honor is the person closest to the bride and takes care of every aspect of the wedding. Here is a list of duties or the tasks that she is expected to perform.The maid of honor is the most important of all the bridesmaids, since she has to take care of many wedding activities, before and during the celebrations. She has to be with the bride 24x7, after the wedding date has been finalized. This is because the bride needs assistance in all the wedding arrangements. She needs a helping hand or sometimes, an honest opinion about things. She is the bride's right-hand who needs to fulfill all her duties on, and before the wedding day.Firstly, discuss with the bride about the kind of wedding she wants. This will give you an idea of what preparations you need to do.It is suggested that you maintain an errand checklist on your mobile, in a notebook, or a diary.Shopping for a wedding takes a lot of patience. Hunting for a wedding gown is very important when it comes bridal shopping. Help the bride find a dress that goes with the wedding theme. You will be required to be with the bride for each of the bridal gown trials. Give her honest opinions about the dresses, and suggest any alterations, if required. You will also be expected to take care of the bridesmaids' dresses.Search for bridal shops which have the kind of bridal accessories that you are looking for. Help the bride shop for all the accessories such as shoes, jewelry, makeup, hair accessories, etc.Next thing to be done is to start maintaining proper communication with the other bridesmaids as well as the groom's best man. You will be the bride's messenger, who will communicate with the bride and groom's parents, friends, and relatives.This would be necessary only if you do not wish to hire a wedding coordinator for making all the arrangements. In case, you are taking help from a wedding coordinator, he would take care of all these arrangements. All you would have to do is communicate with the coordinator.A bachelorette party is an expected affair that has to be taken care of by you. You will also have to throw a bridal shower and buy a gift for the bride.Book a hairstylist and a makeup artist for the wedding day.The photographer is usually chosen by the bride and groom. However, you might want to run a check on him prior to the wedding day.The wedding decorations and wedding catering arrangements will also have to be looked into. The bride will need your help while deciding the venue, menu, and the interiors of the reception. You are also expected to help the bride with the seating arrangements, wedding invitations, and sorting out the guest list.The rehearsal dinner is another important event for the bride and the groom. The maid of honor is supposed to look out for any minute details that might have been missed out initially. Help the bride fix them before the wedding day.Your most important duties begin on the wedding day. You will have to escort the bride everywhere, help her get ready, put on her dress and makeup, take her to the beauty salon, if needed, etc. Your most important job of the day would be to take care of the groom's wedding ring. During the wedding, you must hold the bride's bouquet, hand her the ring, straighten her gown, take care of the bride's possessions and gifts, etc. During the wedding reception, you are expected to deliver a speech in honor of the newly weds; and make a toast to them.Last but not the least, your most important duty would be to support the bride mentally and emotionally. The bride may get nervous, anxious, or over-sentimental due to the stress of the wedding. This is when you should reassure her about the wedding and ensure that she feels gratified.

Command history of sea airHelicopter Anti-Submarine Squadron ThreeHSC-9 was established as Helicopter Antisubmarine Squadron Three (HS-3) on 18 June 1952, at the Naval Air Facility Elizabeth City, North Carolina. The Tridents commenced operations flying the Piasecki HUP-2S Retriever helicopter, and later transitioned to the Sikorsky HO4S and Sikorsky HSS-1 Seabat helicopters. In 1959 the squadron received the Sikorsky HSS-1N Seabat giving it the capability to operate at night. HS-3 was the first Atlantic Fleet operational squadron to operate the gas turbine powered HSS-2 Sea King in 1961. In September 1962 in compliance with the 1962 United States Tri-Service aircraft designation system the HSS-2 Sea King was redesignated the SH-3A Sea King. In 1967 the squadron received the SH-3D, and finally the SH-3H TACNAV equipped helicopter in 1979. Additionally in 1982, HS-3 became the first squadron to deploy with the AQS-13E Sonar Data Computer.HS-3 was first deployed aboard USSAntietam and USSLeyte in late 1956, flying the HSS-1 Seabat. This was followed by assignments to USSLake Champlain, USSTarawa, and USSValley Forge, in 195860.From 1960 to 1970, HS-3 was assigned to Carrier Anti-Submarine Air Group 56 (CVSG-56). CVSG-56 was first assigned to Valley Forge before moving to USSRandolph, in 1961. HS-3 then transitioned to the Sikorsky HSS-2/SH-3A Sea King. In 1962, on board USSWasp, HS-3 participated in the naval blockade of Cuba. The Air Group was reassigned to USSIntrepid in 1962 until she was deployed as an attack carrier to Vietnam in 1966. The Tridents began an active role in the space program on 24 May 1962, when then CO Commander J. M. Wondergem picked up Lieutenant Commander M. S. Carpenter from his Aurora 7 spacecraft and delivered him to Intrepid. Astronauts Carpenter, Grissom, Young, Collins, Gordon, Conrad, McDivitt, Scott, and Schweickart all ended their space journeys with rides aboard HS-3 helicopters.In 1967, CVSG-56 and HS-3 again deployed aboard Randolph, having received the SH-3D Sea King. Two years later, CVSG-56 was reassigned to USSYorktown and then to Intrepid in 1970. HS-3 was then reassigned to Carrier Air Wing Seventeen aboard USSForrestal. In the early 1970s, the squadron participated in the relief of Tunisian flood victims, rescuing or relocating 630 people while transporting over 43,000 pounds (20,000kg) of food and medical supplies. In recognition of this humanitarian support, the squadron was awarded the Meritorious Unit Commendation. In July 1976, a detachment from HS-3 went aboard Forrestal and participated in the United States Bicentennial celebrations in New York Harbor. Squadron personnel participated in some of the publicity events leading up to the actual ringing of Forrestal's bell on July 4, 1976. HS-3 won the Battle "E" in 1973 and 1978 as well as the Isbell Trophy in 1974 and 1978. In 1979, HS-3 received the SH-3H Sea King.From 1984 to 1991, CVW-17 and HS-3 were reassigned to USSSaratoga. HS-3 was awarded two Navy Unit Commendations for operational accomplishments and outstanding maintenance efforts during the 1985-86 Mediterranean and Indian Ocean deployment on board Saratoga. The Tridents won three consecutive Battle "E" Awards for 1985, 1986 and 1987. They also won back-to-back Isbell Trophies in 1986 and 1987. In 1990, the Tridents enforced United Nations sanctions against Iraqi trade while deployed on board the Saratoga in the Red Sea. HS-3 conducted an actual Helicopter Visit, Board, Search and Seizure (HVBSS) with a special forces boarding team to "take down" a hostile merchant ship during Operation Desert Shield/Operation Desert Storm. HS-3 was awarded a Navy Unit Commendation for the combat support they provided in the campaign. The squadron returned home in early 1991 and was the first Atlantic squadron to transition to the Sikorsky SH-60F/HH-60H Seahawk.After transition to the SH-60F and HH-60H Seahawk in 1991, HS-3 was assigned to CVW-8 aboard USSTheodore Roosevelt, in 1993. Preparations for the squadron's 1995 Mediterranean deployment were interrupted when the Tridents were called on to support Operation Uphold Democracy during the military intervention in Haiti. HS-3 provided the sole Navy maritime SAR, CSAR and special operations support. The Tridents deployed one week after the completion of Uphold Democracy and flew in support of Operation Deny Flight over Bosnia-Herzegovina. During this deployment, HS-3 also flew the first actual CSAR mission by an HS squadron since the Vietnam era to search for the crew of a downed French Mirage fighter. In 1997, HS-3 deployed aboard USSJohn F. Kennedy to the Mediterranean Sea and the Persian Gulf.The Tridents again deployed aboard Theodore Roosevelt in March 1999 and headed straight into the first of two conflicts. During Operation Noble Anvil, HS-3 supported combat operations against Serbia in Kosovo. Once a cease-fire was agreed upon, Theodore Roosevelt transited the Persian Gulf to support maritime interdiction operations and enforce no-fly zones over southern Iraq. In August, HS-3 conducted a successful HVBSS to a freighter violating UN sanctions. The operation seized $3.5 million in Iraqi contraband. During this demanding deployment, HS-3 lifted over 1,800,000 pounds (820,000kg) of cargo and completed over 2,000 small deck landings. HS-3 was awarded the Battle "E" for 1999.In April 2001, HS-3 deployed aboard USSEnterprise. In response to the terror attacks on the World Trade Center and the Pentagon, the Tridents conducted sustained operations in support of Operation Enduring Freedom in Afghanistan.The Tridents deployed in 2003 aboard Theodore Roosevelt after a compressed inter-deployment readiness cycle. During the course of the deployment, HS-3 conducted ASW, logistics, and SAR operations in support of Operation Iraqi Freedom.In September 2008, after completing a lengthy training cycle, the Tridents deployed aboard Theodore Roosevelt to the Arabian Sea in support of combat operations in Afghanistan. Along the way, HS-3 took part in a historic visit to Cape Town, South Africa. Theodore Roosevelt was the first U. S. carrier to visit Cape Town in over 40 years. Once arriving on-station in the CENTCOM AOR, the Tridents flew plane guard, anti-terrorism force protection, anti-surface warfare, logistics, and anti-submarine warfare in direct support of Operation Enduring Freedom and maritime security operations during the 2008-2009 deployment. The Tridents also detached three HH-60H aircraft to USSSan Antonio and subsequently to USNSLewis and Clark for two months. This detachment supported Combined Task Force 151 counter-piracy operations in the Gulf of Aden. The Tridents assisted in apprehending 16 suspected pirates and paved the way for future operations in the region. In 2008, HS-3 won the CNAF Aviation Battle Efficiency, Commander, Naval Aviation Safety Center Safety "S", and the CAPT A. J. Isbell Trophy.Helicopter Sea Combat Squadron NineOn 16 April 2009, the Tridents returned home to Naval Air Station Jacksonville, Florida, from their final cruise as a Helicopter Anti-Submarine Squadron. During this year, the Tridents received the Jimmy Thach award for excellence in Anti-Submarine Warfare, an end cap to the squadron's legacy as an Anti-Submarine squadron. In May, the squadron started its permanent duty station change from Naval Air Station Jacksonville to Naval Air Station Norfolk, Virginia and airframe change to the MH-60S Seahawk. On 1 June 2009, the Tridents were redesignated Helicopter Sea Combat Squadron NINE (HSC-9).On 13 January 2010, the Tridents sent two helicopters in support of Operation Unified Response, providing humanitarian aid and disaster relief following the devastating Haiti earthquake. Squadron personnel spent three months deployed aboard USSCarl Vinson and USSBataan off the coast of Haiti, delivering over 280,000 pounds (130,000kg) of disaster relief supplies and completing 240 MEDEVAC missions on the beleaguered island. The Tridents also brought ashore over 556,000 pounds (252,000kg) of sustainment supplies and made 1,300 passenger transfers in support of the Operation.HSC-9 was reassigned to Carrier Airwing EIGHT embarked in USSGeorge H.W. Bush, deploying to the Mediterranean Sea and the Persian Gulf in 2011, 2014, and 2017.------History of diplomatic relations of honorary consulateRussian-French contacts began on August 9, 1717, when Hans Christoph Shleynits, the first Russian ambassador to France, was appointed by Tsar Peter I of Russia and presented his credentials to King Louis XV of France. France responded by sending its first ambassador to Russia, Jacques de Campredon, in September 1721. Since then, relations were fairly constant between the two countries, although they were severed and restored many times. Overall, relations between France and Russia have been very close, and French was even considered the unofficial second language of Russia in the 18th and 19th centuries.Relations between the two countries were severed in 1733 with the start of the War of the Polish Succession and resumed in 1738. Poor relations between the revolutionary government of France and the Russian Monarchy led to relations being severed in 1792. The Napoleonic Wars marked the start of new conflicts between Russia and France, and relations were not restored until the conclusion of the War of the Second Coalition between Russia and France in 1800. The War of the Third Coalition in 1805 led to the disruption of diplomatic relations once again, which were not restored until the signing of the Treaty of Paris on November 20, 1815. Russia and France conflicted over different views on the Revolutions of 1848 and the French support of revolutions in multinational countries. This led to the Crimean War on March 27, 1854, which ended with a Russian defeat on March 30, 1856. Relations between the two countries improved after that, and remained uninterrupted until the October Revolution of 1917.French Prime Minister douard Herriot sent a telegram to Alexey Rykov, the President of the Council of People's Commissars of the USSR, on October 26, 1924, informing him of the French recognition of the establishment of the Soviet Union. When Germany declared war on the Soviet Union on June 30, 1941, the Vichy France government broke off diplomatic relations with the Soviet Union, but never officially entered a state of war. Relations were reestablished on October 23, 1944, with the Soviet recognition of the new Provisional Government of the French Republic. Since then, relations between the new nations remained unbroken, although they were strained at times during the Cold War.After the breakup of the Soviet Union, relations between France and the new Russian Federation were warm, and France recognized Russia as the successor of the USSR on February 7, 1992.