Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honor of Frigyes Riesz; this article will describe his theorem concerning the dual of a Hilbert space, sometimes called the Fréchet–Riesz theorem. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem; this theorem establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; the isomorphism is a particular natural one. Let H be a Hilbert space, let H* denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H the function φ x, for all y in H defined by φ x = ⟨ y, x ⟩, where ⟨ ⋅, ⋅ ⟩ denotes the inner product of the Hilbert space, is an element of H*; the Riesz representation theorem states that every element of H* can be written uniquely in this form.

Riesz–Fréchet representation theorem. Let H be a Hilbert space and φ ∈ H ∗. There exists f ∈ H such that for any x ∈ H, φ = ⟨ f, x ⟩. Moreover ‖ f ‖ H = ‖ φ ‖ H ∗ Proof. Let M =. M is closed subspace of H. If M = H we can trivially choose f = 0. Now assume M ≠ H. M ⊥ is one-dimensional. Indeed, let v 1, v 2 be nonzero vectors in M ⊥. There is nonzero real number λ, such that λ φ = φ. Observe that λ v 1 − v 2 ∈ M ⊥ and φ = 0, so λ v 1 − v 2 ∈ M; this means that λ v 1 − v 2 = 0. Now let g be unit vector in M ⊥. For arbitrary x ∈ H, let v be the orthogonal projection of x onto M ⊥. V = ⟨ g, x ⟩ g and ⟨ g, x − v ⟩ = 0, so that x − v ∈ M and ⟨ g, x ⟩ = ⟨ g, v ⟩, thus φ = φ = φ + φ = ⟨ g, x ⟩ φ + 0 = ⟨ g, x ⟩ φ. Hence f = φ g. We see ‖ f ‖ H = φ

69th Tony Awards

The 69th Annual Tony Awards were held on June 7, 2015, to recognize achievement in Broadway productions during the 2014–15 season. The ceremony was held at Radio City Music Hall in New York City and broadcast live by CBS. Kristin Chenoweth and Alan Cumming hosted the ceremony. Of the ceremony's most winning productions, The Curious Incident of the Dog in the Night-Time and Fun Home each won five Tony Awards in the play and musical categories, respectively. Both An American in Paris and the Lincoln Center revival of The King and I took home four Tony Awards. Shows that opened on Broadway during the 2014–15 season before April 23, 2015 were eligible for consideration; the Tony Award nominations were announced on April 2015 by Mary-Louise Parker and Bruce Willis. The musicals An American in Paris and Fun Home had the most nominations with 12, followed by Something Rotten! with 10, the revival of The King and I with 9 and Wolf Hall Parts One & Two with 8. This is the first year; the award was presented to an educator who demonstrates "a positive impact on the lives of students, advancing the theatre profession and a commitment to excellence."This year, the number of nominees in several categories may be increased, depending upon the number of eligible candidates.

This change affects performance categories, Best Direction of both plays and musicals, Best Choreography. The Creative Arts Awards portion of the awards ceremony was hosted by Jessie Mueller and James Monroe Iglehart; the Creative Arts Awards include Best Lighting Design of a Play, Best Lighting Design of a Musical, Best Costume Design of a Play, Best Costume Design of a Musical, Best Orchestrations. There were performances at the awards ceremony from musicals which have been nominated for a Tony Award, those which have not been nominated, including: An American in Paris, The King and I, On the Town, On the Twentieth Century, The Visit, Fun Home, Something Rotten!, It Shoulda Been You, Finding Neverland and Gigi. The presenters included: Tommy Tune received the Lifetime Achievement Award; the Tony Honors for Excellence in Theatre award recipients are: Adrian Bryan-Brown, Gene O'Donovan, Arnold Abramson. The Cleveland Play House received the Regional Theatre Tony Award. John Cameron Mitchell received the Special Tony Award for his return to the Angry Inch.

Stephen Schwartz was given the Isabelle Stevenson Award. The new education award was presented to Corey Mitchell, Performing Arts Teacher and Theatre Director, Northwest School of the Arts, North Carolina. 4: Bob Crowley 2: Paule Constable, Natasha Katz, Lisa Kron, Japhy Weideman During the tribute Josh Groban sang the song "You'll Never Walk Alone" from the musical Carousel. Drama Desk Awards 2015 Laurence Olivier Awards Obie Award New York Drama Critics' Circle Theatre World Award Lucille Lortel Awards Tony Awards Official Site

Boardwalk Chapel

Boardwalk Chapel is a summertime Christian Gospel outreach on the two-mile boardwalk on the barrier island of The Wildwoods, New Jersey which holds 77 consecutive evening services during June and August, open to boardwalkers. In the early 1940s Rev. Leslie Dunn, pastor of Calvary Orthodox Presbyterian Church, New Jersey, walked up and down the boardwalk of The Wildwoods conducting open-air preaching services, he discovered an empty lot towards the southern end of the boardwalk, to be sold at a citywide tax auction. He persuaded fellow church members to allow him to bid, but no more than US$3,000; the Gospel Pavilion, its original name, had been built during the waning months of World War II. Nightly services began in a few weeks before Japan's surrender. Since its inception the Chapel has been operated by the Presbytery of New Jersey of the Orthodox Presbyterian Church, its hand-drawn logo depicts the Christian cross rising out of an Atlantic Ocean wave. Over the years, the chapel has become a landmark for tourists visiting Wildwood.

In 2005 a Washington Post travel article said "People come to Wildwood for its beaches... and for the boardwalk, a three-mile-long human circus of noise, junk food and amusement rides. The boardwalk—part honky-tonk, part family playground—has a few quirks that give it an endearing quality. The... Boardwalk Chapel, between a tattoo parlor and a pizza parlor, has been the voice in the carny wilderness for 61 years." A history of the first fifty years of the OPC titled "The Orthodox Presbyterian Church 1936-1986" references the work of the Boardwalk Chapel on pp. 134–135. "Rev. Leslie Dunn Memorial 2001" on the Boardwalk Chapel website. Official site "Baptism by water: Gospel preached on Wildwood boardwalk", Cherry Hill Courier-Post, July 6, 2014. "Boardwalk Chapel on the Wildwood Boardwalk", Press of Atlantic City, August 22, 2013