query: Show $\pi$ is the orthogonal projection of $W$ iff $\|\pi(u)\| \leq \|u\|$ for all $u \in V$
positive_passages.docid: 17031
positive_passages.text: Idempotent linear operators are projections
negative_passages.docid: 129549
negative_passages.text: Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional
negative_passages.docid: 106081
negative_passages.text: The Projection to a Projective Line is Holomorphic
negative_passages.docid: 19946
negative_passages.text: Let $W$ be a given subspace of $V$ find the orthogonal complement $W^{\perp}$
negative_passages.docid: 152932
negative_passages.text: Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$
negative_passages.docid: 118049
negative_passages.text: Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$
negative_passages.docid: 89468
negative_passages.text: Space Sobolev $W^{m,p}$ complete
negative_passages.docid: 143869
negative_passages.text: Showing that the norm of the canonical projection $X\to X/M$ is $1$
negative_passages.docid: 146099
negative_passages.text: Why is this true: The only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the identity
negative_passages.docid: 83970
negative_passages.text: Proof of an elementary property of Projection Operators
negative_passages.docid: 89089
negative_passages.text: Prove that there is a unique inner product on $V$
negative_passages.docid: 37794
negative_passages.text: Proving the two given vectors are orthogonal
negative_passages.docid: 26658
negative_passages.text: A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.
negative_passages.docid: 69474
negative_passages.text: Orthogonal projections with $\sum P_i =I$, proving that $i\ne j \Rightarrow  P_{j}P_{i}=0$
negative_passages.docid: 76694
negative_passages.text: Proving that $UT=TU$ iff $U=g(T)$ for some polynomial $g$
negative_passages.docid: 106609
negative_passages.text: Show that $e^{iy} = 1 + iy + \frac{\mu_1 y^2}{2}$ for all $y \in \mathbb{R}$ with $|\mu_1| \leq 1$
negative_passages.docid: 146812
negative_passages.text: Explicit construction for unitary extensions of completely positive and trace preserving (CPTP) maps?