query: $A^{\perp\perp}=A$ for a closed subspace $A$
positive_passages.docid: 21459
positive_passages.text: Orthogonal complement examples
negative_passages.docid: 28565
negative_passages.text: Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
negative_passages.docid: 132487
negative_passages.text: Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$
negative_passages.docid: 98949
negative_passages.text: Subspaces and annihilators
negative_passages.docid: 145668
negative_passages.text: Subspaces of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?
negative_passages.docid: 141762
negative_passages.text: The direct sum of two closed subspace is closed? (Hilbert space)
negative_passages.docid: 74764
negative_passages.text: Dimension for a closed subspace of $C[0,1]$.
negative_passages.docid: 83054
negative_passages.text: Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional
negative_passages.docid: 74623
negative_passages.text: Associative rule of subspaces and vector spaces: $(U+V) \cap W = U + (V \cap W)$
negative_passages.docid: 92291
negative_passages.text: $U_1, U_2, ..., U_k$ subspaces of linear space $X$ with same dimensions, prove $X=U_1\oplus V=U_2\oplus V=...=U_k\oplus V$
negative_passages.docid: 70468
negative_passages.text: Row space and column space of $A^T A$ and $A A^T$
negative_passages.docid: 14840
negative_passages.text: $c_0$ is a closed subspace of $l^{\infty}$
negative_passages.docid: 44912
negative_passages.text: Prove that $ \overline {A^\complement} = (A°)^\complement $
negative_passages.docid: 27482
negative_passages.text: How to show $-\sup(-A)=\inf(A)$?
negative_passages.docid: 30356
negative_passages.text: $a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$...
negative_passages.docid: 38896
negative_passages.text: Prove that $A$ is bounded operator on $\ell^p$ and find $\| A\|$
negative_passages.docid: 119387
negative_passages.text: If $A$ is any matrix then $A^*A$ and $AA^*$ are Hermitian with non-negative eigenvalues