# Tagged Questions

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### Closed subspaces of $L^2(0,1)$

I would like to prove that the almost-everywhere constant functions, and the functions whose integral is 0 are closed subspaces of $L^2(0,1)$. It's readily seen that they are subspaces. I'm finding ...
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### Tensor Product of Hilbert Spaces: incomplete?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as ...
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### Convergence of a sequence of unit vectors in a Hilbert space

Let $E$ be a vector space (over $\mathbb{R}$) with a positive definite hermitian form and let $\{x_{n}\}$ ($x_{n} \not= 0$) be a sequence converging to $x$ in the $L^{2}$ norm ...
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### Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1$ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
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### Completeness: $\mathcal{l}^2(S)$

Surely, for countable index sets this is just the diagonal trick: $\#S<\infty$ However for arbitrary index sets how do I prove that the limit will actually have only countable non vanishing ...
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### Show that Y is a closed subspace of l2

This might be a straight forward problem but I wouldn't ask if I knew how to continue. Apologies in advance, I am not sure how to use the mathematical formatting. We are currently busy with inner ...
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### Weakly convergent sequence

Consider a sequence $(x_n)_n$ in Hilbert space $H$ such that $\langle x_m,x_n\rangle=\delta_{mn}$ where $\delta_{mn}$ equals one if $m = n$ and $C$ otherwise. Prove that $(x_n)_n$ is a weakly ...
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Let $H = L^2(0,T;V)$ where $V$ is separable in $H$. We have $y_n(t) = \sum_{i=1}^n c_{i,n}(t)b_i$ where $b_i$ are basis vectors in $V$. Suppose we have the estimate $$\lVert y_n \rVert_H^2 = \int_0^T ... 1answer 332 views ### In a separable Hilbert space, how to show that the orthogonal projection onto a subspace of n orthonormal basis elements converge? Could anyone help me with this problem? I don't know where to start. Let \{ e_n \}_{n=1}^\infty be an orthonormal basis in a separable Hilbert space H. Denote by P_n the orthogonal ... 2answers 2k views ### Weak Convergence implies boundedness and componentwise convergence Let \ell^2 be the set of real number sequences \{a_n\} such that \sum a_n^2 <\infty. Let \langle a_n,y\rangle \rightarrow \langle a,y\rangle for some a\in \ell^2 and for all y\in ... 0answers 239 views ### Diagonal Dominance and Spectral Radius For positive semi-definite matrices, A and B with real entries, Let: X=I-(2Diag(A)-B)^{-1}(A-B) The spectral radius \rho(X) \leq ||X||. As, (2 Diag(A)-B) becomes a better approximation ... 2answers 140 views ### Cauchy+pointwise convergence \Rightarrow uniform converges (for an operator in a Hilbert space) Suppose that the sequence of operators in a Hilbert space H, \left(T_{n}\right)_{n}, is Cauchy (with respect to the operator norm) and that there is an operator L, such that ... 2answers 463 views ### Weak convergence in Hilbert spaces Definition of the problem Let \mathcal{H} be a Hilbert space, and let \left(x_{n}\right)_{n\in\mathbb{N}}\subset\mathcal{H} be a sequence. Prove the following: If ... 2answers 106 views ### Net of Projections I am trouble proving the following proposition in Conway's functional analysis book. H is an arbitrary Hilbert space, I is an index set. Prop - Let \{P_i:i\in I\} be a family of pairwise ... 1answer 392 views ### Weak convergence Let H be a Hilbert space with inner product \langle\cdot,\cdot\rangle and let V,W be two closed subspaces. For x_0\in H we may define the sequence of projections$$x_{2n+1}=P_W(x_{2n}), \qquad ...
Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...