# Tagged Questions

For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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### Prove of inequality under a Hilbert space.

Let $x\neq y$ when $x,y\in H$ and H is a Hilbert space which satisfy $\|x\|=\|y\|=r$. Show that $\|\frac{x+y}{2}\|<r$. Actually in my question r=1 but as far as i could understand there is a way ...
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### Can I write $H^1$ as $H^1_0 \oplus H^1_{\perp}$?

Let $\Omega\subset \mathbb{R}^d$, with $d\in \{1,2,3\}$ be an open bounded, simply connected domain. Define $H_0^1$ as the subspace of $H^1$ whose member functions have vanishing trace on the ...
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### Properties of Injective Operator on Hilbert Space

I am new to functional analysis and have the following issue: Given an infinite dimensional Hilbert space $H$ and an operator $f: H \times \Omega \to H$, where $\Omega$ is some finite dimensional ...
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### Space filling curves: initial definitions

I am confused on the definition of curve and space filling curve in Chapter 1 of the book by Sagan. I think my confusion comes from notation. Let $\mathcal{I}:=[0,1]$, $\mathcal{Q}:=[0,1]^2$ and $J_n$ ...
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### unitary operator between two Hilbert subspaces

$H$ is a Hilbert space. $P, Q$ are projections. For every $x\in P(H)$, we have decomposition $x = Qx +Q^\perp x$. Then, can we find a unitary operator from the space generated by all $Qx$, $x\in P(H)$...
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### Norms with complex numbers over Hilbert Spaces

Let $H$ be a Hilbert space and $v,w \in H$ ans a be a scalar. Prove that $\|v\| \leq \|v+aw\|$ for all scalar a iff (v,w)=0 for real and complex cases. I want to choose a such that $\bar{a}(v,w)$ ...
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### A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
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### Almost negative definite matrices and norm-distance matrices

An "almost negative definite" matrix $A$ satisfies the property $$v^te = 0\implies v^tAv\le 0$$ where $e=(1,1,\dots,1)$. We know that if $A$ is a simmetric zero-diagonal (hollow) matrix, then $A$ ...
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### Metrics from Operator Norms

Let $X$ be a Hilbert space and $(\cdot,\cdot)_X$ be the inner product on $X$. It is well known that $|x|_X = \sqrt{(x,x)_X}$ is a norm on $X$ and $|x-y|_X$ is a metric on $X$. The norm on $X$ induces ...
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### Showing that the intersection of two closed linear subspaces is the trivial subspace.

I'd appreciate if someone can provide the best way to deal with this problem. Let $\{\alpha_n\}$ be an orthonormal sequence for a Hilbert space H and let $\{\beta_n\}$ be an orthonormal sequence such ...
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### Norm of operator $A$ st. $A^2 = I$?

I'm wondering what can be said about the norm $||A||$ of an operator which squares to identity. All I can think of is that $$1=||AA|| \leq ||A||^2$$ so that $||A|| \geq 1$. But can anything else be ...
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### Unique ground state of Schrödinger Operators

I'm reading a book and there is an argument that the ground state of a Schrödinger operator is unique. The problem is I think the argument is complete non-sense! These are lecture notes by Witten, I ...
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### Weak convergence and strong convergence on $B(H)$

Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence ...
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### If operators $A, B, A+B$ are all closable, show that $\overline{A + B} \supseteq \overline{A} + \overline{B}$.

Let $A, B$ be closable, unbounded, linear operators on a Hilbert space $H$. Suppose further that the operator $A + B$, defined on the intersection of domains $D(A) \cap D(B)$, is also closable. I ...
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### some detail calculation on the proof of equivalence of norms

We say that two norm $\|x\|_1$ and $\|x\|_2$ on a vector space $X$ are said to be equivalent if there exists $K>0$ and $M>0$ such that $$K\|x\|_1\le \|x\|_2\le M\|x\|_1$$ Prove that on a ...
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### Why are function spaces generally infinite dimensional

The other day, I was trying to explain some concepts in Fourier analysis and wavelets to my girlfriend (an electrical engineering student) and obviously, the concept of Lebesgue integration came up in ...
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