# Tagged Questions

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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### Span of a closed subspace

Let $Y$ be a closed set of a Banach Space $X$. Is it true that the linear Span($Y$)is also closed? For the examples I have tried, I see that the result holds true. I understand that the linear span ...
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### Inner product and norm

I have to show that if $\|x+y\|=\|x\|$ then $2x+y$ and $y$ are orthogonal. I think I can use Pythagoras. Thank for any help. Corrected
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### Proving that the function f is constant, mean value theorem, derivatives

Having the following inequality, for a real-valued function $f$ which is twice differentiable: $f(a+h)-f(a)\geq f(a)-f(a-h)$ for any $a \in\mathbf{R}$, $h > 0$. and assuming that $f$ is bounded, ...
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### Question about a counterexample concerning compact operators

Does anybody know if the following is true, Let $H$ be an infinite dimensional Hilbert-space and $K:H\rightarrow H$ a compact operator. Then if $|\mathrm{spec}(K)|<\infty$ i.e the spectrum is ...
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