# 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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### Extreme points of a subset of dual space of continuous function

$K$ is compact Hausdorff, and $C(K)$ denotes the space of continuous functions on $K$. Let $\mathbb{1}\in C(K)$ denotes the constant function taking value 1, and let $S$ be the subset of $C(K)^*$ ...
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### The Schwartz space is dense in $L^p$

Is there any hint to prove that for every $1 \le p < \infty$ the Schwartz space is dense in $L^p$? Thanks so much.
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### Relation betwen dimension of Hilbert space and cardinality of its dense subset

Suppose $H$ is infinite dimensional Hilbert space. Let $A$ be a dense subset of $H$. How to prove that $\mathrm{dim}_{\mathrm{orth.}}\ H \le \mathrm{card}(A)$ ? When we have equality ? I need only ...
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### positive elements of $C^*$-algebra

If $A$ is a abelian $C^*$-algebra and $a,b$ are elements in $A$ such that $0‎\leq ‎a‎\leq ‎1,0‎\leq ‎b‎\leq ‎1‎‎$ then $0‎\leq ‎a‎b\leq ‎1‎$. My problem is:" Is it true if $A$ is not abelian?"
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### “Trivial” extension of functional bounded?

Suppose I am in $\ell^{\infty}$ and define the $\lim$ functional on the subspace of convergent sequences. Is the "trivial" extension which is zero on all non convergent sequences a bounded functional ...
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### States on a $C^*$-algebra

I know that if $A$ is a non-zero and unital $C^*$-algebra then $S(A)$ (the set of states on it) is weak${}^*$ compact. My problem is: Does the same hold if $A$ is not unital?
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### The complement of a first category set in X is a set of second category.

Let X be a complete metric space. Then the complement of a first category set in X is a set of second category in X. What is explain in my class is "if the complement of a first category set is a set ...
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