Intereting Posts

Is $(\mathbb{Q},+)$ the direct product of two non-trivial subgroups?
Proving the sum of two independent Cauchy Random Variables is Cauchy
Prove or disprove: $99^{100}+100^{101}+101^{99}+1$ is a prime number
Number of elements of order $7$ in a group
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Slope of the tangent line, Calculus
Multiple Conditioning on Event Probabilities
Finitely generated free group is a cogroup object in the category of groups
Natural Numbers and Well ordering
Is this function bounded? Next question about integral $\int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac1{||y-x||} dS_y$.
Find $6^{1000} \mod 23$
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Is conjugate of holomorphic function holomorphic?
Evaluate the series $\sum_{n = 0}^\infty \frac{1}{(2n + 1)^6}$ by examining the real Fourier series of the function $f(x) := x(\pi – |x|)$
Proving $\lim_{n\to \infty}\frac{n^\alpha}{2^n}=0, \alpha>1$

Let $f_1,f_2,\ldots,f_n\in L^2([0,1])$, and let $V$ denote their span. Let $P:L^2([0,1])\rightarrow V$ be the projection onto $V$.

Let $g\in L^2([0,1])$. Suppose also that $g\in L^p([0,1])$ for some $1\leq p<\infty$. Is it always true that $\|Pg\|_p\leq\|g\|_p$?

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- (dis)prove:$\sup_{F \in 2^{(L^1(S,\mathbb{R}))}}\limsup\sup_{f\in F}|\int f dP_n-\int fdP|=\limsup\sup_{f\in L^1(S,\mathbb{R})}|\int fdP_n-\int fdP|$
- Questions about Fubini's theorem
- Sobolev spaces - about smooth aproximation
- Completion of a measure space

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- Zero integral implies zero function almost everywhere
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- metric and measure on the projective space
- Is a $L^p$ function almost surely bounded a.e.?
- Measurable Cauchy Function is Continuous
- Is there an infinite countable $\sigma$-algebra on an uncountable set

The answer is no.

That seems to be a overkill, but we can use the fact that $||f||_p \to ||f||_\infty$ as $p\to \infty$ and $||f||_p \leq ||f||_q$ if $p \leq q$. If we can find $g\in L^2(0,1)$ such that $||Pg||_\infty > ||g||_\infty$, then for $p$ big enough, we have $||Pg||_p > ||g||_p$.

I tried $g = \chi_{[0,1/2]}$, $h= g+ 2\chi_{[23/24, 1]}$ and project $g$ onto $h$:

$$Pg= \frac{\langle g, h\rangle}{||h||^2_2}h$$

and seems we have $||Pg||_\infty = 3/2> ||g||_\infty.$

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