Let $Tf(x)=\int_{[0,x]}fdm$ for $0 \leq x \leq 1$. Prove that $C_0((0,1])=\{f \in C[0,1] : f(0)=0\}$ is a closed subspace of $C[0,1]$
92774
Prove that a linear subspace of $C([0,1])$ is closed
10477
Finding the closure of A=$\mathbb Q \cap [0,1]$.
134065
A question about an $n$-dimensional subspace of $\mathbb{F}^{S}$.
66409
Show that the linear operator $(Tf)(x)=\frac{1}{\pi} \int_0^{\infty} \frac{f(y)}{(x+y)} dy$ satisfies $\|T\|\leq 1$.
14759
Prove that $ f(z) = 0 $ for all $ z \in \overline{D}(0,1) $.
136056
$ f:\mathbb C\rightarrow \mathbb C$ defined by $f(z)=0$, if $Re(z)=0$ or $Im(z)=0$ and $f(z)=z$, otherwise.
43431
Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
79061
show that {$(x_1,x_2,x_3,x_4)\in\mathbb{F}^4$:$x_3=5x_4+b$} is a subspace
121771
Proof $f(x)\equiv 0$
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
12467
Why does this proof work: Closed unit ball in $C_0$ is not compact
30569
Prove that the only operator on $\mathbb{C}$ for which his inner product is zero is zero
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
58647
Show that $f$ is integrable and $\int_{I} f=0$
93606
Showing that $C[0,1]$ is not complete
141762
The direct sum of two closed subspace is closed? (Hilbert space)
138054
Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$