$A$ is convex $iff$ for all $s,t gt 0$ , $(s+t)A = sA + tA$
Let $X$ be a TVS ,and $A subset X$.
Then $A$ is convex $iff$ for all $s,t gt 0$ , $(s+t)A = sA + tA$.
If for all $s,tgt 0$ we have $(s+t)A = sA+tA$ then of course $A$ is convex.
I'm not sure about the other direction. It is always true that $(s+t)A subset sA+tA$ ,so assuming $A$ is convex I need to show the other inclusion.
Any ideas?
Thanks for helping!
functional-analysis
add a comment |
Let $X$ be a TVS ,and $A subset X$.
Then $A$ is convex $iff$ for all $s,t gt 0$ , $(s+t)A = sA + tA$.
If for all $s,tgt 0$ we have $(s+t)A = sA+tA$ then of course $A$ is convex.
I'm not sure about the other direction. It is always true that $(s+t)A subset sA+tA$ ,so assuming $A$ is convex I need to show the other inclusion.
Any ideas?
Thanks for helping!
functional-analysis
add a comment |
Let $X$ be a TVS ,and $A subset X$.
Then $A$ is convex $iff$ for all $s,t gt 0$ , $(s+t)A = sA + tA$.
If for all $s,tgt 0$ we have $(s+t)A = sA+tA$ then of course $A$ is convex.
I'm not sure about the other direction. It is always true that $(s+t)A subset sA+tA$ ,so assuming $A$ is convex I need to show the other inclusion.
Any ideas?
Thanks for helping!
functional-analysis
Let $X$ be a TVS ,and $A subset X$.
Then $A$ is convex $iff$ for all $s,t gt 0$ , $(s+t)A = sA + tA$.
If for all $s,tgt 0$ we have $(s+t)A = sA+tA$ then of course $A$ is convex.
I'm not sure about the other direction. It is always true that $(s+t)A subset sA+tA$ ,so assuming $A$ is convex I need to show the other inclusion.
Any ideas?
Thanks for helping!
functional-analysis
functional-analysis
asked Jan 4 at 21:46
user123user123
1,304316
1,304316
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Let $sx+tyin sA+tA$. Then $$(s+t)left(frac{s}{s+t}x+frac{t}{s+t}yright)in(s+t)A$$
So easy. Thanks!
– user123
Jan 4 at 21:56
add a comment |
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1 Answer
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1 Answer
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Let $sx+tyin sA+tA$. Then $$(s+t)left(frac{s}{s+t}x+frac{t}{s+t}yright)in(s+t)A$$
So easy. Thanks!
– user123
Jan 4 at 21:56
add a comment |
Let $sx+tyin sA+tA$. Then $$(s+t)left(frac{s}{s+t}x+frac{t}{s+t}yright)in(s+t)A$$
So easy. Thanks!
– user123
Jan 4 at 21:56
add a comment |
Let $sx+tyin sA+tA$. Then $$(s+t)left(frac{s}{s+t}x+frac{t}{s+t}yright)in(s+t)A$$
Let $sx+tyin sA+tA$. Then $$(s+t)left(frac{s}{s+t}x+frac{t}{s+t}yright)in(s+t)A$$
answered Jan 4 at 21:53
SmileyCraftSmileyCraft
3,321516
3,321516
So easy. Thanks!
– user123
Jan 4 at 21:56
add a comment |
So easy. Thanks!
– user123
Jan 4 at 21:56
So easy. Thanks!
– user123
Jan 4 at 21:56
So easy. Thanks!
– user123
Jan 4 at 21:56
add a comment |
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